o— FORAERONAUTICS ‘sn——n-~r. ; &—+=J~—. - …/67531/metadc56233/m...20 NACATN337’3...
Transcript of o— FORAERONAUTICS ‘sn——n-~r. ; &—+=J~—. - …/67531/metadc56233/m...20 NACATN337’3...
A’!.4CATV,.
NATIONALADVISORYCOMMITTEE ~ ~_
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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 [email protected],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
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