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NATIONALADVISORYCOMMITTEE ~ ~_

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+=J~&—~—.TECHMCAL NOTE 3375

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A THEORY FOR PREDICTING THE FLOW OF REAL GASES IN SHOCK

TUBES WITH EXPERIMENTAL VERIFICATION

By RobertL.TrimpiandNathanielB.Cohen

LangleyAeronauticalLaboratoryLangleyField,Va+

WashingtonMarch 1955

-.b

TECHLIBRARYKAFB, NM

KNATIONALADVISORYCOMMITTEEFORAXRONMJTICS111111111

C10bh4bL

. TEClID71CALNOTE3375

A THEORYFORPREDICTINGTEEFLOWOFREALGASESIN SHOCK

TUBESWITHEXPERIMENTALVER13?ICNION

By RobertL.TrimpiandNathanielB. Cohen

Thenotiinearcharacteristicdifferentialequationsapplicabletoa qpasi-one-dimensionalunsteadychannelflowwithfrictionandheattrsmsferarelinearizedandintegratedinfunctionalformforthepar-ticularstudyof smallperturbationsfromidealshock-tubeflows.Iftheequivalenceofunsteady-@ steady-flowboundarylayersis assumed,theproblemofdeterminingtheperturbationsintheunsteadyflowreducestoanevaluationof thedragof a flatplateintheeqtivslentsteadyflow.

Forairat initiallyuniformtemperature,thetheoryevaluatedwithan equivalentsteady-flowturbulent-boundary-layerskin-frictioncoeffi-cientpredictsthatshockattenuationincreaseswithdistanceandthataveragevaluesaf staticpressure,velocity,density,andMachnmiberat a fixedpositioninthehotgasincreasewithtime,whereasaveragesonicspeedsimultaneouslydecreaseswithtire-eat a fixedposition.

Experimentalmeasurementsof theshockattenuationwithdis@ceandstatic-pressurevariationwithtimeat a fixedpositionfordiaphragmpressureratiosfromapproximately4 to 18 gavegoodagreemsntwiththetheoreticalpredictionswherea valueof O.0~1x (Reyaoldsnuriber)-l/~wasusedfortheskin-frictioncoefficient.

INTRODUCTION

Theshocktubehasbecomea coxmm aerodynamictestingfacilitybecauseof itsrelativeinexpensivenessandversatility.Ina shockt~e it ispossibletoobtainunsteadyflowswitha widerangeof flowparameters,suchasReynoldsnumiber,Machnuniber,andtemperature,thateithercotidnotbe obtainedinsteady-flowapparatuswiththepresenttemperaturelimitof-own alloysor couldbe obtainedonlywithmassiveandcostlyequipment.

Thevariousstatesoftheflowof a perfect,nonviscid,nonconductinggasina shocktubemy easilybe determinedtheoreticallyby application

i

2 NACATN3375

of thebasicequationsofmomentum,continuity,andenerg. (See,for ‘example,refs.1 to4.) ThetheoreticalstatesSQdeterminedareconi-calina distsmce-timesense(untiltheshock-tubeendeffectsinterfere); fthatis,parametersareinvariantalongrayswhichhavethesameratiosofdistancefromthediaphragmstationtotimeelapsedsincediaphragmburst.Thisperfect-gasflowisalsocharacterizedby tworegions,of’equalconstantvelocityandpressurebutofdifferentdensityandtem-perature,separatedby a contactsurfaceorentropydiscontinuity.Oneof theseregionshasas itsleadingboundarytheshockwaveadvancingdownthetubeintothelow-pressweairat rest; whereastheotherhasas itstrailingboundarytheexpansionwaveprogressingintothehigh--pressuresection.Therefore,itistheoreticallypossibletoobtainsteady-flowaerodynamicdataintheshort-durationsteadyflowsthatexistwhileeitherregionispassinga fixedpointalongtheshocktube.

Unfortunately,theflowof a realgasintheshocktubedepartssignificantlyinmanycasesfromtheafore~ntionedtheoreticalflow.It isintuitivelyobviousthata realflowdiffersfromthetheoretical.flowinthata pressuredropisreqybedof a realfluidflowinginatubeanda furtherpressurechangeisreqtiredtoaccountforheattrans- bferbetweenthetubeandthefluid.Inaddition,theflowintheregionbehindtheentropy@iscontinui@hasbeenfoundtobe quiteturbulentanderraticsoastomakeitofverylimitedvaluefornmsttesting vpurposes.Althoughtheflowbetweentheshockwaveandtheentropydis-continuitydoesnotdegenerateintolarge-scaleturbulence,this flowis stillaffectedby viscosity,heattransfer,otherimperfectaseffects,andnonidealdiaphragmburst. ‘7(Seerefs.1, 3, andk. Thenonidealconditionof diaphragmburstmaybe minimizedby theproperchoiceofdiaphr~ materialforeachparticularinitialsetof condi-tions.However,theothereffectsareunavoidable,anditistobeexpectedthattheirmagnitudewillincreaseinimportanceinthehigh-temperatureandhighMachnuuiberrmge,wheretheshocktubeappearstobe otherwisemastadvantageous.

Theattenuationin shockstrengthastheshocktravelsdownthetubeisthemostobviousandeasilymeasureddeviationfromperfect-fluidtheo~. Thisattenuationhasbeenthesubjectof severalexperi-mentalandtheoreticalstudies(refs.3j 5>f5jand7).

Thetheoryofreference3 maybebrieflyoutlinedasfollows.Theunsteadyflowof thehotgasbetweentheshockamdentropydiscontinuityisreducedto a quasi-steadyflowby choosinga coordinatesystemfixedtotheshock.Theboundary-layerproblemisthenreducedtoa laminarsolutionsimilartotheBlasiussolution,exceptthatthewallvelocityisnonzero.Theunsteadyboundaryconditionoftherecedingentropydiscontinuityis ignoredasistheentirecold-flowregionbetweentheexpansionandentropydiscontinui~.Theskin-frictionandheat-transfer 4effectsobtainedfromthissolutionarethenaveragedacrosstheassumed

P

I?ACATN 3375 3

. one-dimensionalflowandtheresultingchsngesinaveragemomentumandaverageener~ requirethegenerationofrovingwaves.Thesewavesare

4 assumedtobe generatedattheentropy&lscontinui@andta travelunclwmgedin strengthuntiltheyovertake-theshock.Theshockattenua-tionisthendeterminedfromthewaveswhichhaveovertakenit sinceflowwasfirstinitiated.

Thetheoriesofreferences6 and7 arenuchlesselaboratethanthatofreference3 andarebasedon theassumptionthatthemassflowthroughtheshockat a certaintim isthessmeas theMss flowat theentro~discontimrityevaluatedatthesametime. Themassflowattheentropydiscontinuityisdeterminedfromtheboundary-layerdisplacementthicknessandfree-stresmconditionscorrespondingto anunattenuatedshock.Thisboundary-layerdisplace~ntthiclmessisdeterminedinref-erences6 and7 fromanextension,to a circulartubeandrectangulartube,respectively,of theRayleighproblemof theinstantaneousaccelerationto constantvelocityof a flatplateina fluidat rest.

Taanunpublishedanslysismadeat theLangleyAeronauticalLaboratiryjd Messrs.PaulW. Euber,DonaldR.MWarland,andPhilipLetinecollaborated

on a theoryforshockattenuationbasedon themass-flowdecrementduetodisplacementthicknessat theentro~discontinuity.Inordertodeter-

> minetheshockattenuationat a giventixre,thedisplacementthiclmesswasevaluatedin thistheoryfora t- smallerthemthisgiventimebyan intervalequalta thewave-traveltimefromtheentropydiscontinue@to theshock.Thisdispl.ac=ntthiclmesswasdeterminedfromempiricalflat-platesteady-flowdataat a distancefromtheleadlngedgeof theflatplateequalto thedistsmcefromthediaphragmstationto theentropydiscontinuity.

Noneof theaforementionedtheorieshavegivengoodagreementwithexperimentallydeterminedshockattenuation.Reference8 reportstheuseof a chrono-interferometerto obtainthetimwisevariationofdensi~inthetwotheoreticallyconstantdensi~regions.A densityrisewithtimewasnotedforbothregions.Comparisonof theexperimentaldatawiththetheoryofreference3 gavepoorcorrelation.By employingvari-oussimplifiedflowmdels,thesforenentionedtheorieshaveignoredthefactthatunsteadywavessrecontinuallybeinggeneratedby theeffectsof frictionandheattransferintheentireflowregion.Theresultofthesewavesovertakingtheshockis at,tinuation,andtheirmtion sLongthetuberesultsinvariationsinpressure,densi@,andveloci~atgivenpoints.Therefore,inorderto obtaina satisfactoryunderstandingof theflowof a realgasin a shocktube,a theorymustrecognizeadtreatthewavesystemintheentireflowfield.Sucha theoryhasbeenderivedin simplifiedform,andthistheory,togetherwithexperimentalcorrelationobtainedat theGasDynamicsBranchof theIangleyLaboratory,ispresentedinthispaper.

4 NACATN 3375

SYMBOLS

A

a

B

Cf

Cv

CP

D

D( )Dt

%

h

J

1

M = U/a

areaofflow

velocityof Bound

constantdefinedby equation(D4)

localskin-frictioncoefficient,2T/pu2

coefficientof specificheatatconstantvolume

coefficientof specificheatat constantpressure

hydraulicdiameter,4 x AreaPerimeter

~()+ub()convective‘erivative’at ax

heataddedperunitmassby frictionaldissipation

walJheat-transfercoefficient

heataddedper

fixeddistance

unitmassduetoheattransferandheatSources

alongshock-tubeaxis

b

v

N

n

definedinequation(53)●

‘% y l/nreciprocalofveloci~exponentinboundarylayer,— =u ()E P

NA(!ATN 3375

.P

JP’

Pr

P

*8

Q

Q’

i R

%J

r

s

T

t

u

u*

us

‘%

v

w

x

Y

4

2CVcharacteristicparsmter,~ a + U

aSeffectivecharacteristicwaveparameter,P - - -7R

Prandtlnumber

staticpressure

staticpressure_diately behindshock

Pcva-ucharacteristicparameter,T

aSeffectivecharacteristicwaveparameter,Q - — -7R

gasconstant

Reynoldsntier

recoveryfactor

entropy

temperature

time

free-streamvelocity

vshearing-stressWIOCLty, T p

shockvelocity

velocity

veloci~

veloci~

distance

distance

inboundsrylayerat y

ofrayinx,tplotoffigure1, ~/~

ofrayinx)tplotof figure1, IXw -%

alongshocktubefromdiaphragmstation

fromsurface

NACATN3375

ratioof specificheata,cp/%;assumed= l.kOforcomputations ●

boundary-layer

characteristic

thickness;alsoindicatesdifferentialquantity v

a( )ao+(u*a)Fderivative,X-

differencebetweenwalltemperatureandadiabaticrecoverytemperatureof flow

coefficientofviscosity

coefficientofkinematicviscosity

distanceflowhasprogressedalongsurface

densi@

wallshearingstress

dP&@C&@pB>%#&$fsp influencecoefficients,definedinequa-tions(12),(13),and(26)

Subscripts:

Subscriptsnotincludedonthesymbolsdefinedaboverefer,ingeneral,tovaluesatpointsorwithinregionsshowninfigure1.Bceptionstobe noted,

o perfect-fluid

t attimet

x atdistancex

exp experimental

however,areas follows:

value

ref referencevalue

theor theoretical

NACATN 3375

‘iHEoHY

ThebasicstepsfollowedinderitingThecharacteristicdifferentialequations

thetheoryarefirstoutlined.fora q~si-one-dimensional

unsteadychannelflowarelineari~edon thebasi{of theperfect-fluidflowina shocktube. Thetermsof theseequationsfora particularlaminaof fluidintheshockMe areexpressedas functionsof thedistancethelaminahasmovedalongtheshockt@e. A transformationofvariablesof integrationisthenmadewherebythethe inte~alofthecharacteristicequationisreplacedby a distanceintegralalongtheparticlepath. Thevaluesoftheflowvariables,suchaspressure,velocity,temperature,andsoforth,maythenbe foundfromthevaluesof theintegratedcharacteristicparameters.

Onemethodofevaluatingtheresultingintegralsistoassumetheequivalencyof steadyandunsteadyflowsbasedonparticle-flowtime.Applicationof thismethodreducesthesolutionforthecharacteristicequationstothesimplecomputationoftheskin-frictionintegral(totaldrag)ofa flatplatein steadyflow.

●

Thecharacteristicegyationsfora quasi-one-dimensionalunsteady4 channelflowas derivedinappendixA are:

5P -aD 10&&A bg ‘D: m%f—= ——et Dt

+=a—.—+;E% 7 Dt D

Furthermore,undertheassumptionfrictionandheattransferisthesame

DE 2u2cftia_&+T7

(1)

(2)

thattherelationbetweenskinforb@h steady=d unsteady

flows,theconvectivederivativeofentropymaybe expressed(seeappendixB) as

~~

(R– M2+~Qpr)

-2/32YucfDt 7-IT

. Forapplicationto thestudyofequationsarelinearizedby assuming

(3)

flowsina shockt~e, theprecedingthatallcoefficientsanddifferential

8 mcA TN 3375

operatorson theright-handaideareconstantat theirrespectiveperfect-fluidvalues.Inotherwords,thelinesnpkingup theperfect-fluidcharacteristicsnetworkareusedaspathsof integration,andthevariousderivativesandcoefficientsareevaluatedon thebasisofthevelocities,temperatures,andsoforth,oftheperfectfluid.Eqm-tions(1)to (3)thenbecom,iflinearizedonthisbasis(subscriptoreferstoperfectfluid),

(3a)

where

(M%)o’ao&T_tisfurtherassuredthat,althoughthevelocityof theleading

edgeoftheexpansionfaniscorrectlyusedas -ae,thefluidandcharacteristicvelocitiesmy be approximatedby Uc and (U* a)a intheentireregionfromtheleadingedgeoftheexpansionwavetotheentropy.discontinuity.Thisassumption,whichgreatlysimplifiesthecomputations,introduceserrorsthatincreaseinmagnitudeastheexpansion-fanregionincreaseswithshockpressureratio.

Eqyations(la)and(2a)areexpressionsforthederivativesalongchuactertsticsof Slope~=(Uta)o andequation(3a)evaluatesa

NACATN 3375 9

. derivativealonga characteristicof slopeU. (particlepath).Hthelen@h ~ is titroducedanddefinedas thedistancethata particle

* intheperfectfluidhastraveledsinceitsaccelerationfromrestbyeithertheexpansionwaveor shockwave,thetermsof equations(la)to (>) maybe evaluatedasfunctionsof ~ aloneineitherregionaor ~. (Seefig.1.) It shouldbe notedthatthefunctionalrelation-shipmaybe differentforthedifferentregions.Whentheseassumptions,togetherwiththedefinitionof ~,areappliedto theregiona betweentheexpansionzonesmdtheentropydiscontinuity,thefollowingrelationsareobtainedfromtheplotofdistanceagainstt- of figure1:

x = -a~ti+ Uo(t- ti) (4)

g = Uo(t- ti) (5)

Substitutingequation(k)intoeqyation(5)gives

~= ‘0 (x+ act)U. + a~

Therefore

ao_do~_” ‘O doax d~ &-— U. + ae d~

and

ao_dio>_” aE doat Lag& U. + ae d~

(6)

(7a)

(n)

Equations(7a)and(7b)maybe usedtoobtaintheconvectivederivativeasfollows:

D(~=U,+Uo&Uo~Dt at (8)

NACATN 337510

andsincethecharacteristicderivativeisexpressedas

o-w+ M_).m Dt au ax (9)

thisderivativema-ybe written,forregion& offigure1,by substi-(8)intoequation(9)astutingequations(~a)and

50_Ml

80=&t

d( )~ d( )+ ‘O —o~- Uo + a~ a= d~

Similarly,forregionp,whichliesbetweentheentropydis.

(lo)

continuityandtheshockwave,

50 aMs-MP~lUdo—= (n)bt Glfs- % 0 d~

Whenequations(%), (8),(10),and(11)aresubstitutedinequa-tions(la)and(2a)andtheareatermisneglectedbecauseof theaveragingprocessdescribedinappendixA, thefollowingeqmtionsarefoundforregionsa and ~,respectively.Inordertoavoidneedlessrepetition,theGolutionsforboth ~P/8tand bQ/bt arepresentedinoneequationwithbothplusandminussignsindicatedforcertaintermson theright-handside.TheuppersignsapplytothecharacteristicswithslopeU + a andthelowersignstothosewithslopeU - a. !Thus,

f

--’J

(12)

r

G

NACATN 3373 11

1—a’s

(13)

Theseequationsmaybe integratedalongthecharacteristiclines.However,foreaseof computation,sincetheslopesof theselinesareassumedconstantforeachregiondueto thelinearizationprocess,integ-rationmayproceedalongparticlepathsby a suitablechangeofvariable.Forexample,fromthegeometryof theaccompanyingsketchofregiona, “thefollowing

x(t)= -actf

x(t)= -ac~

X(t)= Utd+

equationsapply:

It(X,t)-f- U(t- tf)

+(U+aa)(t-%)

(U- ~) (t- td) bi

o x

Solutionsof thefirstandsecondequation6andofthefirstandthirdeq=tionsyield

U+aa+ae(t-t~)= ~+a (t- tb)

E

12

and

NACATN 3375

U+a~- aa(t- t~)= (t

U+ae

Therefore,fromthedefinitionof thek = U(t- ti),thefollowingrelationsare

Along t~,

length~ inregionu asobtained:

Along ttd,

(14)

(16)

(17)

Thereforeequation(12)isintegratedinregiona withthechangeofvariablespecifiedinequations(14)to (17):

or

(18)

NACATN 3375 13

. and

4

or

Qt-Qi=ac

Similarrelationshipsmay

(19)

be foundforregion~. Fromthegeometryoftheaccompanyingsketch,itisevidentt& x “By be expressedas-follows:

X(t)= U~tj+ U(t- tj)

x(t) . Utf+ (U+ ap) (t - tf)

X(t)= Ustk+ (U- a~)(t- tk)

o

Solvingtheaboveequationsyields

aB(t- tf)t- tj= u-us

d\, (X,t)

t/

f

//

and

Consequently,sinceinregion. apply:

U* - U + aB(t

U*-U

x“

+t

- tk)

p, E= u@- tj),the followingrelations

14

Along tft,

anddO?2g ttk,

Ml= # J“’ - ‘$-MB-1

l+u~-ME= P U(t- tk)

u% - MP

NACATN 3375

(20)

(21)

(22)

d~ u% - MP8t=— (23)Ul+uM~- MP

Therefore,equations(13)and(20)to (23)canbe usedtoobtain

and

@pp UM6-Mp

JJ

g(t)=— cfPd~

D OM~-MP-l tf(24)

r

u

(25) N

P

NACATN 3375 15

. Theentropyatanypoint(x,t) inregiona or ~ maybe foundsbrplyby integrationalonga particlepath(since~ istheproductof U

● multipliedbyflowtime,d~ = U dt),sothatfromequation(>),

St-sbR

st-sJR 1

(26)

Itisnownecessazyto considerthetransmissionandreflectionofthecharacteristicsat theentropydiscontinuityseparatingthe

% fluidwhichwasinitiallyat a highpressurebehindthediaphra~fromthatwhichwasinitiallyata Iowpressureaheadof thediaphragm.Theseeffectsareimportant,evenina linearizedsolution,becauseofthei largedifferencesin v whichmayexistinthefluidsoneachsideofthediscontinuity.

Ifthesubscripto referstoconditionsina perfectfluid(thatis,immediatelyafterdiaphra~burstinrealfluid)andthenotationoffigure1 isfollowed,itand 7 isthesameinboth

i=showninappendixC that,if A<< 1a and “Pj

2s E+-sc As—=R R ()

.+(l+A)=+(l+A)o

Then,

(27)

Pc - Pao+ Qf- Q@.+=

[ 1

%3./’% Sf - spo Sc- s%=1+

‘% + Q% 7

(-)

2a~o - ~

R R

aao

(28)

16

Thesecondtermontheright-handcharacteristicvaluewhichwoiLdoccurrefractionsevenwithoutanychangein

NACATN3375

sideindicatesthechangeinthe .

asa resultofreflectionsandentropy.Thethirdtermevaluates P

theadditionalvariationduetothechangingentropy.

Thereflectionsat theshockwaveof theincidentwavesareofhigherorderandmaybe ignoredinthissolution.

Thefollowingeqyationsresultfromthedefinitionof P, Q,and S/R andsubsequentlinearizationwiththelimitationto smallperturbations:

-lP+Q~=~ a.

27 s-soa 7-1 .-

P()

R—=Po z

(=1-1-

(=1+-

u P-Q—=—% 2ae

e

27 S.SO

a-a. ~-—

)

Re

a.

-+++”””7-1

2ya-~s-so+..s~+ —— -

R . . .7 -l%

(29)

(30)

(31)

HACATN 3375

2 s-s..3 .—. .,-,P

()

a’—=—PO aQ

— Re

(acP )

-Po+ Q-~ S-SO— -— +.....a. aE R

17

(32)

Consequently,thedescriptionof theflowat ~ desireddistanceandtm maybe determinedlycotiiningequations(29)to (32)withequations(18),(19),(24),(25),(26),and(28)withtheproperlimitsof integrationdeterminedby geomtryon thex,tplot.

Itistobe notedthatthevaluesof P, Q,and SIR mustdSCl

be determinedat theintersectionoftheentropydiscontinuitywiththecharacteristicpassingthroughthepoint(v or w) atwhichtheflowpropertiesaretobe determined.Thisstepisreqyiredto computethetransmissionandreflectionsofthecharacteristicpmarmtersasdeter-

% minedby equation

InregionG3

eqmtionsbecome:

Pv -

(28).

atthepoint ~ = W% = M&c% (seefig.1),the

(33)

(34)

(36)

18 NACATN 3375

Qd-Gk=l+%-9%

(P.-,q)+ (%-%)+

p% + %0

7-

.

C/(37’)

(38)

It shouldbe notedthattheneglectoftheexpansionfaninthederivationisreflectedintheterm 1 + ~ -limitsof integrationoftheaboveequations.thepoint w is intheexpansionfanmakethenegativeandgiveonlymeaninglesssuwers.

EquationssimilartothoseforregionG

~ appearingintheValuesof ~ suchthatfactor1 + * - ~

areobtainedforregion~

Cfa% (39)

.-

V

NACATN3375

(PC-%)+(%-%)+

‘%+%0

1-

(43)

(44)

Thesolutionto thesmall-perturbatio~problemof shock-tubeflowhasnowbeenreducedto theevaluationof integralsof theform

J’cf(~)d~ where ~ issimplya lengthformedby theproductof the

flowtires?mltipMedby thefluidveloci~. Amyanalyticor graphicalmethd givingcf asa functionof ~ maybe usedin theevaluationof theseintegrals.

b ordertoevaluateeqyations(33)to (~), it isnecessarytohow thestateof thebound~ layerata pointintheshocktubewhichisdenotedbya correspondingpoint ~(x,t)of thecharacteristicsdiagram.Theboundq-layerproblemisa “hybrid”of theRayleighproblemof theinstantaneousaccelerationof a flatplateandtheBlasiusproblemof thesteadyflowovera semi-infiniteflatplate.It is

d

20 NACATN 337’3

similartotheRayleighprobleminthat,ata giveninstantof time,alltheflowbetweentheshockandexpansionwavehasbeenacceleratedfromzerovelocityto a constantveloci~butisdifferentfromtheRayleighprobleminthatthefluidhasbeeninmotionforvaryingtimedurations.Itis similartotheBlasiusprobleminthat,alonga particlepathinx,tcoordinates,theboundsqlayerincreasesrearwardfromtheshockor “leadingedge,”butisdifferentinthattheleadingedgeisnotsta-tionary.A solutionto thishybridproblemforlaminarboundarylayerswasfoundinreference3 forregionp by reducingtheunsteady-flowproblemto a steady-flowproblembychoosinga coordinatesystemfixedto theshockwave. TheproblemthenbecametheBlasiusproblemnmdifiedfora nonzerowallvelocity.Ina similarmannera solutioncouldbefoundforregionCL iftheexpansionwavewereassumedtohavezerothicbessandthecoordinateGystemwerefixedtotheleadingedgeoftheexpansionwave.

Forlaminarflows,thesolutionsto theRayleighproblem,theBlasiusproblem,andthemcdifiedBlasiusproblemofreference3 allshowtheskinfrictiontobe inverselyproportionaltothesqusrerootofthetimethatafluidparticleinthefreestreamoutsidetheboundarylayerhasbeeninmotionovertheplate.Itshouldbe notedthattheconstantsofpropor-tionalityaredifferentineachoftheabovecases.No solutionstotheturbulent-boundary-layerequivalentoftheRayleighormodifiedBlasiusproblemareImowntotheauthors.Onemightassume,however,thatthestateof theboundarylsyerata pointin a turbulentflow(aswellasin a laminarflow)overa flatplatewithzeropressuregradientmaybeexpressedas a fictionofthetimeintervalthatthefluidhasbeeninmotionovertheflatplateorofthedistancetheouterfluidhasmovedalongtheplate.

Intheabsenceofa modifiedBlasiussolutioninregiona andofmy solutionwhatsoeverfortheanalogousproblemof turbulent-boundary-layerflowineitheru or j3,anapproximatemethodofevaluationfortheintegralsmaybe obtainedby extendingtheabove-mentionedtime-dependencyassumptiontitherandassumingthatthelinearizedunsteadyflowina shocktubehasthesamepropertiesasanequivalentsteadyflowdefinedinthefollowingmanner:Thepropertiesof a laminaoffluidintheshocktubewhichhasbeeninmotionwitha veloci~ U.fora time t areequivalenttothoseofa laminaof fluidwhichhasprogressedrearwardfora periodoftime t fromtheleadingedgeofa semi-infiniteflatplateina steadyflowwithfree-streamvelocityUo.Undertheseconditions~ beconesthedistancefromtheleadingedgeoftheflatplate.

Theintroductionofthisassumptionreducesthesmall-perturbationsolutiontothesimpleevaluationoftheintegraloftheskin-frictiondistributionalonga semi-infiniteflatplatein steadyflow.Thisintegralmaybe evsluatediftherelations~pbetweencf end ~ is

NACATN 3375 21

. prescribedeitherinanalyticformor as a curveof cf plottedagainst~ orof cf against&. Theformermethodof relationshipwill

* usuallypermitevaluationof theintegralin closedform,whereasthelattermethodmayrequiiregraphicalintegration.

A theoryhasnowbeenobtainedforsmll-p-erturbationflowinashocktubewiththefollowingassumptions:

(1)Variationsfromperfectfluidflowshallbe small.

(2)Averagedvaluesareusedforquasi-one-d-nsionalflow.

(~)Averagedvaluesof p and U whichidenticallysatisfythecontinuityequationareslsoassmd to satis~the~mentumequation.

(4)Thefrictionaldissipationis approx-tedas theprcductoftheaverageveloci~tiresthewall-shesringforce.

(5)Theperfect-fluidchmacteristicnet,velocities,densities,t andsoforthareusedtoobtainlinearizedvalues.

(6)Theexpansionfsmistreatedas a “negativeshockwave”inthat* thefinslvaluesofveloci@,pressure,andsoon,sreassumedtoexist

irmnediatelyaftertheleadingedgeof theexpsmsionfan.

(7)H nUIEriCd evaluationofthetheo~ isperfomd on thebasisof equivalent-flat-platesteadyflow,thefollowingassumptionsarealsointroduced:

(a)me steadysmdunsteadyskinfrictionsareequalforafluidwhichhastraveleda givendistanceovera flatplateinsteadyflowandfora fluidwhichhastraveledthessmedistamcealongtheshocktubeinunsteadyflow.

(b)Theboundarylayeris smallrelativeto thewidthorheightof theshocktube.

in a

ExPERIMENrAIl

Experimentstodeterminehigh-pressureshocktube

APPARA!IWSANDPROCEDURE

thevalidityof thetheorywereperformed.

2 incheshighby 1: inchestideinthe

Langleygasdynamicslaboratory.Airatroomtemperaturewasusedforbothl&&- ad low-pressuresectionsoftheshock-tube.Measurements

. of shockveloci~andpressure-timevariationweretakenatvsriouspointsslongtheshockttie.

●

22 NACATN 3375

Thelow-pressuresideoftheshocktulewasatroomconditions; .whereasthehigh-pressurechamberwasfilledtopressuresof 45,70,95,or 250poundspersquareinchgage. Eitheroneor twosheetsof *softbrassshimstock0.00125or O.CKl~inchthickwasemployedas adiaphragm.Then~er andthicknessof thebrasssheetxweredeterminedsothatthediaphragmwasnearitsbreakingstressattheparticularpressuresof thetest. Consequently,optimumdiaphragmburstwouldresultwhenthepointof therupturingmechanismpiercedthecenterofthediaphragm.Ithasbeenfoundthatimperfectdiaphragmburstswillresultinbothincreasedshockattenuationandscatterofexperimentaldata.

Theshockvelocitywasmeasuredatdistancesof 3.23,5.23,7.23,9.18,and11.18feetfromthediaphragmbyinsertingshock-tubesectionsofvariouslengthbetweenthediaphragmandtheglass-walledtestsec-tion. A blockdiagramoftheexperimmtslapparatusis showninfig-ure2. Velocitywasdeterminedfrommeasurementsofthetimerequiredfortheshockwaveto traversethe1.206feetbetweenthetwolightbeamsoftheopticalsystems.(Seefigs.2 and3.) Eachopticalsystemconsistedofa direct-currentautomobilelampwithlinefilamentdinedverticallyatthefocalpointof a two-inch-diameter,seven-inchfocal-

r

len@h lens.Thelightemittedfromthelenswasmaskedto a verticalslitapproximately0.030inchwideand1 inchhigh. Straylighton the Pupstreamsideofthisbeamwascutoffwitha sharpMife edge. Thebeamthenpassedperpendiculsmlythroughthetest-sectionwindowandimpingedontheedgeofa secondkuifeedgeaboutteninchesfromthetest-sectionwindow.Whentheplaneshockwavereachedtheupstreamedgeofeachbeam,lightwasrefractedoffthesecondlmifeedgeontoaphototube.Theresultingsignalofeachphototubewasamplifiedbythecircuitshowninfigure4. Theoutputpulsefromeachthyratron(2D21)wasusedtotri~eronechannelof aneight-mgacyclecounterchronograph.Thepulsefromtheupstreamsystemtriggeredthechronograph“start”circuitandthatfromthedownstreamsystem,the“stop”circuit.(Seefig.2.) Thetimeintervalcouldthusbe measuredtowithiniiL/8microsecond.

Pressure-timerecordswereobtainedforthefournominalpressureratiosata distanceof 8.13feetfromthediaphragmby employingaflush-nmuntedcapacitor-mepressurepickup(RutishauserElectronicPickupIndicator,!&peST-127A)andassociatedapparatus,m shownschematicallyinfigure2. ThesignsJfromfluctuationsonthepressurepickupwasfedintotheY-axisofonebeamof a four-chsmnelcathode-rayoscillograph,andtheresultingdeflectionofthebeamwasphotographedby anNACAsynchronousdrumcamerarotatingat1,800rpm. A l,~o-cpssine-wavetimingtracewasfedintoanotherchannelof theoscillographinordertoobtaina timebase.

NACATN3375 23

. Theoscillographbesmscouldbe turnedonandoffelectronicallybyapplicationof a suitablesignaltotheZ-axisof theoscillograph.This

● signalwasobtainedfroma piezoelectriccrystslpickup,a preamplifier,andanNACAelectronicshuttercircuit.(Seefig.2.) Thecrystalpickupwaslocated6.4feetfromthediaphragmsothattheoscillographwouldrecordthesmbientpressure(“zerotrace”)beforearrivalof theshockwave. Thetimedurationof theoscill.ographtracewasadjus+xibleup to20milliseconds.

A minimumoffivetestrunswasmadeforeachcombinationofnomi-nalpressureratioanddistanceto themedianpositionof theveloci@-measuringsystem.Thechronographtimeinterval,theshock-tube-walltemperature,andthebarometricpressurewererecordedforeachtest.

Pressure-timerecordsat x = 8.13feet weretaken,slongwithsimultaneousveloci~measure~ntsata medianpositionof 9.23feet.me aforementioneddatawererecordedsmd,in adtition,a staticcali-brationofthepressuregagewaspsrfornedforeachrun. Thiscalibra-tionwasmadeby applyingpressuretotheshocktubeinpredetermined

L increments,andateachpressureleveltheoscillographZ-axiswasener-gizedsothatthestatic-pressurecalibrationtraceof thegagecouldbephotographedtoplacea pressurescaleonthedrum-camera-filmrecord.

G

RESULTSANDDISCUSSION

EvaluationofExperinwntslLData

Theexperime~talshockstrengthsshowninfigure5 weredeterminedfromthetimeintervalAt secondsfortheshocktopasstheslitsofthemeasuringstationa distsmceAl feetapartas follows:

whichbecomes,for

~s 27 () 2AZ/At Y - ~—=— -—

Pm 7+1 % 7+1

7= 1.40 anddryair,

(45)

~=~#’ -0.1667 (46)

Thepressurewasplottedat a valueof x halfwaybetweenthephototubes.

4

24

be

NACATN 3375

Theexperimentaldatarequiredgomeslightad@stmentbeforeit could “presentedinfigure5,eachpartofwhichrepresentsa comnondiaphr~

pressureratioor theoreticalperfect-fluidshockpressureratio.Sincethetestspertinenttoa givennominaldiaphragmpressureratioweremade

*

tiththehigh-pressurechsmiberatthesamegagepressure,variationsinatmosphericpressuxefromtimetotimecauseda variationindiaphragmpressureratio.Thereqyiredadjustmentwasmadeinaccordancewiththerelationsthatfollow(notationof fig.1).

Sincetheequationrelatingshockpressureratioto diaphragmpres-sureratio(seerefs.3,h, and7)maybe expressedinfunctionalformas

then

and

(47)

(48)

(49)

Thereferencevalueof p~/pm foreachnominaldiaphragmpreqsureratiowastakenasthevalueexistingwhenthepressure-therecordswereobtainedby thecapacitorpickup.

Sinceeachofthevelocitysymbolsrepresentsan average(ona givenday)offromfivetotentestruns,inwhichthescatterinanysettskenonthesamedayislessthanthescattersmongaveragesofdatatakenondifferentdays,thereasonforthediscrepancybetweenaveragesatthesamedistanceandssmstheoreticalpressureratiois-own. Temperature,viscosi~,andhumidi~corrections,determinedfromthelineartheoryamdappliedh theseexperimentalpoints,areinsufficienttomsketheaveragescoincide.Thecauseof theapparentlyhighexperimentalpres-sureratioat x = 3.2feet (fig.5)isalsounexplainableatpresent. “

?

.

NACA

from

m 3375 25

Thepressure-timetracesofthecapacitorpickuplocated8.13feetthedianhramnstationareshowninfigxe 6. Theltiea-theow

4 valuescompu~edit x = 8.ofeet areplot~ed(circledpoints)forcom-parisonatmillisecondintervelssfterthearrivslof theshock.Theshockpressureratioobtainedfromconcurrentveloci~dataforeachparticularrunfallson theidtialtheoreticalpointwith@ theaccuracyofplotting.Theveloci~data,obtainedat x . 9.23feet,wereadjustedto 8.13feetby drawinga curvethroughthevelocitypointat 9.23feetpsrslleltotheexperimentalcurvesof figures5(a)to 5(d);therebycompensationwasprovidedforanyscatterintheparticulartest.

EvaluationofId-nearTheory

Constsmtsemployed.-Theequationsof thelineartheoryareevaluatedon theassumptionthatstadardatmosphericconditionsexistinthelow-pressmechaniberunlessspecifiedotherwise.Temperaturerecoveryfac-tors r of 0.90and0.85sreusedfortheturbulentandlsminarboundsrylayers,respectively.A FrandtlnumberPr of O.~ anda valueofthe

L BlasiusconstantB of 8.70areassumed.

* Thetenrperature-tiscosi~relationship

T + 223.2~ 1“5P-ml% ()T+223.2 ~

wasexpandedforcomputationalconvenienceto givethe

(50)

approximation

(51)

whichisconsideredvalidinthetemperaturerangenear520°Rankine.

Relationbetweenef(~) * ~.-VariOUStheoretic~~empirical->andexperimentalrelationsareavailableforrelatingcf h 5 for

evaluationof theintegral$()cf ~ d~. Iftheassun@ionof an

equivalent-flat-platesteadyflowisemployedandtheprofileobeysthelaw

% #—=u ()b

boundsry-layer

(52)

26

it iS shownsmdfor n

NACATN 3375

inappendixD foranincompressibleturbulentboundarylayerand B independentof ~ that

Then,theintegralmaybe evaluatedinclosedformas

(55)

,

t-

(53)

(54)

Similarly,iftheequivalentsteadyflowisusedwithanincompressiblekminarboundarylayer(seeref.9),thesolutionis

()1/2cf(E)= 0.664~Ug

(56)

IfthemovingwallormadifiedBlasiussolutionofreference3 isused,theexpressionfor cf(~) inregionB reducesb

A2Us , 1/2

()cf(g)=.———

Al U U~ (57)

r

*

NACATN 3375 27

. inwhich A2/Al areconstamtsevaluatedinreference3. Then

(58)

Sincereference3 doesnotconsiderregioncc,no equivalentexpressionforthisregionisavailable.Eowever,ifthesamegeneralassumptionswereappliedtoregionm aswereappliedtoregion~,evaluationsof cf(~)wouldagainyieldemequationinwhich cf(~) isproportional

1/2to

()$“

In ordertodeterminea possiblecorrectrelationshipbetweenCf(g)and ~ tobe usedinevaluatingthelineartheoryfortherangeofexperimentscoveredinthispaper,thecurvesof theoretical.andexperi-

k mentelshockpressureratioagainstdistanceof figwe 5(c)wereused.In additiont:pressureratio

● areplottedby

(frictioncf=\

theexperiment~pointsobtainedfo=a no&naldiaphragmof 7.455(shockpressureratio2.523),theoreticalcurvesassuminganequivalentsteadyflowfor (a)lsminsrskin

)()o.664R&/2; (-1/5b turbulentskinfrictioncf= O.058~ ,

n = 7)withandwithoutheat-transfereffects;and (c)an~r-nw

curve(fromfig.8$ofref.10)gitingtheintegratedvalueof cf,whichincludestheeffectoftransitionof theboundarylayerfromlaminarto turbtient.Since,asmentionedprevioudy,themcdifiedlaminarBlasiussolutionof reference3 wasnotevaluatedinregionu,no theo-retical.curveC= be drawnforthecorrespondingcf(~).However,duetothelaminarassur@ionsof reference3,theattenuationin shockpres-sureratiowouldbe proportionalto @ if cf(~)wereevsluatedinregiona withsiudlarassumptionstothoseusedforregion~. Con-sequently,inordertodeterminewhethersucha nmdifiedBlasiustreat-mentcouldpossiblyyieldthecorrectrelationbetweencf and ~,a

h-s %ocurveof the~e —=—- (Constaat)6 wasfittedb theexperi-Pm pm

mentaldataat x = 9 feet andtheconstantwasfoundtobe 0.c626.

Thisresultingempirical curve,Pvs—= 2.523- o.0626~,iS dso plottedPm

& infigure5(c).

28 NACATN 3373

Itisevidentfroman inspectionoffigure5(c)thatbestagreementbetweentheoryandexperimentisobtainedeitherfromtheturbulentskin-frictionlawincludingheat-transfereffectsorfromtheexper-ntallydeterminedskin-frictioncurveofreference10. Thereasonforthesimi-larityof theshockpressurecurvesobtainedby thesetwomethodsis

.simplythat,for ~ < 20x 10b,theturbulentskin-frictionlaw

( )‘1/5 fora valueof nCf= o.0581& = 7 whenintegratedclosely

approximatesthecurveinreference10exceptatlowvaluesof ~,wherethelaminarandtrsmsitioneffectsareapparent.Theselat$ereffectsresultintheinflectioninfigure5(c)of thecurvebasedonthedragcurvefromreference10. Inasmuchasthesetwoskin-frictioncurvesRSUJ_tin 8hDSt thesametheoreticalshock-pressureattenuationeithercouldbe usedforfurtherevaluationofthetheory.However,sincem smalyticclosed-fomnintegralismoreconvenientforcomputation

()-1/5inthiscase,theskin-frictionlawcf= 0.0581~ wasusedfor

evaluationofthelineartheoryfortheotherfiguresinthispaper.

Itisalsoevidentfromfigure5(c)thattheequivalentsteady-flowlaminar-bound~-layerskinfrictionpredictsshockattenuationmuchlessthanthatmeasured.Furthermore,theempiricalcurve(atten-uationproportionalto @ - whichmightbe saidtorepresenttheshapeofhminarsolutionsin general,includingthemdifiedBlasiussolution-doesnotindicatethetrendofmeasuredshockattenuation.Thus,itmightbe inferredfromtheseconditionsthattheboundarylayeristurbu-lentinmostof theshock-tube-flowregionsa and ~ atthesevaluesof ~.

Comparisonof TheoryandExperiment

Shockattenuationwithdisteacefromdiaphragm.-Thetheoreticalshockpressurelossandshockattenuationcoefficientsarepresentedasfunctionsofperfect-fluidshockpressureratiosinfigure7. Theserelationsareusedtopredicttheattenuationoftheshockwithdistancetraversedfromthediaphragmstationinfigure5 smdthetheoreticalresultsarefoundtobe ingoodagree~ntwiththeexperimentaldata.Althoughthetheoreticalpointsarecomputedforstandardatmxphericconditions,theexperimentalpointsareadj%.tedto thenons~d~d atmos-phericconditionsexistingwhenthepressure-timerunsweremade. Inordertodeterminethemagnitudeoferrorsointroduced,thetheorywasevsluatedforthenonstandard(T= 542°,p = 14.7lb/sqin.) conditionsforthediaphragmpressureratioof 17.915wheretheerrorintroducedwouldbe largest.Theadjustedtheoryindicatedby thedashedcurveoffigure5(d)showsa slightincreaseinattenuationwhichagreessomewhat

.

b

c

●

NACATN 3375 29

. betterwithexperimentthanthe“standsrdatmmspheretrtheory.Theadjustmentfornonstanddcortditionswouldbe evenlessfortheotherpressureratios.(Seefig.7(b).)

Q

Useofthelineartheorywitha valueof cf= 0.0%~-1/5 appearsin figure5 tooveresthatetheattenuationslightlyatthelowershockpressureratios(below2.5)withtheoppositeeffectappsrentathigherratios.Thistrendcouldbe dueto anyof a numberof simplificationsmadein thetheoryandcomputations.TwoofthemoreobviousfactorsaretheReynoldsnumberandcompressibilityeffects.As thediaphragmpressureratioincreases,theReynoldsnumiberof theflowperunitlengthincreases.Since,forthecomputationsof thisanalysis,cf isasswd

proportionalto ~(

‘2/(~3) whichiS equal to ~-1/~ fOr n = 7)

andsincesteady-flowexperimentsingenerslindicatea trendof cfproportionalto smallerpegativepowers(largern)with ~ increasing,thereisthepossibili~thatvaluesof n shouldbe increasingsome-whatwith & anddiaphragmpressureratio.Compressibili@wouldtendto reducetheratioof thecompressiblecf b theincorqressiblecf

kandconsequentlywouldtendtoreducetheattenuationastheflowMachnumbersincreasedwithdiaphragmpressureratio.

●

Theneglectof thespreadingoftheexpansionfanmustalsobe con-sidered,sincethisomissionmustintroduceanerrorwhichincreaseswithshockpressureratio(expsmsion-femsize).

Otherpossiblefactorsopento questionaretheuseof constsmtPr)theevaluationofheattrsmsferon thebasisof v of thefluidinthecenterof theflow(forexample,notat so~ intermediatetemperaturebetweenthewallandstresm),andtheuseof r = 0.90(0.85).Ibweva,becauseof theassumptionsintheone-dimnsionalaveragingprocess,inthelinearizationprocedure,andintheapplicationof sneqtivslentsteady-flowboundarylayer,thesepointssreminorrefinementstothetheoryintheexperimentalrangeconsidered.Nevertheless,suitablecorrectionsshouldbe appliedashighergoodcorrelationisdesired.

Thecurvesof figure7,whichwereand cf= 0.0581R&/5,Summarizeshocklinesrtheoryforshockpressureratios

diaphrqgnpressmesareusedif

computedon a basisof a~= ~attenuationpredictionby theup to10. For cf proportional

~ ~-2/(n@)tiis curveisofthew

(59)

30 NACATN 3375

where4 istheshock-pressureattenuationat a distsmce2 fromthe .diaphragmstation.Accordingtothisequationtheattenuationataconstantnumberofbytiaulicdiametersfromthediaphragmwouldvaryas

2 P-—

()

a> n+3theshock-tubeReynoldsnumiber~ . Thisrelationshipisdls-

mcussedmorefullyIna subsequentparagraph.

Variationofpressuretithtimeata givendistance.-Fressure-ti~curvesobtainedexperimentallyby thecapacitorpickuplocateda distanceof 8.13feetfromthediaphragmstationsrecomparedinfigure6 withtheoreticalresultscomputed(seefig.8)fora distanceof 8 feetand

‘1/5.Theerrorintroducedby thevariationindistanceCf= o.o~qis about1 percentoftheattenuationpressurechangeandmaybe ignored.Theexperimentalshockpressurewasalsoobtainedforeachrunby com-putingthepressureat 9.23feetfromsimultaneousvelocitymeasurementsandtheninterpolatingbackto 8 feetalonga curveparalleltotheexperimentalcurveof attenuationagainstdistance.Thesevelocitymeasurementsfallon thetheoreticalmeasurementswithintheaccuracyoftheplottingof figure6.

z

Boththeoreticalamdexperimentalcurvesshowa pressureincreasing #withtire?.ThisincreasehasalsobeennotedwhenNACAminiatureelectri-calpressmegages,aswellaspiezoelectricpressurepickups,havebeenemployed.Theinitielpressureriseacrosstheshockdeterminedfromthevelocitymeasurements,aswellasfromthetheory,appearsto fallbelowtheaverageoftheinitialdampedhigh-frequencyoscillationsofthecondenserpickuppressure-timecurve.However,thedynamicresponseoftheRutishausersystemisnotknown,whereastheshock-velocitymeasuringsystemhasa knownhighprecision.Consequently,theuseof theinitialshock-velocitypressurepointinconjunctionwiththeRutishauserrecords,afterthehigh-frequencyoscillationshavesubsided,appesrsa logicalchoicetogivea betterrepresentationof theactualpressure-thephe-nomena.Suchsm “experimental”curveagreesveryfavorablywtththepres-surerisepredictedby thelineartheory.

Variationofotherparameterswithtimeata givendistsmce.-Fig-ures9 to 12 showthepredictedvariationwithtimeof averagesonicspeed,fluidvelocity,density,andMachnumberat x = 8 feet.Thechangeof theseparameterswithtimeincreaseswithshockstrength,sincetheskinfrictionandthetemperaturedifferencebetweenwallsandfluidincreasewithshockstrengthforlowvaluesof shockpressureratios.Theseaforementionedquantitieswerenotmeasuredintheexperimentsofthisinvestigation;however,thephenomenaofdensityincreasingwithtimewerefoundexperimentallyby theuseof a cbrono-interferometerandarereportedinreference8. Thebehavioroftheaverageparticlevelocity -

NACATN 3375 31

.hasnotbeenascertained.An increasein speedof theleadingedgeof themixingzonewhichreplacestheentro~discontinui~betweenthehotsmd

* coldgasregionsofthetheoreticalflowhasbeenreportedinreference7andelsewhere.However,theapparentvelocityincreaseoftheleadingedgeisalsoattributabletoturbulentmixingsinceit ishewn thatthewidthofthezoneincreaseswithtime. Consequently,thevelocityof theleadingedgedoesnotgivea trueindicationoftheaveragepszticleveloc-i@ dueto themaskingeffectofmixing.An increaseinMachnumberwithtimewasfoundexperimentallyinreference3. Also,an attempttomeas-uresoundspeedwasreportedinreference3,butthereportedresultofconstantsoundspeedwith*2 percentvariationovera 100-microsecondintervaldoesnotappearsignificsmtinviewof theprocedureemployedand,inaddition,thepredictedlinear-theoryvariation(fig.9) islessthm scatterof thedataofreference3.

CorrelationofWaveSystemandFlowPhenormna

Wavegenerationandidentification.-Theresxetwowaysto considerpressurewaves:Thenmrefamiliarmethodisthatof an observerin a~tiedpositionwho (withwavesassumedtobe comingfromonlyonedirec-tion),upondetectingan increasing(ordecreasing)pressuretithtime,

. recognizesthearrivalof a compression(orexpansion)wave. Theothermethodisthatof anobsemer,travelingwiththespeedofthewave,who (wavesagainassud tobe comingfromonlyonedirection),upondetectingan increasingpressurewithtime,recognizesthegenerationof compressions.Thismovingobservermayalsodeterminethetypeofwa’vewith.whichhe is associatedby ascertainingat a giveninstantoftimewhetherthepressureislower(orhigher)~ead of thewave,inwhichcasehe is travelingwitha compression(orexpansion)wave.

Thestationaryobservertilldetectwavesonlyinunsteadyflow,whereasthemetingobserverwilldetectthegrowthofwavesin a steadyflow. AS snexampleof thelattercase,considertiepress~edI’OPofa steadyflowof airthrougla frictionzone. Itmaybe treatedas asystemofdowmstresm(P)waveswhich,althoughof zerostrengthbeforeenteringthezone,becomestrongerexpansionwisethefartherintothezonetheytraverse.Simultaneously,thereis a trainofupstream(Q)waveswhich,alsoof zerostrengthinitial@,growintostrongercom-pressionsastheytravelup intothefrictionzone.Thesetwosystemsofnmvingwavescombineto forma standingwave,invariantwithtine,or a pressuredropalongtheflow. Consequently,a fixedobserverwouldrecognizenom-wingwavesin theflow. Itmightbe notedthatthesetrainsofwavesarethereflectionsofthewaveswhichproducedthesteadyflowfromqtiescentair.

Cognizanceshouldalsobe tskenof thefactthatthetwoobsenerswoUlddisagreeon thetypeof supersonicQ wavesin a flow. Considera unsteadycompressionwavenmvtngupstreamagainsta supersonicflow

●

32 NACATN 3377

.butactuallybeingwasheddownstreamrelativetoa fixedcoordinatesystem.Thetravelingobserverlooksahead(upstream),seeslowerpres-smes,amdsorecognizesa compressionwave. Thefixedobserwer,how-ever,firstencountersthetrailingedge(highpressure)andthenat alaterinstanttheleadingedge(lowpressure)of thewaveandapparentlyrecognizesanexpansionwave,sincethepressurefallswithtime. Con-sequently,insupersonicflowthefixedobservermustmakeadditionalvelocitymeasurementsinorderto interpretthewavesigncorrectly.

Theunsteadywavesgeneratedi.nthevariousregionsof theshock-tubeflowcm besttionswhichresultequation(31)

16Qla—— -.—ac M 7 a~

Itisevident

veloci~wavesbut

be understoodby considerationofthefollowingequa-fromequations(1),(2),(3),anddifferentiationof

.

a[[

}= ~%Jcf (7- l)MT ~M+ Pr-2/3~D a~

●

(60)

~sthattheterm ~ isnotconnectedwithpressureor

St

{1PmerelyaccountsforthechangeinQ

=~a~u due

to traversalofthecharacteristicthroughfluidofdifferententropy.Thepressme-wavegenerationisrepresentedby thechangeinthedifferenceofthesetermsandisa resultoftheskin-frictiondragandheattransferasexpressedinthefirstandsecondterms,ontheright-handsideofequation(6o).

E equation(6o)isnowexaminedfrom

viewpoint,whichismwe appropriatesince

respectivefi,withinthebraces

thetravelingobserver’s8( } representsthechangeF

withtimeasnoticedby thetravelingobserver,thefollotingfactsbecomeobvious:

(1)Upstream(Q)compressionwavessregeneratedby theeffectsofbothheatadditiontothefluidandskin-frictiondrag. .

●

jKI?ACATN 337’3

.(2)Dwnstream(P)

s (3)Downstream(P)dragif M<~.

7-1

(4)Downstream(P)

33

compressionwavesme generatedby heataddition.

expansionwavesaregeneratedby theskin-friction

compressionwavesaregeneratedby theskin-frictiondragif M >~.

7-1

Thislastresult(item(4))wasunexpectedsincethedragisusuallyconsideredashavinga brakingordecelerating(exp=sion)effectonthefluidaheadof it.

Thefollowingfactsmightbe notedat thispointconcerningtheheat-trsmsferandskin-frictioneffects.Expansionwaveswillbe gen-eratedin a forshockpressureratiosup to about5.9whenairatinitisllyuniformtemperaturethroughoutisusedonbothsidesof thediaphragm.Forpressureratiosabovethatvalue,compressionsaregen-

. crated.Also,heattransferisto thefluidinregionu belowshockpressureratiosgxeaterthan7.5;abovethisratiotheadiabaticrecoverytemperaturebeginsto exceedthewalltemperaturefor r = 0.9.

n

For M<~ thefollowingconditionsthenapplyfor T. = T= =7-1

‘wall“ Inregionm heatisbeingtransferredto thefluid.Thus,theeffectsofheat-transferandskin-frictiondragareadditivewithregsrdto thegenerationof Q compressionwaves.However,inregsrdto thegenerationofP waves,theeffectssxein oppositionsincetheheattransfertendsto generatecompressionsandthedrag@ generateexpansions.Inregionp heatisbeinglostfromthefluidto thewalls. Theheattransferanddragsre,therefore,cumulativewithrespecttotheproductionof P expansionssadinoppositionwithrespecttoQ waves.Thestrongerwavesgeneratedwill,consequently,be Q ccxnpres-sionsinregionm andP expansionsinregionp,wherethedragandheat-transfereffectsarecumulative.

Solutionforshockpressureratioof 2.6.-Theexclamationfortheflowbehaviormaynowbe consideredwiththesepointsinmind. me plotof x againstt fora shockpressureratioof 2.6is showninfig-ure13. Adjacenttovariouspointsareindicatedthevaluesof P’, Q’,a’, U, and p. Theprimedquantitiesreferto thevaluesthattheparam-eterswouldhaveif an imaginarylaminaof fluidwithentropyequaltothatof theunperturbedflow,butwithvelocitiandpressureidenticalwiththelocslperturbedflow,wereinsertedintheflowat thesepoints.Inotherwords,let

. (61)

34 NACATN 3375

andsoom Thensince,by useoftheimaginarylamina,theentropytermmskesno contribution,P’ representsonlythepressureandthevelocitywaveeffects.

Thereasonforthedecreasein shockpressurewithtimeisnowevi-dent. JustbehindtheshocktheovertakingP expsmsionwavesaregrowingin strength(P’ decreasing)inproportiontothet- titervalthatittakesfortheP characteristictotraversetheregionsa and ~;and,of course,thistimeintervalincreasesasregionsm and ~ divergewithtime.

Infigure13,thevaluesof P’ on the P sideof theentrowdiscontinuityaregraduallydinrhdshingwithtimefromthevalueP’.= Ppo= 6.7471amasweakexpansionsaregeneratedinregiona.Theseexpsmsionsareweakbecausetheskin-frictionexpansioneffectisohlyslightlystrongerthantheopposingheat-transfercompressioneffect.Thevalueof P’ slowlydecreaseswithtimealongtheentropydiscon-tinuityastheregiona inwhichthewavesaregeneratedbecomeslager.

Inregion~,however,boththeskinfrictionandheattrsmsfercombineto generatestrongP’ expansionswhichincreasein strength(P’ decreasing)rapidlywikhdistancetraversedin p. Thus,thefirstP characteristicshowninfigure13haeexperienceda decrementin P’from Po = 6.5451of only0.00> attheentropydisconttiuityandDf0.1167whenitovertakestheshock.ThesecondP characteristicshownhasa decrementin P’ of 0.0105attheentropydiscontinuityandof0.2032whenitfinallyovertakestheshock.Notethemuchgreaterexpansiongeneratedinregion~.

Thecauseofthevariationofpresswewithtimeata givenstationisnotsoobviousbutmaybe describedin a simplemannerforthepar-ticularpressureratioof2.60asfollows(theexplanationisnotgen-eralsincethesignsof someofthewavesmaychangewithdiaphragmpressureratio):At a givenvalueof XP,astimeincreasestheinci-dentdownstreamcharacteristicshavetraversedlongerdistancesin ~md shorterlengthsin 13.Nowsincethedragandheattransferdonotreinforceoneanotherin m,whereastheyhaveanaccumulativeeffectin~ onPwaves,theeffectofdecreasein traversaltimeof ~ nnrethanoffsetstheeffectof theincreaseinthatof a.

Iffigure13 is againused,thefollowingtabulationcanbe madeforthevaluesof thevariationin P’ alongthecharacteristicforthefourcharacteristics.Subscriptsdenoteregionsresponsibleforthevariations.

.

.

NACATN 3375 35

FCharacteristicsline(fig.13)

1234

Z-o.C@-.0105-.0179-.0232

5P’P

-0.1113-.0602-.0255-.C@17

mP ‘

-0.1167-.0707-.ok34-.0279

aValueincludesreflectionatentropydiscontinui~.

Consequently,eventhoughexpsmsion6arebeinggeneratedin a and p,to a fixedobserverata givenx~,theincidentdownstreamwavesa’ppesxas compressionsinthatthevalueof thepsrameterP’ isincreasingwithtimesimplybecauseitsreductionfromtheconstsmtvalue Pc isdecreasingwithtime. In otherwords,theexp~sioneffectgeneratedalongtheP characteristicsup to a given

-X9 isdecreasingwithtime

althoughthenetexpsmsioneffectup to theshockwaveisincreasingwithtime. tiaddition,theincidentQwavesinregion~ arecompres-

. sionwaves(forexample,bag effectis lsrgerthanheat-transfereffectforthisparticularcase)thestrengbhofwhichincreaseswithtrav-ersaldistance(time)fromtheshockto a fixed xp. me resultoftheseincidentwavesisthusa pressurerisingwithtime.

Theaverageveloci~isincreasingwithtimeat a given x~ because

()

aP‘the“effectivecompressions,”thatis ~ > 0, downstreamarestrongerx

thantheupstreamcompressions.Theaverageveloci~of theentropydiscontinti@decreaseswitht- becausestrongerexpansion=(lowervaluesof P’)sreovertakingitfrombehindwhilesimultmeouslystrongercompressions(highervaluesof Q’)aremeetingitfromahead.I?athwavesaredeceleratinginfluences.Theaverageveloci~justbehindtheshockis,of course,decreasingwithtimeanddistance.Forthisparticularcase,theperfect-fluidvelocityisneverattainedin regionp fort>o.

Althoughtheprimedquantitiesoffigure13 givecorrectvaluesofpressureandaveragedveloci@,thevalueof a’ isnotanindicationof thecorrectaveragesonicspeedsincetheprimedquantitiesareslwaysevaluatedinan imaginaryfluidlaminaof constantentropy.Iftheentropycorrectionwereapplied,itwouldbe foundthattheaveragesonicvelocity(temperature)isdecreasingwithtimeinregion~becausetheeffectof theheattransferredoutof thefluidis greaterthanthercotiinedeffectofthecompressionwavesandfrictionaldissi-pation.Inotherwords,althoughthestrengthof theincident“compression”

36

wavesincreaseswithtimeata givenx, thefluidhasbeeninmotionforincreasinglylongerperiods

NACATN 3375

arrivingatthat xandmoreener~ is

lost by heattrsmsfertothewalls,sothatthenetresultisa decreaseintemperature.

AssessmentofVariousAssumptions

Thefavorablecorrelationbetweenexperimentandlineartheorysti-stantiatestheuseof a small-perturbationapproach.However,thelimi-tationsof theassumptionsusedintheanalysismustbe recognizedandweighedcarefullybeforeapplicationof thetheoryto a particularproblem.

Oneof themostprecariousassumptionsintroducedwastheaveragingprocess,wherebythequasi-one-dimensionalflowwastreatedas a pureone-dimensionalflowinwhichaveragedvaluesof p and U weredefinedasthosesatis~ingthecontinuityequationintegratedovera crosssec-tionof theflow. Itislmownthatthisassumptiontillintroduceerrorsin themomentumequation.Theseerrorstillbe smallerfor“full”veloc-ityprofilesand,iftheboundary-layerprofileisassumednotto change 9

withflowtravel,willincreaseinmagnitudeastheratioofcross-sectionalaveragevelocitytomaximumvelocitydecreases.Theerrorswillalsoincreaseaathevelocityprofilechangesshapewithflowdis-tance.Forexample,consideranincompressiblesteadyflowof fluidenteringa pipewitha purelyone-dimensionalrectangularvelocitypro-filewhichlaterbecomesa parabolicvelocityprofilesomedistancealongthepipe. Theflowhasa constantaveragevelocity,yetitrequiresapressuredropin excessofthatneededtoovercomefrictionto accountforthechangedvelocityprofile.Consequently,theaveragingprocessemployedinderivingthebasicequationsmaylimittheapplicationofthetheorytoboundaxylayerswhicharesmallrelativetotubeheightandwidthorwhichhavenearlyconstantshapesalongthetubelength.

Inaddition,iftheskin-frictionintegralistobe evaluatedbytheuseof smequivalent-flat-platesteadyflow,thereis introducedtheassumptionof a shock-tubepotentialflowboundedby a viscousbound-arylayer.Evenwithouttherestrictionsof’anaveragingprocess,thisassumptionrequiresa boundary-layerthicbess b smallinrelationtothedimensionsoftheshock-tubewidth.!Ihisrestrictionwasviolatedinthenumericelevaluationofthetheorywhenappliedtotheexperi-mentalresults,‘sincethetheoreticalvalueof b at theentropydis-

continuitywasapproximately1 inchtithe1.$-by 2-inchshocktubefor

x = 8 feet fortheworstcase‘(shockpressureratio1.94).Thefactthatagreementwasstillgoodbetweentheoryandexperimentevenundertheseconditionsindicatesthatthisboundsryassumptionmaybe violatedmarkedlywithoutseverelypenalizingtheaccuracyof thetheory.Of

.

.

NACATN3375 37

> course,whentheboundszylayerdevelopsto sucha degreethatitfillsthetubeandso-called“yipeflow”arises,thenthecharacterofthe

4 flowwillbe soalteredthata newapproachbasedonpipeflowwouldprobablybe mre applicablethantheboundary-layerandptential-flowapproachoftheequivalent-flat-platesteadyflow.

Theassumptionof a flowmdel inwhichveloci@andthermalboundsry-l~ereffectsareincludedby averagingacrossthechannelareaisopentofurtherqyestion.Thistiel presumesinstantaneoustr-ferofheat,mmnentrm,andsoforth,fromthefluidadjacentto thewalltothatinthecenteroftheflow. Inreality,however,eithermolecularormacroscopicmotionoftheparticlesortransversepressurewavesisnec-esssryforsucha transfersothattheinfluenceisfeltatthecenterlaterthanatthewalls.Sucha delayisusuallyof secondorderandcanbe ignored.

Theaccuracyof theapproximationforthefrictionaldissipationasequalh theproductofwallshearingstressandaveragedvelocitycsmnotbe verifiedunlessthevelocityprofileisknown;however,thisapproxi-

h mationshouldbe god formmy casesandisexactforbothtrueone-dimensionalflowsandPoiseuil.leflows.

Thelinearizationofthedifferentialequationspreventstheappli-cationof thetheoryto largedeviationsfromthetheoreticalflow. Thedegee towhichthisrestrictionmaybe stretchedis stillunlmownsincetheexperimentaldatacomparedwiththeoryhadvariationsfromperfectfluidflowonlyup tofifteenpercent.

Intheanalysisitwasassumedthattheleadingedgeof theexpan-sionwavetraveledwitha velocity-a= andthatimmediatelybehindtheleadingedgethefluidandsonicvelocitieswere U and ~, respec-tively.Sincethesonicsp-eedvariesfrom a= to ~ andthefluidvelocityfrom O to U as theexpansionwaveistraversed,theabove-mentionedassumptiontillintroducesmerrorwhichwillmagnifyas theexpansionfanbecomeslargeratthehigherdiaphragmpressureratios.Furthermore,ifa mre exacttreatmentweredesiredfortheexpansionregion,itwouldbe necessaryto considersmequivalentacceleratingsteadyflowwitha favorablepressuregradient.

Inthenumericalevaluationof thetheo~,itwasassumedthatthefunctionalformof theskfn-frictiondependencyon theflowlength~wasidenticalfortheunsteadyflowintheshocktubeandthee@vslent-flat-platesteadyincompressibleflow. Thisassumptionhasno rigorousargument.So= supportmaybe foundinthesimilarityof thevaluesof Cf obtainedfromtheRayleigh,Blasius,=d modifiedBlasiussolu-tions. Forlaminarboundsrylayersallthreesolutionsgivefunctional

38 NACATN 3375

relationships,cf= (Constant)#~, buttheconstantsUg

differ.No com-

par~lesolutionsarelmownforturbulentboundarylayers.Untilsomefurthertiowledgeisobtainedaboutunsteadybound~ layers,it appearsthatthisassumptionof theequivalencyof steadyandunsteadyflowsmustbe acceptedprincipallyonthebasisofagreementbetweentheoryandexperimmtinthisramge.

Whenthenumericalevaluationof thetheoryisbasedon incompressi-bleboundsry-l~ertheory,thereis,of course,a sourceoferrorasdis-cussedearlier.However,thiserrorcambe eliminatedby theuseofclosercompressibleboundaqy-layerapproximationsintheevaluationofthecf(~)integrsls.It shouldbe notedthattheeffectof thecompressi-.bilitycorrectionisnotaslargeasmightfirstbe expected.SincetheMachnuniberinregionp cmnotexceed1.89forair,althoughitmaygoto infinityinregiona, thecorrectionto cf forcompressibilitywillbe muchlsxgerin a. Eowever,Va becomesincreasinglylessthan VPas ~ increases,sothattheinfluenceofregiona decreases.Inaddition,thecompressibilityeffectwillbe lesson theshock-waveattenuationthanon thevariationoftheflowparameterswithtimeata #

givenstationsincetheimportanceofregioncc on 5P’ changesasdiscu~sedintheprevioussection. .

Theeqyivslent-steady-flowMel doesnotgivea truepictureof theregionneartheentropydiscontinti~evenifturbulenceisneglected.Since v isdifferentoneachsideofthediscontinuity,themodelyieldsdifferentboundsry-layerthicknessesor animpossiblediscontinuousbound-arylayeracrossthediscontinuity.Thiseffect,plustheneglectedheattransferacrosstheentropydiscontinui~,apparentlyis smallwhenexsminedh thelightof the

Applicationof

Thecompromiserequired

experimentalremil.ts.

TheoryinShock-TubeDesign

betweenaerodynamicandmechanicaldesignof shocktubesforuseathightemperaturesandMachnumbersbecomesevidenton scrutinyofthefunctionalformoftheequationrelatingtheratiooftheattenuationshocklossorpressurerise(orfall)withtimeto theanfbientpressure.Thisequation,for cf proportionalto

R#/(n+3), is

4—=P.

NACATN 3377

>At a givenvaluetions,theratio

of P~pmY inordertohaverelatively2/% shouldbe large;butinsulating

39

longflowdura-problemsusually

* dictate& ~n theatmosphericrangeso that 2,consequently,becomeslsrge.

Theninordertominimizethetimewisevariationsat a given x, theratio Z/D shouldbe as smallaspossible,a requirementprescribingmaximumavailableD. If D islarge,mschanicdstrengthof theshock-ttiewallswilllimitthepeskshockpressuressothat,forlargevaluesof Pp/Pmfitwillbe necessarytomake pm small.A smallvalueof pmwillinturncause V. to increasesothatthereciprocalof theReynolds‘@er ‘m/@ risesandwithit theattenuationandflowfluctuations.Conse~ently,thedesignof theshocktubewillbe a compromisebetweenmechanicalandaero~amicdesign,withthe Z/’Dterm(whichhasa morepowerfulexponentthantheReynoldsnumber)probablybeingtheprincipalaerodynamicconsideration.

* CONCLUDINGREMAR16

.● Comparisonof theoreticalandexperimentalresultson theflowin

shocktubesappearsb substantiate,intherangeoftheexperiments,themethodof anelysisemployed.Theapplicationof thebasicconceptstomuchhigherpressureratiosandlsrgerperturbationsappearslogical.However,in suchsm application,refinementofthelinearizedproceduremaybe requiredto compensateforthecrudityof saneof theassmnptionsor itmayevenbe necesssrytorevertto a step-by-stepintegrationofthebasicnonlineerdifferentialcharacteristicequationsandto employsimultsmeousl.yskin-frictioncoefficientsbasedonlocalvelocities.Inaddition,thebasiccharacteristicequationsthemselvesmayrequirefurthermodificationas theshockstrengbhincreasestovalueswheredissociation,ionization,andrelaxationeffectsbeccmetiportant.Belowtheshockstrengthswheretheselast-mentionedeffectsoccur,thetrendspredictedby thelinesrtheoryandevaluatedinthispapershouldapplyalthoughtheirmagnitudeisdependentontheformof skin-frictionlawasszmed.

Althoughanequivalentsteady-flowturbulentboundary-layersolu-tiongavegoodagreementwiththeexperimentaldata,applicationof thetheorytocaseswheretheflowReynoldsntier islowerthanthatoftheseexperimentsmayrequiretheuseofhninarboundsry-layersolu-tions.SuchlowerflowReynoldsnumberswouldariseeitherfromshocksweakerthanthoseof theexperimentsadvancingintoairatatmosphericpressureorfromshocksof thesameorhigherpressureratioadvamcingintoairbelowatnmsphericpressure.It shouldalsobe notedthat,at

ko NACATN 3375

highReynoldsnunibers(>2 x 10?),theevaluationoftheskin-friction‘1/5becomesinaccurateand. coefficientas 0.0581x (Reynoldsnumber)

shouldbe replacedby a moreappropriaterelation.

Thetheoreticallypredictedtrendsinthehotgasflowforairatuniformtemperaturethroughoutinitially~ asfollows:Shockpressureratiodecreaeeswithdistsace;staticpressureandaveragevalue6ofdensity,fluidvelocity,andMachnurriberincreasewithtimeata fixedpoint;andaveragesonicveloci~decreaseswithtimeata fixedpoint.

,.

LangleyAeronauticalLaboratory,NationslAdvisoryCommitteeforAeronautics,

WrigleyField,Vs.,December7, 1954.

.

NACATN 3375 41

is

APPENDIXA

DERIVATIONOF CEWRA~ERISTICEQUATIONSFORQUASI-

ONE-DIMENSIONALCHANNELFLOWWITEARRA

Thegeneralexpressedas

Ifand U

a loge Aa-t

averaged

FRICTION, ANDKEATADDITION

formofthequasi-one-dimemional

&pA) + &)UA) = O

CHANGE,

continuityequation

(Al)

+Ub logeA blOgep+ublO~p+bU_oax + at ax G-

(AZ)

valuesacressa sectionoftheflowareemployedfor pwhenchsmnelflowssreconsidered,

onlythephysicalcross-sectionalareaandthiclmessisof no concern.

Theidentities

az= 7RT

theareatermboundsry-layer

d ~= d(Heatadded)R RT

combinedtiththeequationsof stateandenergyfamilisrrelationships:

P = (Constmt)a2c~/Re-S/R

P = (Constant)a2cp/Re-S/R

A representsdisplacement

(A3)

(A4)

yieldthefoil.oting

(A5)

(A6)

42 NACATN 3375

Now,ifallthefluidunderconsiderationhasbeeninthesamestateatanyprevioustime,equations(A7)and(A6),maybe differentiatedwithrespectto x and t withno contributionfromtheconstantterm.

a logep 2CVa 10gea a ~

ax ‘— ax ‘K(A7a)

R

a loge ps

2CVa 10gea a ~-—

at ‘— R at at(Am)

a loge~ 2CPa 10gea a;=— .— (A7c)

ax R ax ax

a loge~ 2C alogea a:.2

at R at ‘—at(A7d)

Themomentumequationisthenwrittenby useoftheskin-frictioncoefficientandhydraulicdismeteras

&J+u 2u2cfw+A&+_= oat ax P ax D

(A8)

Thisequationisnotexact,sincetheeffectivevalues of p and Uwerechosento satisfythecontinuityequation;however,forltfulll~velocityprofiles(thatis,profilesinwhichthevelocitydeviatesonlyslightlyfromitsmaximumvaluein a largepartofthecrosssection),equation(A8)isa goodapproxtition.

Equations(A8)and(A2)csabe re=ranged,titersubstitutingfromequation(A7)andemployingtheconvectivederivative

D()_ a( ) +UaoDt at ax

(A9)

.

.

●

✎

NACATN 3375

● to give

.

D~2~Da+a$-a — +aD logeA oRDt Dt Dt =

43

(Ale)

(Jw

Equations(AIO) and(All)maybe addedandsubtractedtogivethefollowingequationsinoperaimrform:

(A12)

(A13)

Theassumptionhasbeenmadeinequations(A12)and(A13)that cv/RisConstsrlt. Now,fromthedefinitionof thederivativealonga charac-teristicof slope 6x/bt equalto U t a as

equations(A12’)and(A13)maybe writteninthefinalform:

(A14)

() Dloge A ab; D:5P 52cva+u =-a 2U2cf—= .— + -—+-. —-—b-t bt R Dt y 5t Y Dt D

(A15)

44 NACATN 3375

(m6)

NACATN 3373 45

APPENDIXB

EVALUATIONOF CONVEC3ZIWEDERIVATIVEOFENTROPY

Theconvectivederivativeof entropymay

D;

()

1 DJ + ‘Hf—= —— —D-t ETDt Dt

be expressedintheform

(m)

where DJ/Dt representstheheataddedperunitmassby heattransferandheat6ourceswhile DEf/Dtrepresentstheheatmassby friction.

Theheattransferperunitmassfromthewane. as &h6/pDwhere e isthedifferencebetweenthetherecoverytemperatureoftheflow;forexample,

.

e = Tw~ -(

Y-)

1 #Tflow1 + r —

2

dissipatedperunit

maybe approximatedwalltemperatureand

(B2)

Theassumptionisnextintroducedthattherelationbetweenheattransferandskinfrictionisgivenby a modifiedformofReynoldstanalogywhichisassumedto applyforturbulentaswellasforlaminarboundarylayersinbothsteadyandunsteadyflow.SubstitutionoftheperfectgaslawintothemodifiedformofReynolds’analogyresultsinthefollowingequationforthewallheat-transfercoefficient:

-2/3l+@upr Cfh=–27-1

Theheat-transfertermmaythenbe wittenas

(B3)

DJ 27 RO ucfPr-2/3.= — —

D’C7-lD(B4)

46 NACATN 3375

Thefrictionaldissipationperunitmassisassumedtobe theproduct doftheaveragevelocitymultipliedby thewall-sbesringby themassintheflow:

DHf UT(Perimeter)& 2u3—= .—cfm pAh D

Althoughequation(B5)isnotexactingeneral,itmationformostcases.Itmightbe notedthatequationa trueone-dimensional-flowmodelwherethevelocityis

forceanddivided

(B5)

isa goodapproxi-(B5)isexactforconsideredcon-

stantacrossa crosssectionoftheflowwitha dis~ontinuityinvelocityatthewall. It isalsoexactforincompressiblePoiseuilleflowsince

thedissipationfunction,r(\@12~ dy,reducestothevalueof equa-

J wdy

tion(B7).

Therefore,undertheconditionspresumedtions(Bl),(B4),and(B5)naybe combinedtoderivative

to applyevaluate

D:

(

_9PrM2+ 1

)

-2/3q Ucf—=Dt 7-lT D

.

above,equa-theconvective <

(E6)

I?ACATN 3375 47

APFmNmxc

L~ SOLUTIONFORINTERSECTIONOF CHARACTERISTICS

ANDJH’JTROPYDIsCONTmY

Thelinearizedsolutionfortheintersectionofthecharacteristiclinestithanentropydiscontinuityisobtainedinthefollowingmanner(seefig.1 fornotation):

28P= Pf -Pc=Qf-Qc=—

()

afl7-@z-

2Pc+Qf=—()

%?—+17-~

acac

Intheseequationsthevaluesof 7 sxeassumed. everywhere.

Dividingequation(Cl)by (C2)andapplyingpressuresarealwaysequalto onesnotherat anyofthediscontinuityyields

Sf-sc2CP - ~bP=e

pc + Qf Sf-sc

2CP + ~e

(cl)

(C2)

tobe identical

theconditionthattheinstantonbothsides

(C3)

Now,ifthesubscripto denotesconditionsattime t = O immed-iatelyafterthediaphragmburstwith

AS=sf-sc (C4)

or

48

and

NACATN 3375

substitutingequation(C5)inequation(C3)andexpandinggive~

As.

(c6)

(C7)

or

1

5P Pci-Qf

1 1

,+,5-+ . . . . .—= (c8)5P0 ‘% + Q~o % apo2

(-)-1

a%

5P

[

—= l-i-8P0

Pc - p%) + & - QBO

1

+. ....‘% -I-Qpo

(@)

.NACATN 3375

.P= - Pa. Qf - Q~o

Stice and areof order A, thefirst-order‘% + Q~o ‘% + QPO

or Mnesrizedsolutionbecomes

8P m (P=- Pao

)(+ Qf -QPO)—=1+

~=wo+

p% + Qk

R J

.

NACAm 3375

APPENDIXD

RELATIONBETWEENSl!CEN-llRICTIONCOEFFICIENT

ANDVELOCITYPROFILE

Therelationbetweenskin-frictioncoefficientandvelocityprofileforan incompressibleturbulentboundarylayeron a flatplatemaybefoundinthefollowingmanner.Letthevelocityprofileintheboundery

l/nlayerbe oftheform

()%= zU5

where ~ isthelocal

velocitvatthedistanceY fromtheplateand ~ isthethickne~s.Then,equating-thedecrease,inthestresnrwiseglayergives

dT =—

—wallshearingstressT todirection,ofthemomentum

nti

Evaluationofthisequationforconstantp yields

T‘=(n+l~(n +2)p$

d5+ (2- r12)5 ~—U (n+ l)2(n+ 2)2‘~

streamwise

boundery-layertherateofintheboundsry

(Dl)

(D2)

Thefollo~ relationsexethenassumedbetweentheshe~ing-stressvelocity~ and-thebound~-~ayerthic~ess b (seeref.10):

d.;=— (D3)

(D4)

*

.

.

.

TheparameterB maybe assuneda constautfor n invariantwithE,butitsformfor n= n(?) isnotspecified.

NACATN 3375 51

Substitutio~of equations(D3)and(D4)intoequation(D2)resultsinthefollowingdifferentialequation:

.

a2 x

()

l/11 n+3~n+l~ . 1 UV (n+l)(n +2) -.— (2 - Da) —~n+lg

‘~@B n n(n+ l)(n+ 2) d~

(D5)

Forthecaseof n and B independentof f,equation(D5)maybeinte=ated~bY ass~i% n ~ -1. Thisintegrationwithapplicatio~oftheboundsryconditionof 5 = O at ~ = O produces

n+l

[ 1—n+ln+3—v (n+2)(n +3) ~~ ~n+35=—W’ v

(D6)

Substitutionof equation(D6)intoequation(D2)resultsintheexpressionfortheskin–frictioncoefficient:

L J

(D7)

NACATN 3375

REFERENms

1.Lobb,R.K.: A StudyofSupersonicFlowsina ShockTube. UIIARep.No.8,Inst.ofAerophysi.cs,Univ.ofToronto,May1950.

2.Huber,PaulW.,Fitton,CliffE.,Jr.,andDelpino,F.: ExperimentsJ-InvestigationofMovingFtressureDistuxbsmcesandShockWavesandCorrelationWithOne-DimensionalUnsteady-FlowTheory.NAC!ATN 1%3,lgkg.

3.Hollyer,RobertN.,J&.: A StudyofAttenuationintheShockTube.Proj.M720-4(ContractN6-oNR-232-ToIV,Off.ofNavalRes.),Univ.ofMichigsmEng.Res.Inst.,July1,1953.

4. Geiger,F.W. andMautz,C.W.: TheShockTubeasam InstrumentfortheInvestigationofTransonicandSupersonicFlowPatterns.Proj.M720-4(ContractN6-oNR-232,NavyRes.Project),~iv. ofMichiganEng.Res.Inst.,June1949. (WithA&dendumbyR.N.Hollyer,Jr.) .

5.Emrich,R. J.,andCurtis,C.W.: AttenuationintheShockTube. Join.ofAppl.Phys.,vol.24.,no.3,M=. 1953,pp.360-363. .

6.Donaldson,Colemandup.,andS~ivan, RogerD.: TheEffectofWallI!rictionontheStrengthofShockWavesinTubesandHydraulicJwnpsinCharnels.NACATN1942,1949.

7. Glass, I. I.,Martin,W.,andPatterson,G.N.: A TheoreticaladExperimentalStudyoftheShockTube. UTIARep.No.2,Inst.ofAerophysics,Uml.v.ofToronto,Nov.1953.

8.Mack,JohnE.: DensityMeasurementinShockTubeFlowWiththeChrono-Interferometer.Tech.Rep.4,LehighUniv.Inst.ofRes.(Froj.NR-061-063,ContractN7-ONR-39302,Off.ofNavalRes.),Apr.15,1954.

9.SchlichtiW,H.: LectureSeries“EoundaryLayerTheory.”PertI -Iand.narFlows.NACATM1217,1949.

10.Schlichting,H.: LectureSeries“I!mndmyLayerTheory.”PartII -TurbulentFlows.NACA!IMI-218,1949.

.

+

aE.-1-

, ,

Characteristics

—— — ––Porticle paths

~+~

c

w

//

/

/

o

Distance from diaphragm,

Flgore1.-Dist.ante-tlmsplotof

x

shock-tubeflow.

uVI

Shock tube

Light

sources

R P,J_li slits LA

@sc1E}ectronlc

shuttercircuit l-l

system

I

F@re 2.-

r1000-cps

Oscillator

Schermticdiagramoftestequipmsnt.

~

Photot ube

rnpiif!er

7Thyratrcm

I

uPlwtotubeand

amplifier

PThyratron

t8-mega- :

cycle

k?Y!Kl

(Allpower.ol~e re~-latedat115volts,altenm~~ current.)

.

● .

Figure3.-P85316.1

Photographofmsafiu.r~stationinshock~ubes.

Phototube (931A) Amplifier (6AK5) Amplifier (2D21)

Push torwset switch RIO

6

I ::=

R5I l--’-c) output to

%5cmmter

NE-2

v

+250v

+900

~ Dynode+ Anode< Photocatho@

RI 22K R6 22K 2W Cl 300 ppfd

R2 22 K R7 470K l/2w Cz 300 ppfd

R3 22 K R8 15K 2W C3 0.031 ~fd

R4 100 K R9 1.2 megohm l/’2w c4 16 pfd 450v

R5 IOOK 1/2 W RIO 3.3 K 2 W C5 0,001 pfd 600 V

Figure4.-Circuit~agraaofphototieandan@ifiers.

. . .

, . .

1.96

Perfect fluid

me\~>m1.92

0- ----

z - Lir@ar theory, cf = 0.05131 R;l/5: 1.88 \

mL 1.

2u v

O Avemge from independent \: 1,84 — %velocity measurementsL -a o Avercqe from vekity meawre- 1 0x ments concurrent with ----

measurements: 1,80 —

pressure \

V Average of pressure pickup +

measurements

1.760 2 3 4 5 6 7 8 9 10 II 12

Distance from diaphragm, X, ft

(a) p6 .45 lbl.clti.gage.Experimental&& correctedtoPG/Pm- 4.M1.

Figure5..TheoreticalandexperimsntilshockpressureratioplottedagaingtdistanceI’romdiaphragm.

2.28Perfect fluid

2.24 \~8\

m \

a>

- 2.20 \ Q0,- /Linear theory, cf. 0.0581 R~l’50. /

~ 2.16 \

3UJ om J7.@ \ wL O Average from independentn 2.12 — velocity measurements \ ~

x O Averoge from velocity measure-V \ 0

0 ments concurrent with-co-l

pressure measurements o2.08 — ~ Averoge of pressure pickup IT

measurements

2.040 I 2 3 4 5 6 7 8 9 10 II 12

Distance from diophragm,

(b) p. = 70lb/sqin. gage. !&perimsntal

PE/Pw= 5.764.

Fiwe 5.-continued.

. , .

, .

!2

I

Oist once from diophrogm, x,ft

(c) P. = 95ti/aqin.~~e. ~rimental.PE/Pm= 7.455.

Figure5.-Continued.

datacorrectedb

60 NACATN3375

.3,60

Perfect fluld

3.56\

o Average from independent

3.52 velocity measurementsO Average from velocity rn8asure-

ments concurrent with pressure

3.48measurements

v Average of pressure pickup

3.44

@ o\

c? 3.40y .\- -Linear theory, cf =0.0581 ~-16; To= 520° R

0- \ -.-

? 3.36\ .

\ \@ h.z

k

: 3.3 z n \

: ‘1/5: T .542” R—+Linear theorv. C<= 0.0581 R~ — \\\I . 1 ..m I I

x,> \\6 3.285

3.24

3.20

\

3.160

3.1~ 2 3 4 5 6Distance from diaphragm, x, ft

(d) p~ = 250lb/sqin.gage.ExperimentaldatacorrectedtoP~/Pm= 17.915.

.

●

Figure5.-Concluded.

.

.

NACATN3375 61

:U 4a<

n— 0

Time —(a) pe/pm Z 4.061.

= c“ ,5co .—

— o

Time—(b) pe/pm= 5764

o

Time—

(c) pe /pm= 7.455.

Time_

(d) pE/pO=17.915.

. Figure6.-CO~arisOnofexperimentalandtheoreticalpressure-timecurves.~~ = 8.13feet;xtheOr= 8.ofeet. Circlesrepresenttheoreticalvalues.Thesine-wavetimebasehasa frequencyof1,000cps.

N

.5

.4

.3

./

,.2/

.1

/ ““

/

/

0,2 3 4

Perfect fluid

,--’&

/

5 (shock pressure

---- -/

/ -/

/ 7-----

,--

7 8 9 I

ratio, Palo/Pm

(a)S&mkpressure-lomcoefficient.

FigureT.-vartitionofshockpressure-lossandshockpressuxe-attenuationcoefficientswithperfect-fluidshockpressureratio.aC=&; n=T;

8’

)

.

Cf= o.058q-l/5.

.

t . ,

-,00

T—- .07. 2 L ——

\

?, In --- .\ .

2!..,{

f\

\

1

a- a“\

1, / ‘..0

a’n--.05

‘..\

;.2.cz0 -.04“

E,-E2 -.03.~

:s: -.02 -— — — — — — — — —?n

206 -.01

?

‘1 2 3 4 5 6 7 8 9 10Psrfect. fluld shock preswre ratio, ~OIPm

(b) $hockpressure-attemation coefficient.

Mgme 7.-Concluded.

44

40p /pm~o

,---’

3.6

~8\> 3.6”y

a 3.2 — —o 3.4’-’ _.--- Y.- L c — — — _ .0. 3.2

3.0y/ N — —_ . — —

.:

_.--- Y. 2.s.----*k 2.40= 24- “:0 2,2 ~ “

20

2.0

1.751.6

1.31-

1. 1?5$03 .004 .005 .006 .007 .008 .009 .010 .011 .012 .013

Time from diaphraqm burst, tv, sec

P

FQwre 8.-Theoreticalvariation ofx = 8 feet; ae = ~ = I,H7 feet

static pressure ratio with time. u:

per second; Cf = 0.058~-1/5.

r

.● ☛ .

.?0’.:0 1,3. p /pa

‘o.“ — 40—.—0

; 1,2 33:>

.2:m~ 1,100.a)>

a .W3 ,0

b.yB4— — — —

— =Lo-.—— — — — —1.75

I 151,25

4 ,005 006 .007 .008 .009 .010 .011 .012 cTime from dlophrogm burst )tv, sec

FiP 9.-Theoreticalvariation of average sonic-velocityratio

with time. x = 8 feet; ae = ~ = 1,u7 feet per second; Cf =

‘%/%0.058H&5.

Lo ~.f’ Pm —

3.6 N— —

~ —.9 — — — — — — — — — — — — — — — —3.4 ---

3.2=- ‘

~w 8 3C -\

3“ 2.8

0 .7.-0 2.6~ “.

.> 2.4.-:“

6

z>I

29

u.-2c

52.o -

.mQ.? .4a 1.75

31.5 -

Time from diaphragm burst, tv, sec

Figure10.- Theoretical variation of average particle-velocityratio I&/ae

with tw. x = 8 feet; aE = ~ = l,IJ_’7feet per second; Cf = 0.0~~-115.

, , . .

4 , + .

~. 1Pm

4,0~

3.6 /3,4 ~3.2Y

30-2.8

3 .004

.

---

/ ‘/

/

-----‘--- ----- / -

/ ‘~~ ~

---- / —/ ‘---- ~

/ A

-------/ ————_

— —

‘,4- — — —

2.2—2.o— ‘

1.75

1,5-

1.25-

.005 .006 ,007 .008 ,009 ,010 011 .012 .01

Time from diaphragm burst, tv, sec

F%ure 11.- Theoreticalvariationofaveragede~l~ ratiowith-t&.x G 8 feet; tie .

% = 1,1.17Teetper second; cf . 0. 0731#/5.

3

.9 —

a —

.7 ---

>:>

~ ,6 —

.2E2 .5 —z“

2

.4 —c.:.><,

3 —

2 —

.1 —.003

)ad ----~ “

.0

~ “3.6- ~

–34 // A

3i ~---- ————_

30- ‘

2.8 -

2.6-

24

2.2

1. – . – -.

2.0

1.75—

I,50

-1.25-

.004 .005 .006 007 .008 .009 .010 .011 012 .013

Time from diaphrogm burst, tv, sec

Figure 12. - T!mretical variation of average Mach number with %Inbs.

x = 8 feet; a~ = M . 1,11.7 feet pm second; q = O.O~l& +.

. .,

. .

Ic

c

#=6.5219 aa

b

Q’=5.0938ama’=1,1622a-u= .711 Clmp= 2,624 py

P’ = ‘6.5272am/

/

LP

@ Q’=5.0938ama’=1.1618amU= .71Eamp= 2.618Pm

A P’= 6,5451 am

Q’ ~ 5.0603 am,.~ a’. 1.1606am

1 I 1

-2w-

2 ‘4-6x, ft

P’= 6.3419am

{

p’=6.5172a- Q’= 5.0609a~, Q’c 5.09% aa a’= 1,1403 am

a’= 1.1617 am U = ,640 ~

U M .709 a= p = 2,298 k

p =2.617 PO

[

P’ = 6.W17 a.Q’ = 5,0887 an

a’ M 1.1590 am

U m .706 a=

\

[

U= .684 G“II=2422 Pm

@

1 1 I (

10 12 14 16

Figure13.- Characteristicplot for shock pressure ratio of 2.60 showinggenerationofvaveOIDregionp. s