—— _&, ,.-

m3N CuPY:REIQ’BNm --—-.-~.y:.AR-WC@-woIIJ)

KmrIAmama 3.lKm NAc/4&

a ,

NATIONALADVISORYCOMMI’M’EE13’71a

c. I-—..

FORAERONAUTICS

TECHNICAL NOTE 3712

---- -.

BOUNDARY LAYER BEHIND SHOCK OR THIN EXPANSION

WAVE MOVING INTO STATIONARY FLUID

By HaroldMirels

Lem& FlightPropulsionLaboratoryCleveland,Ohio

Washington

my 1956

.

1---”...- . . . ,----- . ... . .. ..... . .... . .. .._—-.

https://ntrs.nasa.gov/search.jsp?R=19930084483 2018-03-26T14:28:59+00:00Z

ItGH LIdRARY KAFB,NM

NACATN3712

,,

$

suMMARY. . . . .

JIW!RODUCTION. .

ANALYSIS. . . .

TABLEOFCOI&N17S0=

. . . . . .

. . . . . .

. . . . . .CoordinateSystems. . . .LsminarBoundaryLayer. .Integral.solution. . . .Jhterpolationformulas.W~ surfacetemperature

TurbulentBoundaryLqjer.Boundary-lqwrsolution.Wallsurfacetemperature

CORRECTION@’DAM.....

CONCLUDD!W3ImmIUw . . . . .

APPlmDlmSA-

B-

c-

D-

SYMBOLS. ...*...

WAVEJ3ELATIONS. . . .

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LAMINAR-BomY-IAYERSOUJTIONImmENcm ToTmmmwmm . . . . . . . . . . . . . . . .

EVALUATIONOF ~ . . . . . . . . . . . . . .

ImFEREmEs.. o........ . . . . . . . . .

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1

1

22391215191924

26

29

30

33

36

38

40

..-.- .. . . . ..— ——- ——,

. NATIONALADVDORYCOMMITIZEF(R

TIZCllNICALNOl!lZ3712

,.

*

BOUNRARYLINERBEHINDSHOCKORTHINEXPANSION

WAVE

Theboundary~erintoa stationaryfluid

MOVINGINTOSTATIONARYFLUID

ByHaroldMirels

sUlmmY

behinda shockorthineq%nsionwaveadvancinghasbeendetermined.Lminarandturbulentbound-

aryl~erswere&nsidered.Thewdl surfacetemperaturebeldndthewavewasalsoinvestigated.Theassumptionofa thinqymsionwaveisvalidforweakexpansionsbutbecomesprogressivelylessaccurateforstrong~ion waves. .

Thelam3nar-boundary-lsyerproblemwassolvedbynumericalintegra-tionexcet fortheweakwavecase,whichcanbesolvedanalytically.

7ExtegralK&rm&n—Pohlhausentype)solutionswerealsoobtainedtoprovidea guidefordetemdningeqmessionswhichaccuratelyrepresentthenumer-icaldata.Analyticaleqressionsforvariousboundary-layerparametersarepresentedwhichWee withthenumericalintegrationswithti1 percent.

Theturbulent-boundary-~erproblemwassolvedusinginte~almeth-odssimilartothoseemployedforthesolutionofturbulentcompressibleflowuvera semi-~initeflatplate.Thefluidvelocity,relativetothewall,wasassumedtohavea seventh-powerprofile.TheBlasiusegya-tion,relat~ turbulentskinfrictionandboundary-werthiclmess$wasutilizedina formwhichaccountedforcompressibility.

ConsiderationoftheheattransfertothewallpermittedthewaJJsurfacetemperature,belxlndthewave,tobedetermined.Thewallthick-nesswasassumedtobegreaterthanthewalltkrmal-boundary-lsyerthiclmess.Itwasfoundthatthewalltemperaturewasuniform(asafunctionofdistancebehindthewave)forthelaminar-boundary-~rcasebutvariedwithdistancefortheturbulent-boundary-layercase.

INTROZ?JCTION

Ifa shockoreqansionwaveadvancesintoa stationaryfluidboundedbya wall,aboundary-layerflowisestablishedalongthewallbeklndthewave.Thisboundary-layerflOWtioftm ~ort~t ~ s-es ofPh~~

\

.-—. —.—- .. ——. —

2 NACATN3712

involvingnonstationarywaves{e.g.,shocktribestudies,initiatimofdetonationstudies,etc.).

M references1 to3,thelaminarboundarylayerbehinda shockwaveisstudied.!Chepurposeofthispaperistocuclxmdreference1 toincludethecaseoftlxinexpansionwaves{i.e.,eqymsionwavesofzerothickness)andthecaseofturbulentboundarylayers.Thisextensionwasmotivatedprimarilybya desiretoprovideboundary-layerinform-ationforstudiesofshockattenuationina shocktube(e.g.,ref.4). CNCDulco

Theassumptionofa zero-thicknessexpansionwave{referredtoaaa “negativeshock”inref.4)becomesprogressivelylessaccurateasthestrength[pressureratio)oftheeqansionwaveincreases.Thus,theapplicabilityoftheboundary-layerresultsinshockttieswhereinstrong~im OCCI-Umaybe questioned.Nevertheless,theassumptionofazero-thicknessexpansionwavewillberetainedforrangesoftheparsm-eter~ue {symbolsdefinedinappendixA) correspmdingtostrongex-pansionwaves.Thereasonforre-~ theassumptionistmfold.Ftist2a qualitativeestimateoftheboundary@er intheconstsnt-pressureregionbehindstrongexpansionwavesisobtained.IXmaybe %possibletomoMfytheseresultstogivebetteragreementwithmoreac-curateesthetesoftheboundaryleyer behinda strongexpansionwave. .Second,theresultsaredirectlyapplicabletootherphysicalproblems

.

suchas{1)theboundarylayerona semi-inftiiteflatplatewithtan-gentialblowingand(2)theboundary-r behinda planarflamefrontpropagatmina stationarycmimstiblemixture.

An independentinvestigationoftheboundarylayerbehindanex-pansionwave,tsk3ngintoaccounttheftiitethicknessofthee.qmnsion

urrentl.yinprogressattheNACALangleylaboratory.wave,isc

A shockorqansionwaveisconsideredtomovewithconstautveloc-ity}paralleltoa wall.,titoa stationaryfluid{fig.1). TheboundarylsyeralongtheWSJ3.behindthewaveistobedetermined.Jnthecaseofaneqansionwave,finitethiclmessofthewavew3JJ-be ignored.BothLm5narandturbulentboundarylqpxswill.beconsidered.

CoordinateSystems

IA %,7 be a comdjnatesystemfixedwithrespecttothewallandlet ii,~bevelocitiesparallel.to =,$ asindicatedinfigurel(a).Theflowisunsteadyinthe =j~-coordinatesystem.Let x,y representa coordinatesystemmoi3ngwiththewave(fig.l(b)).Thevelocitiesparalleltothex-andy-coordinatesqredenotedby u and v,respec- titively. b thiscoord=tesyst=,theflowW ste*O

.-. . -—

NACATN3712.

3.

,.

.

_-Assumethat attimet = O thetwocoorMnateIf UA isthevelocityofthewaverelativetothe

ux arerel.atedbyx == -fidt.!J!hesxial.u=;+ Notethatthewallmoveswithx)y-coordimtesystm. Thetransformatimslsobeqyressedas

1

~=x-’%ty.y

systemscoin~ide.wall,thenx and

velocitiesarerelatedbyvelocity~ = -;d illtherelat~ thetwosystemC=

(1)

EquationsdeftiingconditionsacrosstheshockandeqansionwavesaresummarizedinappendixB.

LaminarBoundaryLsyer

ThePrsndtlboundary-layerequationsapplyfortheflowb thevi-cinityofthewall,exceptatthebaseofthewavewherethebuundary-layerassumptionsbreakdown.Byassuminglaminarflowand d@x = 0,theboundary--regpationsfor x >0 are

(2a)

(i?b)

P=@ @d}

Theboundaryconditionsfor x >0 are

U(x,o)= ~ u(x,=)= ue

V(x,o)= o 1T(x,+ = Te

J

[3)

. . ........_—.—. — .— — ——-—. .

4 . NACATN3712*

Theseeqpationscanbetransformedtoa formsuitablefornumericalin-tegration.Tmaccordancewithreference5,itwillbeassumedthatpsnd k areproportionalto T andthat ~ aud a we tidependentof T. Thesestatepropertiesw5Jlbedefinedtohavethecorrectnumer-icalvaluesatthewall.Theuseofa mesnreferenceteqerature,ratherthana wsJJreferencetemperature,isdiscussedinappendixC.

Frcmequation(2a],a streamfunction~ existssothat

p=Pw $

~=-f 1

CAcoUICD

(4). .Pw-

Deftiea s~ity parameter

andassumethat~ canbeexpressedintheform

* = -W f(q)

Thevelocitiesarethen

Assumethat~ snd k areproportionaltoofpropotiionalitysothat p and k have

Tthe

??)}

(5)

(6)

(7)

andchoosetheconstmtscorrectnumericalvalues

w

.

Q

.- . .. - .- —- —.

NACATN3712 5

atthewall;sidlarly,assumethat cP and a areconstantattheir

wallvalues. Thus,

k=

CP=

Una w JThemomentumequationcannowbewritten

1ft?f+ ff”= ()

f(o)= o

JJ

f’(o)= u Ue

f’(=)=1

andtheenergyequationbeccmes

T(x,O)-‘%T= T=

JSinceequations(10)areltiear,T canbeexpressedasthelinearsuper-positionofthesolutionforzeroheattransferplustheeffect”ofheattransfer.Thatis,

(U)

.— — —... —.- —. —..

s

6

where

-2awP’=%r“+ a z(f“)2

‘()}—- 1ue

J!IAC.ATN3712 .,

>

and

Equations(13)and(14)

r(ao)= rt(o) x c)J.

}

‘“+‘W@’=0“[o] = 1

s(a)= o

can also beexpressedtnquadraturefore:

{13)

.

(14)

Wkrest IIW nowbeexpressedaaBoundary-layerparametersof

.W= (Il$)w.uefmq’ (Ma)

%’ ()‘k% r= -S1(O}{TW-Tr) =* ~ (Mb)

w w

.

#,

—. . .-. .—_ -—. . -_—— . . —. .. .-. .______

NACATN3712

(H’)f”sdilo

6*=r‘e ‘e—dxuzp‘e

oe

{16d)

whereq equalstheheattransfe.inthe+y-direction.Equation[16.)resultsfromtheexpressionv&e = db*/dx,whichcm bededucedf.OIUthecentinuityequationsndthedefini.timof 5*.

Equations{9)to{15)canbe titegratedsnalyticaldyfor

l’+.$”e-11<<1 (i.e.,weakwaves).Theintegrationswereconductedinreference1 andtheresultsaresumma.tiedasfol.lows:

. . ... ..— ..-. — .—

“

[“sd,=~+o(+)

NMIATN3712

(17f)

Equations{15)aretitegniblefOr Cw = 1. - 3X3Sd.tSShOW

2

()

%—-f’r = 1- %

%1—-‘.

m

J[()25!-1 (f-q]-1-f“(o)‘erdq=ldm

7-

()

2%1—-‘.

0

..=%1—-‘.

J*sdq’’E(t

{17i)

(Ma)

(18(I)

Equations(9)havebeenintegrat~nmeric~ (~tiganIBMcard-progrsmnedelectroniccalculator)for 1+#~ = 1.5,.2, 3,4,5,and6,andtheresultssrepresentedinreference1. Therange16 u&e&6

.

wco(nco

-. ..—. . ..—- --—— —-- --——-

NACJlTN3712

coverstheentirerangefromveryweaktoverystrongshockwaves.Equations(15)arealsoevaluatedinreference1 for Uw= 0.72 and%/% = 2,4,and6. Jhordertoextendthenumericalcalculatimstothecaseofa thin~ansionwave,equations-(9)have‘beenintegratedfor~/ue=S0,0.25,0.50, and0.75[table1). ThecaseU#Ie = O isthewell-knownBlaaiwproblemforflowpasta semi-infiniteflatplateaudisincludedhereinforthesakeofcompleteness.Equations{15]wereintegratedfor ~~ = O,0.25,0.50,0.75and Uw= 0.72(tableI).135xxuappentiB,thelimitingvalueof u#e forstrongeqymsionwavesis fi=y-l Thesecalculationsweremadeustigtheintegrationtech-

‘e y+l”nigyedescribediuappendtiB ofreference6. Theresultsofreference1for ~~~ = 2,4,and6 arealsoincludedintale I.

titegralsolution.-Thenumericalintegrationsofequations(9)and{15)havetheUnitatimthatonlyspecificvaluesof u#~ and

‘W csnbetreated.Theanalyticdependenceoftheboundary-l~erpa-rmeterson uJ~ and IJwarenut-licitlydefinedbythesespecificSolutions.tiordertoobtainapproxhnateanalyticsolutions,aninte-gral{K&& -PohlhaMen)meth~ofapproachwillnowbeused.There-sultswillbeusedlaterinthissectiontoobtatianalyticexpressionswhichaccuratelythisreport.

titegrating

Let y=~view.The~= q/l&nomialill

gives

representthenumericaldatapresentedintableI of

eqy.ations(9)withrespectto q,intherangetheWegralformofthemomentumequation

J’mf“(o)= f’(1-f’}d~o

betheedgeoftheboundarylayerfromacorrespondingvalueof q isdesignatedThevelocityf~ isnowrepresentedbya~ intherange06 ~< l.l Usingthetwo

{19)

Pohlhausenpointofng” Definefourth-degreepoly-

tiom on f‘ notedineqpatjions(9)plusthebounm

‘f‘“)&O = O,inordertosatisfyequations(9),and

%eferences3 and7 alsousedintegralapproaches

boundaryCOJdi-conditions

f51@o@l$fourth-degreepolynomialsinstudiesofthelsminarboundarylayerbehindmovingshocks. Inreference8,a normalizederrorfunctionwasusediusteadofthefourth-degreepolynomial.Thisresultedinverygoodagreementwiththeexactnumericalintegrationofthecorrespondingboundary-xewtions.

,

. -- .-— —— —.. . -.———--

.

10 NACATN3712 .

(f’’)g=l= (f‘“)g~ = O,inordertogeta smoothjoin= withtheex-=ternalflow,gives

for 0< ~<1,

( )(%%T12Cf’.—-)

- 2C3+ C4‘e ‘e

for 1< ~,

f’=1}

Equation(19)canbewrittenaa

J’1f“(o)= llg f’[1-f’]d~ .0

Stistitutingequations(20)intoequation(21),noting

u-

11315ng=z74+1155

‘e

(20) 04ECD

@l)

(22)

which,

titiestogetherwithequations(20),deftiesf‘. Thefollowingquan-areofinterest:.

(23a)

.

Equations[23)arec7

ared,intableII,withtheexactintegrationofequations(9)for % = (),1,and6. Itisseenthateqpation(23a) .

-..

—. . —.. . . — -.. -—— .— -..

NACATN3712r u

agreeswithin4 percentwiththeexactintegration.Equation[23b)agreeswithin2percentat ~ue = O andwithin10percentat1+.$s= 6. TableIIservestoindicatetheaccuracyoftheintegralmethodofSoluticmofthemomentumequation.

Stistitutionofequations(20)and(22)fitoequaticms(15)gives

.

where

.

@4a)

(24b)

(24C)

@d)

(24e)

(25a)

(25b)

(25c)

(25d)

(25e)

[25f}

.— .—— - ——

12 NACATN3712

Equations(24)definethefunctionaldependence,on Ow)~ ‘d %/”e~ofthev=iousquantitiesnotedtowithintheaccuracyofthetitegralmethod. Equations(25)canbe integratedanalyticallyfor Uw= 1. ForotherPrandtlnumbers,numeric~titegratim~ oramro~te aic~integrationswouldberequired.

Equations[24c)to (Me)indicatethatthefunctionaldependenceof

J“rd’J J“s “) ad “(0’m “w m ‘i&pm*nt ‘f ‘iUeocJ

arimnofequation(24d)withtheexactsolutionfor s‘(0)at‘e= 1 (i.e.,eq.(17h))indicatesthatK(aw)shouldbeproportional.

1’2 Howevercomparisonofequation(24d)withtheexactsolutimto Uw .tionfor u#ue= O‘(ref.5)indicatesthatK(@ shouldbeproportional

●

wcoVlCD

113for Uwto aw intherange0.6< aw<1. ThediscrepancyisduetotoImitationsoftheintegralmethd. Henceeqmtions(24)sad(25)wouldhavetobemodified3fgreateraccuracyisdesired.Possiblemod- .ificationsareindicatedinthenextsection.Comparisonofeqpa%ions(24c)end(24d)withequations(17f)and(17i)indicatesthat,for ~/ue .near1,1(@ isa constattifleJ(cW)variesd~ctlY ~ l/@w.

hterpolationfommilas.-Equations(23)ad (24)canbeusedasaguidetoobtainam.1.yticexpressionswhichaccuratelyrepresentthedataobtainedfromtheexactnumericalintegratimsofequations(9)and[15).Thevariousboundary-layerpammetersareassumedtod~nd on u#eand Uw asfOllows:

(26a)

(26c)

*

.

_. —-. ..—— .- ——————.———-- -. --— .—-. —-———-——

13

J00

s d?

o

s’(o) = C9

C7

r{O)=

Thecaseofanexpansionwave

(26e)

(26f)

anda shckwavearedistinguishedby0< ~j~~ 1 ‘d

3writ separateformulasfortheranges1< Ue< 6. Theresultsare,

(27a)

1.217

%1.326—%

[

w

‘d’=11%’”W:”2[

m

sdq=

Jo

.

1o217

m{””’-0”36”n)

(27c)

(27d)

- ---- —... — .—

%1+ l.665—‘e

=-F=J0““’=F=’”W:”MI0“Sd’=i-(ow

NACATN3712 ,:

.

S1{o)= -0.489

F=’”w:”4m”022(tiue’0.39-0.023(u#e]

do) = bw)

(27a’)

{“b‘)

(27c‘)

.

(“d’) r.

[27e’)

{“f‘)

Equations(27]wereobtainedbym&cMng equations26) attheendpointsoftheintervalsO < (U#Ue)<1 and [1~ ~~~) < 6,withtheexactintegrationsofecyzations{9)and(15)(table1). Equations

[127 agreewithtableI towithin1 percent.Theaccuracyofe ations

r)27 canbeestimatedfromtableIII,whichcomparesequations27withtableI fortheintermediatepointsu#e = 0.25,0.50,0.75,2,and4.

Eqyations[27]me basedm numericalsolutionscorrespondingtoaws 1 ad aws ..72.Theyshouldgivereasomibleesttitesforothervaluesof Uw ofthisorder(i.e.,0.6< crw<1). An exacta~rtisalofequations(27)for Uw other.than1 and0.72requiresfurtherfives-tigationoftheexactintegrationsofequations(9]and(15).Suchautivestigationisconsideredbeyondthescopeofthisreport.

P.

—. -. —. .— .—— .-. .— —

I

.

.

NACATN 3712 15

Wallsurfacetemperatie.-Untilnowithasbeenassumedthatthewallsurfacet~eratureisconstsmtbehindtheshockor~im wave.Thiscanbeshowntobetruefora l.smhaxboundarylayer.Themagni-tudeofthewalltemperaturewillnowbedetermined,assumingthatthewallthicknessisfgeaterthenthewallthermalbouddarylayer.2

Inthesteadyx,y-coordinatesystem,thewallhasa uniformveloc-ity ~. However,thereisa themalboundary@yer inthewallwhichiss~ tothatintheadjacentfluid{fig.2]. Assumethatbeforetheappearanceofthewave,thefluidandthewall.wereinthermalea@.-libriumatthet~eratureTb.

Whena -tity isdiscontinuousacrossy = 0$thesmscri~s wand O- willbeusedtoindicatequantitiesevaluatedat y = @ endy a 0-, respectively.Neglectthevariationofstateproperties(suchas k,p,etc.) intheWSJJ.ad mite the~fusivitYofthe~ a

()k

so-=x “ Byusingboundary-werassumptions,theenergyequation\ ~/ ()-

forthewallbecomes

a= ao- ii%.X ~$ (28a)

for x >0,

Equation(28a)canbeobtainedfromequation(2c)byassmd.n$thewalltobe equivalenttoa fluidwith p = v = O. Assume(tobeverfiiedlaterh thissection)thatTw isconstmt.Thesolutionofequations(28)iSthen

%he problemofthewallsurfacetemperatureassociatedwiththeladnarboundarylayerbeh~ amov~ shockWM SOIVe%fi@?efi~tW)h reference8. Thedevelopmenth reference8 issomewhatmorede-tailedthanthatgivenherein.Theresultsareessentiallyh agree-mentwiththisreport.

-.. —__ —.. _ _-

16

Theheattransferh the+y-directionat y = 0- is

%-“()‘k?$o-

Frm equation(Mb],theheattransferredintothefluidis

% = -s‘(0)(TW-Tr)~’(g)

NACATN 3712 ,,

.

Equations{30)and(31)~ beeqmted,yield3ng

~ -~ =A [(Tr/~)-1]Tb l+A

where- —-

~= -s’{0) %2 %-%——TL%#o-(–(v2-uYcw

Equatim {32a)issolvedfor’Tw byfunctionsof ~). ThechoiceTw=god firstapproximation.Notethat

iteration{since

(30)

(31)

(32a)

[32b)

A and Tr areiteration,isa

~ isa constant,whichiscon-

.

sistentwithand,therefore,substa.utiatestheoriginalassumptiontothis”effect.

TheratioTr/Tbcanbeexpressedasa flmcticaof ~/~ andr(0)byuseofequatim(12)andequatioxis(B3),(lM),(B7),and(B8)ofappendtiB. Theresultsare(assumingCp,e = Cp,w)

.

. . . ..-— .—. .— —..—. .——-

NACATN3712

for u@e >1,

Tr ()%—-1. ~-1Tb e

u)%to

for 4 Ue< 1,.

1.

Tr—-1.Tb

=

a

2

(>)[-1 y+l%

2uue y-l~-

17.

(338)

1=-3 (~ )-1 forr= 1.4;r(0)= 1

)2 ( )]%21 +~r(0) —-1 -

r %1 (33b)

\

Itshouldbenotedthatfor Tr/!%>1 theheattmmfer isfromthefluidtothewall,whereaafor T~~ < 1 theheattransferisfrmthewalltothefluid.Thus,fortheflowbehinda shockwave(eq.(338))theheattransferisalwaysfrm thefluidtothewall.However,fortheflowbelxlndanexpansionwawe(eq.(33b))theheattransferisfromthewaU tothefluidforrelativelyweakexpansionwavesandfromthefluidtothewallforstrong_sion waves.WhenT = 1.4 and r(0)= 1,thechangeinthedirectionofheattransferoccursat pe/~ = (2/3)7.Thischangeh heat-transferdirectionisa consequenceofthefactthatthestagnationtemperature(relativetothemill)behinda tti cqansion

&y-l

()waveislessthanZ& for p~~ > ~ andisgreaterthanTb

2r

()3- ~ r-lafor P~~C ~+1 Theboundary-layerrecoverytemperatureTr

. .. —--- ___ ——. .

18

egyalsthefree-stresmstaguatimtemperature(relativefor r(0)= 1 andissome-t lessthanthestationr(0)< 1. Ifthefinitethicknessofthewaveistaken

NA.CATM3712

tothewall)temperatureforintoaccount

f&-a verystrong~ansionwave,itis~ected thattheheattransferwillbefrm thewa2J.tothefluidfortheearlystagesofthe~amionandfromthefluidtothewallforthefinalstagesoftheexpansim.

Theorderofma~itudeof(~ -TJ/Tb will. nowbetiteminedo

Equations(17h),(27e),and(27e’) indicatev-r

‘e-s’(0)~ ~= 1“Therefore,A canbeapprciximatedas

EFkisk~~a istobe&intmm.yreference

rao- %‘w-c (34)

proportionalto T and ~ isconstant,thenthevalueofhdependentoftemperaturefora givengas.Assumethefluidandthewalltobecasttim {carbon,approx.4 percent)bothata temperatureTb= 530°R. Us@ thepropertyvaluesof9 evaluatedat530°R givesA ~ 0.0004.Stistitut~equa-

tions(33)intoequation(32a)yields,withA = 0.0004,r(0)= 1,andy = 1.4,

for u#e > 1,

~-Tb $()-1

%z().0004— 3 (35a)

TJfor u Uee 1,

ThewaXlwillremaintitlxlnO.1percentofitsoriginaltemperatureforshocksUp to K ~ 3 md forECU~ion waves.Thewallwillremainwithin1 percentofitsoriginalvalueforshocksw to B$~ 9 andwithin10p~ent ofitsoriginal.valuefor 1%~ 27. Similarresultswouldbeobtainedif&hergasesandwallmet& sreused.Theimpor-tantparameter1$.& issmallbecauseoftherelativelylowheatcon-ductivityofgasesccmparedwithmetals.Thefactthatthewallr~essentiallyatitsoriginaltemperature(exceptpossiblyforextremelystrongshockwaves) justifiesthedisregsrdofthevsriationofwallstatepropertieswithtemperaturesinequation(28a).

.

.

.

.

.

-. .—— — . ..— —.——... .— - .. . . .—— ____

NMA TN3712

Thethicknessofbefound.Definethe

19

thewadlthermal.boundaryQmr ~ canreadilyedgeofthethermalboundarylsYertocorrespond

Itisofinteresttoc~ 4 withthethermal-boq-xer thick-m nessh thefluid~. Forconvenience,considerthe-caseofweakwaves.U)Oam Equations(n) and(17$),togetherwiththepreviouslydefinedcriterion

fortheedgeofa thermalbouudarylayer,give

(37)

whichisoftheorderof1 forsaneatiandmetalwaJlconibinations.

Thesolutionforthewallsurfacetemperaturepresentedhereinisvalidprovidedthewadlthicknessis(atleast)greaterthan~.

TurbulentBoun&myLsyer

Previously,itwasassumedthatthebmmiarylayerbehindthewavema laminar.Nowtheturbulent-boundary-layercasew513.betreatedus3ngintegralmethods.@ solutionswill.beobtainedbyextendingempirical,semi-~fiiteflat-plate,boundary-m datatothecasewherethewa31ismoving.

Boundary-layersolution.-Theintegralformofthemomentumequa-tionis

de=—dx

whereu/ue representstheawerageboundsry-@yerthiCkIleSSby ~ =d

(39]

velocity.Asbefore,denotethedefinea similarityparameter

—.—— ____ .—

20

t+ = y~ii Itisnecesssrytoexpressu/~ asa fuuctionstudiesoftheturbulentboundarylayerona semi-ini?tiite

NACATN3712

of ~. kflatplate,

itiscustomarytoassumeu/ue= #7 for OS~<l. Theuseoftheseventh-powerlawsmearstoagreewithavailabledata,evenforhighMachnmiberflom$andtoresultinreasonablyaccuraterelationsforskinfriction(eg.,refs.10and11). Inordertoextendthisproffietothecaseofamovimwa12,itisassumedthatthefluidvelocitiesrelative tothemillh&e a

for O<~T~l>

~eventh-powerprofile.Thatis,

Itisslsonecess~toexpressT/Te(~P~P) ss a tictim d CT.TheformassumedhereinW

(40)

for O<!T<l)

whereTr

b=%-l

()Tr Tec= q-lq

(A%CD

(41)

Fora Prandtlnumberof1,thedependenceof T/Te on u/ue h eqyations

(41)isthesameasthatindicatedinequation{12).WhenthePrandtlnumberisnottoofarfrcm1,equations(41)shouldgiveatleastareasomibleest-te for T/Te.

.

.

—.. .—— .-——--— -. _..—

.

mInmto

.

.

NACA

ness

TN3712

BYusing equatimz[40]ad (Q.),anddisplacementthicknesscanbe

21

theboundary-@yermomentumthick-Qressedas

where16)%J and 18titegral

withN = 6,7,or8.appendixD. Itshould

(42b)

arefunctionsof b and c def~d bythe

1d’ zNdza (43)

o l+ bz-cz2

A methodforevehating~ isindicatedinbenotedthattheiutegrsls~ end(~ - In)

sretabulatedinreference U for the range -l<b~”lo ad - -o~b;~—<1. TheresultswerecomputedusingSimpson’srule.There-sultsofreference11canbeusedtotabulatevaluesof 18 land16.Thehrtegral16 isfoundfmm therelation16= ~ -b% + C18)whichfo~owsfromequation(43).m reciprocalsof 16)%, ~d 18,b~ed “onthedataofreference11,arepresentedh tableIV. Reciprocalssreusedto ermitMmearinterpolation.Whenb isnem -1 or

Tc/{b+ 1 isnear1,thelinearinterpolationbecomesinaccuratesnduseof

be

b

a~enti D isrecommended.

Stice8/~ isindependentofconstant),equation(39)canbe

x {assumingthewald.temperaturetowritten

(44)

ordertointegrateequation(44)Zitisnecessarytohavea relation..betweenTV and ~. Forinc~ressibleturbulentflowpasta semi-infinitefiatplate,theBlasi&relation(ref.

~e 1~4=W

()— = 0.0225—peu: ue~

12)is

[45)

—.. —

22 NACATN3712

.Thisformulacanbeextendedtoa~oximatethecompressibleflowovera smi-infiniteplatebyevaluat~thefluidprop&rtiesata suitable .meantemperatureTm {refs.10,11,and13)giv@

or

where

1~4=W

()

‘m= 0.0225~pmu: ue~

lp‘w ()‘e— = 0.0225Q .-peu: Ueb

Franreference13,a reasonableestimatefor Tm is

Tm= 0.5{TW+ Te)-I-0.22(Tr-Te)

(46a)

(46b)

(47)

Whenthewallismom!.ng,thelogicalextensionofequation(46b)istousevelocitiesrelative

Tw

pe(ue-Uw)z

to thewall..Thus,

() 1/4‘e= 0.0225Q

Ille-a#

or

Equations(48)are~ressed,asindicatedtoensurethat%W hasthesameSigrlas Ue -~, whichistobeexpectedfromphysicalconsidera-tions. Substitutionofequation[4Sb) intoequation(44)permitsthelattertobe integrated,for Tw ‘constant,giving - --

(49)

-.-.—-. . . .______ ____ --

.

10)‘U-Jto

.

.

NACATN3712

Otherboundary-layerparametersofinterestare

23

+=o.w60:~l-,&y,l-uJue,3/5(&j’5(50.,peue

*‘e_ db‘e F

* ()1 TJ-Uu 4f5= 0.0460~ Q e

5 e[~

TheheattransferatthewallcanbeestimatedusingaReynoldsanalogy. Forcompressibleflowovera semi-infiniteflatplate,theReynoldsanalogymsybeexpressedas (ref.13)

Whenthewallmoves,anestimateoftheheattransferfrom

c M(TW-Tr)o-m-2~3%“p’ue-~ ‘w

[!IL)

mightbe obtained

(52)

ItispointedoutintheCORRELATIONOFlZ!lTAsectionthattheexponentof am shouldprobablydecreaseh maguitudewithticreasing#ue.

TherecuverytemperatureTr isyettobedetermined.~ snalogywiththeturbulentflowovera semi-infiniteflatplate,Tr msybeestimatedas(ref.13)

where

(53b)

Equation(53a)issolvedbyiterationsincec and r{O)arefunc-tionsof Tr. FortheMminsr-boun@-layer~~e, r(0)increasedwith

.— ---- —. . .. —

24

snincreaseh U.#Ie.Thisestinater(0)atthehigher

W TN3712

suggeststhatequation(53b)mightunder-valuesof U&e. However,equation{53b)

isprobablywell.withintheaccuracyoftheotherassumptionsreq@?edtosolvetheturbulent-boundary-layerequations.

Wallsurfacetemperature.-Forconvenience,itwasassumedduringthecourseoftheturbulent-boundary-hyersolutionthatthewallsur-facetemperaturewasconstsntwith x.-The

be i&estigatedbysolvingforthewsdllsminarboun&my@er .

validityofthisassumptiontemperatureaswasdonefor uwula

Eqyatingequations(52)and{30),plus

~ -Tb B[(T~~)

equation(46b),gives

-1]..%-= 1 +-B

()‘B & ‘0(B2)(54) .

where

IhJ3iethe laminar-boundary-l.s#ercase,B varieswithx. Therefore,T#~ varieswithx,whichcontradictstheoriginalassumptionsunderwhichequations(54)werederived.However,equatioqs(54)cambeusedtoestablishtheconditionsumderwhich~ isnearlyequalto Tb. Theproceduretofollowisto evaluatetherightsideofequations(54)assuming~ = ~. (Eqs.(33)areapplicable.) Iftheresultingvaluefor(~ -Tb)/~ M _ (e”g*>less~ 0=1‘orW ‘iWs ‘f xunderconsideration),thentheass-ion ofa constantwalltemperatureisreasonablyaccurateandthesolutionindicatedbyequations(54)isvalid.

Theconditionsunderwhichthewalltemperaturebehinda strongshockwave(~ >>1, ~ue ~ 6)remainsessentiallyatitsoriginalvaluewZllnowbedetermined.Assumep isproportimalto T,Tw

~ isconstant,a = 1,y = 1.4,=Tb, ti ~=1/[(1 +b-c) (N-t-1)].

Thelatteristheleadingterminthe~ansionfor ~ whenequations.

(D3]and{D4)ofa~endixD areused.ThefolhwingrehtionsCanthenbeobtained: .

—. .— —— . . . . .. — —

NACATN3712 25

.

-fEic)

.

.

~ 351~2()

-—23 for (u#e) = 6

EvaluatingB for ~ue = 6 andsubstitutingintoequations(54)yield,for $>> 1,

Assumethef&idtobeairandthewalltobecastiron,bothinitiallyat ~=5w13* Itisconsistentwiththeassumptionsof pa

usingthepropertyvaluesofreference7 for ~ = ~ = 530°R.tion(56)becomes

Kc,cr=l,

OOOU04,

Eqw-

(57)

Thus,when~ islarge,thewdl surfacetemperaturewillremainwithin10percentofitsoriginalvaluefor (xu&e)3~10#<0(104).

‘!l?heprecedingsolutimrequiresthatthewallthicknessbeatleastgreaterthanthewallthermal-boundary-layerthiclmess(eq.(36)).

..—.--— .—.— — —

26 .

- CORRELATION(X?lM!M

Ih ordertocorrelatethebehinda wave,itisnecessaryacterizesthefluw. -

Considerthe t,~-dia@m

Wave

NACATN3712

.

data pertainingtotheboundarylayertodefjneaReynoldsnudberwhichchar-

fortheflowbehinda wave(sketch(a)).particlepath

: (externalflow)

=xSketch(a) ,

Anexpsnsimwaveisindicatedh sketch(a)forconvenience.Whenaparticleoftheexkrnalstreamhasmoved‘arelativetothewall,itisa distancex =behindthewave. Thus,5= X(l-u#le)isspendingtoa givenx)whichcharacterizesment. similarly,(~ -Ue)theveloci@oftothewdl isthecharacteristicvelocity.thenbedefjnedas Re= ~(ue-u#w or

a physicaldimension(corre-theboundary-layerdevelop-theexternalflowrelativeTheReynoldsnunibermsy

Re= (UeX/Vw)(l-U#Ue)2 (58)

wherev isarbitrarilyreferencedtothewalltemperature.Eqyation(58)reducesto Re= {uex/vw)for ~ = O. Eqmtion(58)differsby

J%thefactoru fromtheReynoldsnmiberdefinedby equations(32)ofreference1. For ~ue = 1,theReynoldsnumberofreference1 agreeswithequation(58),butforothervaluesof ~ue, thetwoReynoldsm.mibersdiffer.M particular,theReynoldsnuuiberofreference1be-comeshftiiteas ~ + O. Hence,equatian(58)appearstobeamoreusefulfo?ml. I

Thecorrelationofthelaminar-boundary-~erskin-frictionandheat-transferdataWJ1.nowbe indicated.Thelocalskin-frictioncoef-ficientmsybedeftiedu

‘wCf=

[2+Pw~- Ue)(59)

~mco

.

.,

.—— —________ ____ ____ _____ ....-. .. ___ _____ .- ___ -_ -.

NACATN3712

Therefore,usingequations(16a),(58),md (59),

f.@ . amw-Id‘e-1 I

27

wherecf-@ ispositiveifthewallshearisinthe-f-x-direction.Equation(60)ispositivefor O< ~~< 1 andnegativeforu#e> 1. Themagnitudeof cf-@””at u#e = O to1.128at ~ue = 1equations{16)and{27),theReynolds

for 0< 4 Ue< 1>

&creasesmonotonicdd.yfrom0.664and2.291at ~ Ue = 6. Ustiganalogyc= bewrittenas,

- Ue (Uw)0.65-0.15{U#Ie)q~\.

‘r%,wF~— = 1Ww

for 1< 4 Ue< 6,

%%-ue~Tw-Tr cp,w

{61a)

{61b)

Theexponentof uw isseentodecreasefroma valueof0.65at

4 u =0 toavslueofO.39at U#e= 6.eByusingequticm(50a),togetherwiththedefinitionsof Re end

cf asgiv~byeq@i~ (58)~d (59)>theturbtie.ntskinfrictioncanbeexpressedas

TheReynoldsanalogypraposedfortheturbulentcaseisgivenbyequa-tion(52),whichmqfberewrittenas

#

[63)

... .— . —__ ______

28 NACATN3712

Comparisonofequations(61)and(63)suggests(exceptfortheuseofameanreferencetemperatureineq.(63))thatthe~cment of rJmshould .

decreaaewithincreashg~ue. However,theuseoftheexponent2/3isprobablywithintheaccuracyoftheotherassumptionsinvolvedh theturbulent-boundary-layerSIl&&SiS.

Inreference4,theboundarylayerbehindawavewasestimatedbyusingtheconceptof“auequivalentsemi-infiniteflatplate.” Thatis,reference4 assumed“...thepropertiesofa lsminaoffluidb theshocktubewhichhaabeeninmotionwitha velocity(ue-~) fora timetareequivalenttothoseofa laminaoffluidwhichhasprogressedresr-wardfora periodoftime t fromthelead5ngedgeofa semi-tnfiniteflatplateina steadyflowwithfree-streamvelocity{u=-~) ....”For thelaminarcase,thisis equivalentto replacingeq&bim (60)by

%J1- UeCf+ = 0.664

}1-%/”el ‘

whi’chisaccuratefor due << 1,butunderestimtestheshearbyafactorof1.7at UJue = 1 am bya factorof3.4at u#ue = 6. Fortheturbulentcase,thisisequivalenttousingequation(62),butwith

/( )e /%J5

evaluatedfor ~ue = O. Ifthevarbtionoffluidstate‘e

prop-irties-is

whichhastherespectively.

neglected,equation

p+lfs (= 0.05771

(62)canbeexpressedas

1/5 1- (u#le)

)2%

‘7= 11- (~”e)l(65)

values0.0577,0.0607,and0.0705at u#ze= 0,1,snd6,1/5 for ~f”eThefactthatthevaluesof cpe =0 and

1 &er relativelylittlesuggeststhattheequivalentflat-plateap-proachmsygivereasonableestimatesfortheturbulentboundarylsyerbe-hinda weakwave.Anotherinterpretatimisthat(bycomparisonwiththebminarsolution)thepresentmethodunderestimatesthewallshearforvsluesof ~/~ &herthanthosesatisf@ng~ue << 1. Notethattheseventh-powerveloci~profileandtheBlasiusrelationbetweenshesrandboundsry-~rthiclmess(eq.(45))applytofullydevelopedturbulentpipeflowaswellastotheturbulentflowovera semi-infiniteflatplate.Itishopedthattheseqressionsaresufficientlyuniver-salsothattheycsnbeusedherefiwithoutmodtiication,excepttousevelocitiesrelativetothewall.Thevalidityofthisapproachmustultimatelybeverifiedbyexperiment. ,.

NACATN3712 29

CONCLUDINGREMARKS

mInmto

Theboundarylayerbehinda shockorthinexpansionwavemovingintoa stationaryfluidhasbeenstudied.Bothlamharandturbulentboundarylayerswereconsidered.Thewallt~eraturebehindthewaveswasalsoinvestigated.

Forstrongshockwaves,theboundarylsyerbehindtheshockwillgeneraldyhaveverylargetemperaturegradientsnormaltothewall.Theassumptionsuponwhichthelaminar-boundary-layersoluticmswereobtained(eqs.(8))maythenhavetobereevaluated.ItmaybepossibletoobtainmoreaccurateestimatesofthelsminarboundarylayerbyusingsomemesareferencetemperaturesasdiscussedinappendixC. Thechoiceoftheappropriateemeantemperaturemightbeobtainedfromeither~erimentaldataorbycomparisonwithmoreaccurateintegrationsofthelaminar-boundary-~erequations.

Theturbulent-boundary-layersolutionspresentedhereinrepresentanextensionofempirical,semi-infiniteflat-plate,boundary-layerdatatothecasewherethewa13.ismoving.Thebasicassumptions(especiallythoseincorporatedineqs.(40),(41),and(48))mustultimatelybever-ifiedby~eriment,particularlyforthecaseofstrongshocks.

Itshouldalsobenotedthatthefluidbehinda verystrcmgshockmaybedissociated.Suchdissociationintroducescmplexitiesofthebuundary-layerproblemthatarebeyondthescopeofthisreport.Exper-imentalstudiesoftheboundarylayerbehindverystrongshockswouldyieldfundamentalinformationrelatingtoboundary-layerdevelopmentina dissociatedfluid.

LewisFlightPropulsionLaboratoryNationalAdvisoryCommitteeforAeronautics

Cleveland,Ohio,February16,1956

. . . . ..— . ..-—___ ._ . —.-—

30

A

a

B

b

c

‘f

CPerfc

f

%k

M

P

q

R

Re

r

s

T

Tm

Tr

t

Al?l?lmmxA

SYMBOLS

functiondefinedbyeq.(32b)

localspeedofsound

NACATM3712

functiondeftiedby eqs.(54)

functiondefinedbyeqs.(41)

functiondeftiedbyeqs.(41)

local.skh-frictioncoefficient,‘c4 ; Pw(~ -ue)2

spectiicheatatconstsntpressure

complimentaryerrorfunction

functionof q definedbyeq.(~)

integraldefinedbyeq.(43)whereN=6,7,or8

thermalconauctivi~

kh nuniberinx,y-coordhatesystem,MeS (ue/ae),etc.

~essure

rate ofheattrsnsferin+y-direction

gasconstxmt

.

—

Reynolds

function

function

absolute

absolute

absoluteOR

lxhlle

definedbyeqs.(15)

definedby eqs.(I-5)

statictemperature,%

meanstatictemperature(e.g.,eq.(47)),%

wallsurfacestatictemperatureforzeroheattransfer,

●

,.

\—

NACATN3712 31

velocitiesparallel

velocityof&ll in

velocitiesparallel.

velocityofwaveh

tox,y-sxes

x,y-coordinatesystem

8yT3tem

coordhatesstationarywithrespecttowave(fig.l(b))

coordinatesstationarywithrespecttowall(fig.l(a))

d~fusivity,k/pep

ratioofspecificheats

wallthermal-boundary-layerthicknesscorrespondingtoT-%=099~-Tw “

fluidvelocity-boundary-layerthicknesscorrespontigU-%=099ue-~”

to

fluidthermal-boun~-layerthicknesscorrespondingtoT -TwTe -Tw=O*gg

fluidvelocity-boundary-layerthicknessfromPohlhausenviewpoint

fluidboundary-layerdisplacementthickness,J=(l-&).

similarityparameterforintegralsolutionof

s-ity parameterforintegralsolutionofyp5

sindl.arityparameter,eq.(5)

valueof q correspondingto

fluidboun~-layermomentum

Y= 6

Jw

thickness,o ()pu ~-uw

peue ‘e

. . — .- ___ _

32 NACATN3712 .

v coefficientofviscosity

v kinematicviscosity

P massdensi~

a Prandtlnuniber,pep/k

TV localshearstressexertedbyfluidonwall

9 functiondeftiedbyeq.(46b)

w stresmfunction(eqs.(4))

Stiscripts:

b undisturbedflowaheadofwave

e flowexternaltofluidboundarylqer

m quantityevaluatedat ~

w quantityevaluatedjust

o- quantityevaluatedjust

Superscripts:

f denotesdifferentiation

Specialnotation:

0( ) orderofma@ituae

abovewallsurface,y = 0+

belowwallsurface,y = O-

withrespectto q

.

.

.

-.. —. —.. .— .. .. .. ———— .——. —----- -——-

NACATN3712 33

APPENDIXB

Forconvenience,useful

WAVE-IONS

equatims relatingconditionsacrossa wavearenotedherein.%ockwaves‘andexpansia~ves areconsideredseparately.

Shockwave(~/~ > 1).-Cmsidertheflowinthex,y-coordinatesystem(i.e.,coordinatesystemmovingwiththeshock).Letsubscriptb designatetheundistu??bedflowupstreamoftheshockwaveand

%=% ‘e

%T% Te>ae)Pe———

/’/

/////[//////////// /// -x, u

subscripte theflowdownstreamoftheshockwaveandexternaltotheboundarylqyer.Notethat~ = ~ sothat~ = ~~ = #~ ● ~=>fromordinaryshock-wavetheory,

% (r+ I)@—=ue (T-1)%+2

&=—,

$ }for y = 1.4

+5

(Bl)

4 w‘e=(r +u - (r - W#J5u#ue

=6- (~”e) }

for y = 1.4

(B2)

-——_ . .._ —— —.—

34 NACATN3712

.

$e= (Y +1)(%/:,)-(Y -u

5 1for y = 1.4= 6(u&e) -1

(Y+ l)(~”e)- (Y-1,

(uJuel[(Y+ 1)- (r-W+,/%)]

6(~/~) -1

}for y = 1.4

(%#e)~ - (#ueJ

(B3)

(.JCnco

(B4)

-Sire wave(uJw) c 1). - Theexpansionwaveisassumedtohavenegligiblethickness.Inthe ~)~-coordinatesystem(i.e.,fixedwithrespecttowaU),theactuslaudassumedformoftheexpansionwaveisasfollow8: .

BQansionwave

Actualexpansionweve

The &Lscontinul.tymoveswith

fluid.Conditionsoneithersameaashownfortheactualfollo~ figureapplies:

.-

Assumedqansionwave

veloci@ ;d = -y intotheundisturbedsideoftheassumedacpsnsionwavearethewave.h thex,y-coordinatesystemthe

%> % %T ae~Pee}/-—————

OH x,u(///~///////~////”/’ “’”~

.

--——— .- .. — ..—. .

.I?ACATN3712

ThestatepropertiesacrossthewavearedefhedbytheisentropicT-1 y -1

relationsa~~ = (Te/~)1/2= (Pe/~)~ = (p~~)~. Otherequa-tionsoftiterestare

Pe—=

‘[ 176(u#e) -1=

5u#ue ifor T = 1.4

%—=‘e

[ 11- (U#e} */(r-l)l-y+ =7z-

Me=

((r +1) -k-e;%)Td &

11/7

-)(’/ 1

for T = 1.46 -5(Pe/1#

\2 I

(r +MJ#e) - (r - u\

5

Jfor y = 1.4

6(u#e) -1

\Te

[

1-=1

1- (u#le)2—=

2~

‘[ 16(%J%)-12‘=TW7 1for y = 1.4

35

(B6)

(B8)

- --—..—— .-. -—.- —— _. -—

36 NACATN3712

AePENmXc

LMINAR-BOUNDARY-LAYERSOLUTION

TOAMEANTEM5RATURE

InthebodyofthisY?eJ)Ort,thelaminar-boundary-layersolutionwasobtainedbyassmingthat p and k areproportionalto T andthat C and C areconstant,allreferencedtowa12conditions(eqs.(8)).~p questiannaturallyarisesastowhethertheuseofameanreferencetemperature~ wouldleadtomoreaccuratecorrela-tionswithexact(inregtitofluidproperties)integrationsofthelaminar-boun&ry-@erequations.

If a man reference temperatm isusedinbecome

where

equations(16),they

(cl)

%=-S’(0)(TW- Tr}fsf!q,)

H‘e ‘m ‘m—=—Ue

lim(q-f)+(~-)~$,mf”rd,+Pe he ~+.

o

Tr ()%?“12 Ugr(o)

~=1+~- 2TCe p,m

(C2)

(C3)

(C4)

andthesolutionsfor r and s areconsideredfunctionsof am. Xnreference13,itisfoundthatthechoice

Tm= 0.5(TW-tTe)+ 0.22(Tr-Te) (C5)

.

“

-. -. .— .- ——.—. ..-— ———.

,.

.-

Cn%’to

.

.

NACATN3712 37

correlatedequations(Cl)and(C2)withaccurateintegrationsoftheboundary-layereoyations.Flowpasta semi-infiniteflatplateisconsideredtherein.Equation(C5)shouldalsoprovideimprovedestimatesfor %W and ~ for U#e +.O. X ve/ueistobecorrelatedwithaccurateintegrationsoftheboundary-layerequationsanotherestimatefor Tm mightbecalledfor.Notethatequation(C5\definesthemeantemperaturewhichwasusedintheturtmlent-boundary-l.ayeranalysis.

38 NACATN3712

APPENDIXD

*7

.

Thetu@bulent-boun&&-layeranalysisrequiresevaluationofthetitegral

J’1%=Z%z

o l+ bz-cz2(43)

whereN=6,7,0r8, andQ b=:-l

Thedenoyinatorofequatiau(43)maybefactoredsothat(-C)(X - X)(X - ~) = l+bx - CX2 where

‘ ‘*(b+Fc)

‘=+F-F)

fromphyzic.a,l~ztsb> -1,c%), md 0<

beshownthat f3<0 and X>l. Egpation(43)— < 1. It canthenl~bcannowbewrittenas

where

J1Z%Z%,p= —z-$o

J1

%,xz +Oz

i

(Dl)

,

.

— -. .—. ..— .._ . .-—.- ____ --———- . ..

WA TN3712 39

.

mk!?to

Thejntegrm ~, ~ and ~,~ canbe integratedasfollows:

(D2a)

(DZb)

Thelogarithmtermisveryhesrlyequalendo~ositetothesummationte?nn,inequations(D2a)and(D2b),forvaluesof j3notclosetozeroandvaluesof X notcloseto1,respectively.Hence,eachofthesetermsmustbeevaluatedveryaccuratelyinordertogetaccurateesti-matesof %,B ~d ~,~. Forvaluesof ~ notclosetozeroandval-uesof k notcloseto”1,slternateexpressionsfor lN,$ ad- IN,kareQesirable.Since~ <0,

m

1 L N !m!()

lm‘1-p N+m+ l)! l-B-

1

~[ (1+ 1! ‘

= [N+ 1)(1- j3 + ● -1-N+ 2)(1- ~~ 1(N+ 3)(N212)(1- j3)2 ““”

{D3)

., ..“.”. ,

●

✎� ✎✎✿✿✿✿✿ ✿✿✿ ✿✿✿��

�����

40 NACATN3712

SinceA >1,

‘ T~-(N+l)~(N+m)&+m+lj(~~]‘(N+l)(l-A

1

J[ (1- (N+ 1) (N+ 1)

‘IN+l)(l-A N + 1)(N+2)X- 1(N+2)(N+3)A2 - ““”

(D4)

Eqyations(D3)and(D4)convergerapidlyfor f3<c O ad X >>1.

1.

2.

3.

4.

5.

6.

7.

ImmREmcm

Mireh, Harold.:knhar Boun~ @er wind Shock”AdVSZICillgintoStationaryFluid.NACATN3401,1955.

Donaldson,ColemanM?., andSull.ivsa,RogerD.: TheEffectofWallll%ictionontheStrengthofShockWavesinT@es sndHydraulicJ-s inChannels.NACATN1942,1949.

Hollyer,RobertN.,Jr.:A StudyofAttenuationh theShock‘Mibe.Eng.Res.Jhst.,Uhiv.ofMichigan,J@ 1,1953.(U.S.Navympt.$OfficeNatiRes.Contra&No.N6-ONR-232-TOIV,Proj.M720-4.)

Trimpi,RobertL.,sndCohen,NathanielB.: A TheoryforPredictingtheFlowofRealGasesinShock‘J3ibeswithE@erimentslVerMication.NACATN3375,1955.

chap, DeanR.,andIbibesin,MorrisW.: TemperatureandVelocityProfilesintheCompressibleLsminarBmndaryIqerwithArbitraryDistributionofSurfaceTemperature.Jour.Aero.Sci.,vol..16,no.9,Sept.1949,pp.547-565.

Luw,GeorgeM.: - Cmressfil-e~ BoundaryLsyerwithHeatTransferandSmallPressureGradient.NACATN3028,1953.-

IWamian,RichardA.: SomeViscousEffectsh Shock-TubeFlow.M.s.Thesis,Cornw WT.> Jbne1954.

..

. . ..- —. —__ _.. . . . ... . . . ___ ..__ _______ ____ . . . .-

NACATN3712 41

. 8.

9.

10.

yB

12.c)

. 13,

Rott,N., andHartunianJR.: OntheHeatTransfertotheWallsofaShockTiibe.GraduateSchoolofAero.Eng.,CornellUniv.,Nw.1955. (ContractAI?33(038)-21406.)

Eckert,E.R.G.: lhtroductiontotheTrsnsferofHeatandMass.McGraw-Hill.BookCo.,bc.,1950,pp.266-275.

Tucker,Maurice:ApproximateCalculationofTurbulentBoundsry-LqyerDevelopmentinCompmssibleFlow.NACATN2337,1951.

B*z, D.R.: AnApproximateSolutionofCompressibleTurbulentBoundsry-LqwrDevelopmentandConvectiveHeatTransferinConvergent-DivergentNozzles.Trans.A.S.M.E.,vol.77,no.8,NOV.1955,~. 1235-1245.

Schlichttig,H.: LectureSeries“BoundaryLsyerTheory.” Pt.II-TurbulentFlows. NACATM1218,1949.

Eckert,ErnstR.G.: SurveyonHeatTransferatHighSpeeds.Tech.Rep.54-70,Aero.Res.Lab.,WrightAirDev.Center,Wright-PatterscmAirForceBase,Apr.1954.

..— —— — —.

‘42.

lW2ATN 3712

Ww.iE1.- LAMINARBOUNDARYLAYERBEHINDWAVE(EQS.9AND15)

(a)solutionfor #ue = oSrmWltl0mb6r *. 0.72

n f f~ r“Jv”a ‘

-r1 msn a -s~

0.0 0.0.0:; Oaooo Oj:;: O&m: 0.8477 mm: Goooo I.tltl;:; .0470

::;:;

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.09s9.0211

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z .0375 .1!376 A673 .3357 .a2a4 za64 .3666 .6329 .4166

s .0506 .X4::%6

A650.0R44

.4173 .2082 367a .4470 .7915 .4isaA617 .497a

3.7009

.1147 .3a65a87a .aa4a .7499 .4130

B .1497.457a .ss:; .77 a

~ s16a .a97a .7na7 .4 01.3720 .4s12 .76 a a437 S666 .S679 i

9 aa91 .4167 A&36 .7248.4 6a

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M .2.330 .4606 A344 .7956 .6.239 .aga9 .79az3.% .aasa .603s .4aT4 ae~s .6636 .3z3a

.6877 .3953a4R9 .s4as .3nao

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$%:IA .4s07 .6a44 3797 L0477

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in ..~lso .561S .3618 i~oaz1.6 .s030 .6967 .34a5

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AQ33 . a5so L29a3 a330 la16.9306 aan6

gjji470a a6h9 .s337 13037 als7

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Me$t 9122 Sbiz 13338 :W- 3481?9 143990 .9616 DE06 1s1$4 .30R3 .i7n9 13317 .t176

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g

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4a 2.7839 .997!3 ~069 1.s639&a

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3823 :::: 13::1 +’::: g;:j-0904 .noal .00.4Q 1369S48 X3533 .0996 .001s46 3.3833

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% %:;s $a 9F.! 90

soonmoos ~::~ 38:? 3::4 3

1 70a1 703 1

DO SN:g

4.9 3.6n3aDO 1

.!!599 .0004 1.5686 .001s n036 13704 0009.S.n 3.7oia .0999 400s L5687 .0000 ~oaa 1>70s5a 3.S032 laooo .0002

DO06l.asae

J)019.0009 ooal 1>706

33S.U a

3 $$8:8 f%i000s

ZS68P0014

4.00a 35689 ::882 3:;; $3%: 0003 $tii

5.4 4-lS32 10000 aool 1.s6f19DOO

.0003 aoo9 23706 DOOa DO065.6 4.zo3a6.6

lnooo .0000 3s6894.383a lnooo

.000aaooo

00071.a6e9

13707 DOO1 0004.0001

5.7aoos 13707

4.4n3a 100CO aooo0001 0003

J& 4.6U32 lLIOUO .0000 z:::.0001 DO03.0001 .3003

:;;:;g:::

Dons39 4.A033

pooalnooo aooo 1.S690 .0001 nooa 13707 DOOO nool

&o 4.7a3a6.I

10000 aooo4an3a lDOOO

L36?00000 1.5690

.0000 0001

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6aJlooo 0001

63 :2!3: lnoo8 4100laoo .0008 H%: aeon nooaeon 0008 K/;z; $::: 88:8

w

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.

,—.—. .—. —___ ____ ___ ___ —— ----- — . .. . . ___

.

.

NACATN373.2

Tl&E1.-Continued.IAMIWARBOUNDARY

n f

aoooo:8Ui.0943

. .is4a.17n4

W;

.39s2

.4s90

:Es$i!.6707.7475.aa71.9093.993a1.oao41.16901.a893L36121.4444L531371.6S41L7303

H3Z3anaaea.la13

aaaol:m~

9.6175a7i7aaa170%91603.0167X11663.a165

23;%!~;::

3.716.s3.81643.9164401644.11644.a16431::4.51644.61644.7164%12:ao1646.1164

aa500:938s.37e3.4a05.46al

“8.50 9.54 9.6617.6193.6554

z.68 9.7a 6.7535.78a3ao90.6337.e56a.8765.6948.9111.9as4.9379.9468.9s61

.989s:i?333.9953.9965.99’?4.9961.9986.9990.9993.999s.999.999$.9998.9999.9999.9999>00001.000o3.0000%0000*8888Loooo1.000o

(b)Solutionfor

rg

o.4a9538?8.4a39.4191.41a6.40443943.38a4.3687.35s3

::W;

.?780

.a570a3s6.a14aa951.i7a6.xaaa.1341.I166.lOos.ons7.07a5

-%:8:.0413.0336

.oa70

.oazs

.0170

.0133

.0103

.0079

.0060

.0045

.0033

.ooa4

.0018

.0 138.0 09

.0006

.0005

.0003

.000a

.0001

.0001

.0001

.0000

$8::.0000.0000.0000%::$DOOO.0000

LAYERBEHINDWAVE(EQS.

‘%/Ue = 0.25

77=aoooo:f~fg.a567.3401

.7884

:8?33.965a1.o1591.06a5L1OSO1.1435i.i7aai,a09a1.s3671.a609tiaa21%30061.31661.3303

%34?:L360a1.367a

L3730k3zl+ma49L3a75i.3a96~39iaL39as;;;::

>39s0

l#2Efi3.396a%3964L3966L39671.396a%39691.39691.3969Ii/g?:L3970L3970Z3970

g$%:

L3970

r

. .m6trmmta

*1

.50s0

.64aaf%2.saai.4a66.44a4.4049.36s6.3a78a9ao.SSF13.aa70.lQald7za.14no

.osa6z

44 3.03 4.oaa7aa3a.ola6n14aal17~09ano7aDOS6

$:;3.ooaa.0019.0014DO1l.000a.0006.00040003$8:;.0001noolDOO18n 00

a 00nooo0000

aooooa$n.1396da3aaasaa649.3oo6.33a3.3s93.3a13.3 798.4 90

.4t4a

.4146

.4o96

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.3as9

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5%?asnaSasaao16a7R4~;:;

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4010%%:.06074s0s.0417,034a.oa7a.oaas.olao.0143

::aii$.0069.00s4.0041.0031.ooa4aola.0013DO1O.0 078.0 05.0004.0003.000ano 1

8.00 1nool.0000

43

9 AND15)

7-,0.72 i

86n

Onooo2$88a776L$601

9646S9S4lDa30%047slD69a

ti971ti97ati97aU973la973

a

1.000o3383aS06.5013.7sa4.704a.6s6a.6103.56S0sao93381.397a.3603.3a4a.s913.66oos30aao3aa7a9AS63..13s7a171.1006.oas6

Sz:f43s14.04aa

.03s4:Uzi.0193.01S6.olas.0099.0079.oo6a.oo4a.0037.ooa9.ooaa.0017.0013.0010.0007.0005.0004.0003.000a.0001.0001.0001.0001.0000.0000.0000.0000.0000

U99698+3.4949.4909Aaa4.44;:.4s9s

.34s1

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.oaao

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$8:$pooo.0000

. . —.-—..— . .--t

——. . - .—

—.

44 NACATN 3’712

TABLE1.-Continued.LAMINARBOUNDARYLAYERBEHINDWAVE(EQS.9AND15)(c)Solutionfor u ue= 0.50d

3ru6u ~ 8“, 0.76

n r rt“ r“ xran r -rF 860 s. -6*

aoooo 0.s000 03ae7 0.0000 oa7a7

~

lwooo:Ii:f Z?Zzt

Omooo 3.0000 aa66s::;: ?

JI 7a a696Z 46 .S640 z

.06 1.1647

4::? .94s4.s979

La o .8870 A::f.3590 .s451 .iaz7 .a746 .a310

.4.5!36a

,aa61 .6a97 “-’.3147 .34a5 .aa4i .8s70 .3s49 .77s7 .a490.a906 .6607 soa7 4a37

$.797a .aa?a .4a9a .7a13

m: ::;8; :~:g :j;:.53a9

:~gg; 34:!? .49R.563i :g;g; W:: .5783

.403a .6aa5.774s .asaa .7%53 .5509 .4aaa .6766 .517a .476o

.6!371 .7996 .3.4a7 .77aa .607ag m; 22:8

.4449 .7a60m: 5?8?

.4706m! ti~:g

.4s’s3g:: :48$8 AM!I

.S649 .19al .9401 .4soa1A

.3443 .3047.9945 .8a33 .1746 sa49 Az67 .4397 .aaoa .3071 .3s9arma37 .8999

z.1s74 ma54 .3ass .4a33 .909a .a7as

Mziz .9147 .34 6.3333

.92ao !.za 4 l%~iz 3:;? s~i! 5Z4Z Mg 38::La 1.3600 .9397 .109119

Liaa4 .s666L454S .9499

.35Z6 9771.094s

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aD 3.s499 .9sa7.a3i4

,0816 Lls91 .s019 .s94s . 1.0091 .z37aU M:;: f

.9 62 .069? B!z% :!;: .g!j$;

.ao77.9 26 .osa m:; :18:8 :Iiii

a> sa407 .97ao ,0491 I.*I77a~ L93a7

aloz *oas4 4440.9aa5 .0406 Laa93 .1o67 .ia45 Losoo L0719

as.ias7

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.033s33:3

La39z.9a92

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BS a.334aB9 S4336

.993s .0173 LSS97.9951 .0136 4));: .LOloLS643

L0704 .0340.oasa 1.073S .oa77

.;O;:

a.a3sa .9963 .0106

;La6ao 4336

a-aa9.0713

z.997a .ooa

5L0760 .oaas

p#{;.0479

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s sosaa .999a3.z3aa

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i.08aa .oo7a ,0176.ooa4 .oato

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353 al %:$0014 L679molo Laaoi

.0065 .0167 soa39

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soa53 .0004i.aaai .0005

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.nooa .0006

3s

i.oas44s3ao

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ilLaaaa

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463ao LOOOO aooo Laaaa.0001 .0004 Loa54 .0001 .0003

!:

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4fi3ao Laaaa.0000 Laaaa

.0001 .000a i.oas4 8.0 00493ao

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LOOOO .0000 iaaaa.000a Loa54 .0 00 .0001

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.0001 Loas4 .0000 .00011.000o .0000 Laaaa .0000

pj

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.

——— .. ___ ____ _______ ..__ -_

NACAm 3712 45

.TABILE1. - Continued.LAMINARBOUNDARYLAYERBEHINDWAVE(EQ13.9 AND15)

(d)Solutionfor U#ue = 0.75

-.— .. —- ____ .= —

46 N/WATN 3~2.

TABLE1.- Continued.

(e)

BOUNDARYLAYERBEHINDWAVE(EQS.9 AND15)

Solutionfor u#ue = 2 (ref.1)

Pr8n6tlnumber~, 0.72n r fw f“

0.0 0.0.0:: 2.0000 -3..0191 0.0000.l

aa997 o~::g 0.00001.8964

1.000o 0.ss18-lao91 .oe97 .s922 .095e .9151

.3798 1.798a.a45a

5-.9804 .1780 .s704 .2S61 .1831 .0313

.5!348 1.7oa9 -.9356 .a634.aa79

.8356.4

.4o60 .a6at .7490 .8004.7a05 L61al -.a777 .3447 .7896 .5043 .3331 .671S .7645

.5 .s-774 1.5276 -a203 .4alo .7356 .5758 .3965 .697a

.6 1.oa63.7ai7

1.4503 -.7367 .4916 .6755 .6a09 .45a7 .5a74 .6739.7 i.1677 U804 -.6601 .5560 .62aa.s 1.3oa6 1.3163 -.s834

.:6;: .50aa .46a5 .6aa7.6141

.9S480

1.4316 1.a637 -,5088.5454

.6657 .4049.40a9 .5697

.6a06 .5aa9 .3466 .516a

Lo 1..55S6 La164U

.7111 .4a4i:$:::

.587a .6153 .a996 .4636L6751 1.li’59 .7507 .3676 .s455 .6430L791O

.aasoU4L6 -s135

.4ia7

z.7648 .3155 .4974 .6666 .ai70 .3643

1..9O36IA

&;;2.0137

-a606-ai4a

.0139 J3683 .4465 .6866

.8366 2a6a .3953.i6a9

.7034 .1531 ::%2;

La aza16 LO699 -d74a2aa78

.8594 .i89am54a

3456 .7174 .la74 .a386

z-z 40a .6766 .1570 .2900 .7290

as3a6 1.0417 ‘1116.1063 .ao40

l.a a436a 1.0317 -s879.8909 .la93 .2556.soa6 .1067

.7305.88::

.0064.7463

w as390 i.oa39 -s686 .9iaa.070,s :;:::

.08s8 .75a7 .057a .iai9

a.o 2.6411 1.0179 -.05a9 SW:; .069a as’14 .7570 .0460U a.74a6 1.0133 -.0404 .0554

.ioia~aso .76ao

a.a 2.a438 1.0097 ‘.0306 .9310.0366 .0833

.0441 .ioa4 .76S3 .oa93ag446 1.0071

.0602

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-.0170.0348 .oe3a .7679.oa73

.oa3i .05s3.9381 .0671 .7699 .oiel .0446

a.s 3s4S6 1.0036a.6

-.ola4 .9405 .oa23 .os3e .771s .01413s4s9 1.ooa6 -.0090 .94a4 .016S .04a7

.03s7.77aa

2’7 3346a 1.0010 ‘.006S .9438 .oia7.0109

.0337.oa04

.7737 .0004 .oaa43.4463 ;::;:

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.0064.oaos

.017s.77s0 .0049 .0136

3D 3.646S 1.0006 -.ooa3 .9464 .00ss .01S8 .77s53.1 3.7465 1.0004 ‘.0016 .9469

.0037.0041 .oial

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3JI 3S466 laoo3 -.0011 .9473.ooa6 .0081

.0031 .oo9a .776o .ooao33 3S 466 zaooa -.0007 .947s

.0061.ooa3 .0070

3.4 4.0466 lnool -.0005 .9477.776a

.0017 .oosa.001s

.7763.0046

.0011 .003s

3s 4.1466 1.0001 -.0003 .9479 .ooza .0039 .7764 .0008345 4.2466

.ooa6lJJOOO

4.3466-moos .s480 .0009 .ooa9 .776s

1.0000.0006 .0019

M-J3001 .9480 .0006 .ooal .776S

4.4466.0004 .0014

1.000039

-,oooi .94014S466

.000s .001s .7766 .0003 .00101.0000 -.0001 .94s1 .0003 .0011 .7766 .000a .0007

4.0 4.6466 1.0000 .0000 .94aa .000a .Oooe .7766 .00014> .94t3a .000a

.000s.0006 .7766 .0001

4.2 .940a .0001 .0004.0004

.7766 .000143 .94aa

.0003.0001 .0003 .7766 .0001

4A .946a .0001.000a

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4s .94sa .0000 .0001 .7767 .00004$ .94sa .0000

.0001.0001 .7767 .0000

4.7 .94ea .0000.0001

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.0000.0000 .7767 .0000

49.0000

.

..— .. -.

.,

—.—. .-—..

.

.

NACATN3~2

TABLE1.- Continued.LAMINAR

47

LAYERBEHINDSHOCKWAVE(EQS.9 AND15)

(f)Solutionfor u#ue =4 (ref.1)

Pmndtlnumber~, 0.72

n r fn r’r

rdn r -rIo J

awl s -s10

0.0 0.0000 4.0000 -443623 0.0000 O::;;; 0.0000 O::;:;.1 .3’790 3.6964 -3s84s .0909

1.000o.2582

U1S6.889s

.8 .7198 X3077 +7;:: .1791 .8683 .4847Zloos

.1779 .7811lo8a8 2-8457

LOO;:

::.a6a6 .ao47 .6579 .2509 .67a4

1.a901 aaiae -3D766 .3s9 6 .73ae .769o .3139 .5830 S136

.6 i,5a73 aa3131.7377

-a67i7 .::;:: .6sa9 .8aoa .367%’ .496oiSa44 -W3688

a253

::.570s .8a09 .4134 .4180 .7:3:

asass 1.7768 .s231 .4900:l&:: .7838 .4.517 .349a.a g.;::: 1.6055 .5683 .414s .7a18 .4835 .a893.9 . 1.4664 -La430 .606a

.5S63.3461 .6458 .5o98 .a37a .47s7

a.3a8s 13554 -.9a58i? a,6i95

.6378 .a8s5 .5646 .5313 .1939 .40a6la679 -.771a .6636 .a33i .4a44 .5486

%a a,64a7 11998 -.5957 .6846 .la85.ls70

.4o89 .s6a9L3 a.7600 1~476 -AS47 .7016

:::;.1511 .34o6 .574a

1.4

5ZZ:

aa7a7 1.1079 -.3431 .71s1B306

.iaoi .a804. .5a3a .0798 3a83

l.s as8i9 1.0781 -.2560 .7a58 .094a .aa83 .59033.oa85 ;.oo:g

.06aa-.1890 .734a

.lSas

k:.074a .184a .5959 .0491

3.1933 -s3ao .7407 .0677 .1473zaa6

.6003 .038aL8 3a967 lha80

.097a-J3998

L9 33990.’?458 .0445 .l16a .6o36

1.0195 -D7L4 .7497 .0341.oa94

.0919 .6068.D774

.oaa6 43608

ao 35oO6 1.0135 ..0506 .78a7 .oa60 .0717 .608a .o17aa.i 3J50i8 i.oo9a -.0355

a474.7$50 .0196 .0556 .6097

aa 3.7oa5 1.oo6a -.oa46 m::.0130

.0147 .04a7~367

.6108 .0097a3 3s030 1.oo4a -.0169 .0110 .0326

oa8a.6116

8.4 3s034 1.ooa8 -.0115 .7589.0073

.oo8a .oa47aa16

.61a3 .0054 ~i63

a.s 4.0036 1.0018 -JI077 .7596 .0060 .0186 .61a78.6 4a03a l~ola +:;: .760a

.0040.0044 .0139 .6131

z?.0029

g;;:

4a039 lDooa .7605 .oo3a .0103 .6133 Deal43039 Moos

Do6a

:! 44040-.ooaa :;::: .ooa3 .0076 .6135

1.0003 -.0014 .0016.0015

.0055 .6136 .0011.0050-ao37

3.0 43040 lfiooa -aoo9 .7611 .oola .0040 .61373.1 4.6040

0008 Doa7lAOO1 -.0006 .761a .0008 .ooa9 .6138

Sa 4.7040 1.0001 -.0004 .7613 .0006 .ooaiDO05

.6138 .00049019

53 48040 1.0001 -.000a .7613 .0004 .0015 .6139D014

3A 4s040 1.0000 -.0001 .7614 .0003.0003

.0010 .6139nolo

aooa DO07

3.5 5.0040 10000 -.0001 .7614 .000a .0007 .6139 Dool .000556 8.1o40 1410003.7

-.0001 %;: .00016.8040

.0005 .6139 JJool .00031.0000

X8.0000 .0001

.7614.0003 .6139 J3001 .000a

.00013.9 .7614

.000a .6139 43000 .000a.0000 .000a .6139 J3000 .0001

40 .7614 .0000 .0001 .61394A

4)000 .0001.7615 .0000 .0001 .6139

4a.0000 .0000

.7615 .0000 .0000 .613943

Dooo .0000

4.4

48

TABLEI.-Concluded.LAMINARBOUNDARY

NACATN3712

LAYERBEHINDSHOCKWAVE(EQS.9 AND15)

(g)Solutionfor %/Ue = 6 (ref.1)

Pmndtl-* *8 o.l~n r *1 f“ r +.I Slln s +1

y 0::;: 6.0000 -8~009 ox):;;s.1977

0.9195 CK$o:g (Ml;;: 1.000o i.3a6a

.a-7a7al

lD410.9009 .2603 La99z

4.4383.3 i.4499

-7a6ao .17913.7534 -64078

.2406.2602

.6668 .1738 .7417 i.aa5e.7710 .6671

.4.a4ao

1.7947 =603 -s4454 .33aa .67a7.6a4a Li20a

.9619 .a99i .6183 9963

.s amasa 2.6643 4::;+ .3958 .5616 .9682 .3461

.6 a3307 2.2613 .449a.4a5a .2662

.4871.7

.9118 .3845:3::: xs4ia -2.8161

.3450 ,73a7.49s5 .4004 .8187 .4155

.8 1.6938 -z164a .s29e.2771 .6197

.9 akao9 1.5048.3a39 .7 94

8.4403

-1.6353 .5s86 .a5a6.aao6

.5 83.5ia7

.4600 a74a .4190

343a40k:

1.3631 -~:$;; .6a17 .ao40 .4941 .47553&.1::

.i364 .33Beu35a3

La.5998 .1s94 .4012 “ .4875 ..Y.060 .a71a

laa18 -.64781.3

.6238 .la34 .3ala .496933918

.0818 .alsll~Z66 -.4641

ld.6a47 .0947 .254a .5040

350a4 1.0873 -.3a8a.06Z6

.633o .07ai .1991 .5095.i69a

.0476 A3ao

L6 3.6097 1.0597 -.a304 .6393L6

.0545 .1545 .51373.7146 1s403 -.1597 .6440 .0409

.0360.i2e9 .s168

.ioaa

1.7 3a180.oa70 .0786

La1.oa70 -.1096 .6476 .0305 .090a .si9a .oaoi .0899

39aoaL9

M179 -.0744 .650a .oaa6 .0687 .5a094.0s17 LOi18 -.0500

.0149 .0453.65al .0166 .0517 .saaa .0109 .0340

2.0 4.iaa6 lno77 -.0333 .6536 .olai .0385 .sa3i2.1 4.aa3a

.ooao .oas41.0049 -.oa19 Aim: .0088 .oa8s .sa38

aa 4.3a36.oosa .01s8

1.0031 -.0143 .0063 .oalo .sa432.3 4.4239

,oo4a .0i38l.ooao -.oo9a .6s59 .004s .01s3 .sa46

24 4.6a40.0030 .0101

i.ooza -.0059 .6563 .oo3a .0111 .sa49 .ooal .0073

u 4.6Z41 2.0008 -.0037 .656S .ooa3 .0080 .Sasl4.7a4a 1s00s

.001s .00s3-.ooa3 .6567 .0016 .0057 .sasa

2.7.0010

h8a4a 1.0003.oo3a

-.001s .6s69 .0011 .0040 .sas32.8 4.9a43 1.0002

.0007 .ooa7-.0009 .6S70 .0008 .ooa8 .sas3

2.9 &oa43.0005 .0019

1.0001 -.000s .657o .000s .ooao .sas4 .0003 .0013

3.0 6aa43 1.0001 -.0003 .6571 .0004 .00143.1 6.aa43

.sas4 ~ooa .00093“0000 ~oooa .6571

La &3a43.000a .0010 .sas4 .000a .0006

Loooo -.0001X3

.6571 .000a .0007 .sa54 .0001 .0004s4a43 1.000o -.0001 .6571 .0001 .0004 -sass

3.4 6s243 1.000o.0001 .0003

.0000 .6571 .0001 .0003 sass .0000 .000a

X53.6

.657a .0000 .000a sass .0000 .0001

X7.6s7a .0000 .0001.6s7a

sass .0000 .0001

M.0000 .0001 sass .0000 .0001

.657a .0000 .0001 sass .0000 .00003.9 .6s7a .0000 .0000 .sas5 .0000 .0000

“

.

.4

.

-. - —— - —.—. ._ __ __ —. —— _._. ____ .-_. -_

NACAm 3’712

. TABLEIr. -

,,

.

ImmGEwmmoF EQIJNI!IO19(9’)

uw/+.

o 1 6

,Eq.(9)14. [23)::EQ.(9)’Eq.(23);;Eq-~9)‘Eq.(23)

* ~0.4696:O.&, !0..7W!3:0.7746 ;l!.6m2 L!xm

I I Im-f ; I

\1.2168i1.2379!Q&7979: 0,774s;0-4249 ‘w+l - W%J ,0.38!53

I

. .

———

I

TABLEIII . - CCMPAR1201iOF INTERPOLATIONFORMJLMWITHEXACTINl!=TIOl?S

OF EQUATION8(9)ANo (15)(PRAMDTLlKIMBER~, 0.72)

U’(&

0.25 0.50 0.75 2.0 4.0

(9fy~15) (%j (9pi15) (%j (97h5) (% (9y&l) (%j (9?%) (%j

f“(o)i - (u#le] 0.5727 0.570 0.6574 0.655 0.7312 0.730 1.0191 1.017 1.3541 1.353

g* ‘“M48 ‘“w ‘“SW ‘“w ‘“ffim 0“862 0“6466 ‘“w 0’5013 0“503

J

-r dq 1.3970 1.396 1.2822 1.279 1.1964 1.194 0.8482 0.940 0.7615 0.754

0

J

wE dq 1.1974 1.201 1.0854 1.087 1.C038 1.004 0.7767 0.774 0.6X59 0.610

0

-s1(o) 0.4996 0.502 0.5665 0.569 0.6247 0.627 0.8512 0.856 1.1156 1.122

r(o) 0.8628 0.858 0.8727 0.867 0.8799 0.876 0.8997 0.893 0.9129 0.907

I

.

I?ACATN371.2 51

TABLEIv.-R2r.IPRomLaOFImmRAIaWcuRRlmm ToR8uLmr-Bouml!IRr-LAYElAmLY818

(8) ReclprocslOf ~ (bsse6on~c61 reSUltSOf ref.n)

+1= -. 6 I -.4 -. 2

5.7705.3284.8834.4273.967s.4853.01.12.5072.2441.9701.6821.386.9960.7559

0

bo 2

19.0717.4915.8214.3112.70IL.069.3807.6~6.7885.8694.921.3.8872.7382.0270

4

31.oo28.4325.8323.1820.3417.841.5.11.1.2.2910.869.3637.831.6.I.504.2883.I.51.o

6

42.9639.3435.7432.0628.3924.6320.7916.9114.891.2.841.O.708.4105.8854.2700

8 10

54.88 66.8450.IS 61.0545.62 55.4640.90 49.7036.17 43.8431.43 38.0226.52 32.2821.52 26.0618.91 23.0216Al 1.8.m13.58 16.4410.63 12.867.391 8.8455.397 6.5270 0

0 (.100.200.300.400.500 “.6m.700.750.m.850.8(M.95a.975

1.000L1.8581.8301.7001.5671.4301.2881.I.39.9814.8869.8071.7097.5985.4575.3469

0

3.270 4.5313.036 4.1832.799 3.8512.557 3.5052.310 3.1492.056 2.7841.782 2.4101.516 2.0191.370 1.8141.21.71.5991.054 1.372.87261.123.6418 .8285.3m3 .631.7

0 0

6.9986.4525.9LM5.3424.7764.I.863.6042.9882.7162.3341.9851.6031.161.8764

0

-. 8

2.1752.0271.8791.7261.5701.4081.2403..062.9672.8668.7584.6360.4m5.36080

-. 6

3.6833.4JJ.3.I.352.m52.5702.2781.9761.6201.4851.3221.138.8350.6824.5247

0

-. 4

5.1424.7464.345S.9403.5283.I.072.6752.2261.9921.7481.4911.211.8826.6657

0

-. 2

6kl6.0615.5365.0064.4683.9203.3592.7782.4762.I.631.8341.4781.064.79870

bo I 2 4

36.1633.0629.8426.81.23.6520.4617.2213.9112.2210.488.6856.7834.662s.381o

6

50.1845.8541.5037.1332.7328.2823.7818.1716.8214.41U..919.2826.3544.5830

8 10

64.2078.2158.8471.4353.06 64.6247.4557.7741.&350.88,36.1043.9230.3236.8724.4329.6821.4126.0118.3322.251.5.141.8.3711.7814.278.0449.7335.804 7.0140 01

0 aloo.200.300Au).!WO.600.700.750.800.850.800.850.975

1.000 1

8.W9 22.127.366 20.256.n7 18.366.062 16.475.398 14.564.724 32.624.034 10.663.322 8.6432.852 7.6092.sn 6.5482.170 5.4471.739 4.2781.243 2.965.8279 2.165

0 0

(c)Reciprocalof 18(bsse6onnumerlcslresultsofref.11)

+=1=1=- b6

57.1852.1647.I.O42.0736.9831.8526.6721.3818.681.5.8313.091o.126.8404.-0

8 I 10-. 4

5.7445.2844.8364.3733.8053.4252.8362.43a2.1661.8831.6061.285.83371.6974

0-

-. 2

7.3&36.7846.1845.5744.9634.3383.7013.0432.703I2.3511.9831.5861.I.29.8388

Jo

09.0018.2647.51.86.7706.0105.2414.456S.6503.231.2.8012.3521.8n1.322.9766

0

2

25.1122.8520.7618.5816.3814.15IJ..8O9.5978.4187.2105.9654.6453.1792.2860

4

41.1737.5735.9630.3326.6923.0018.281.5.481.3.561.1.579.5427.386Solo3.5920

0.lal.200.500.400.500.800.700.750.Wo.850.800.850.975

1.000

2.388 4.0922.221 3.7812.054 3.4671.882 3.1491.707 2.8271.527 2.4961.338 2.1561.140 1.8011.035 1.6a.7.82421.424.80451.21.8.6711 .9950.54)23.7284.3734 .5473

0 0 173.2188.2166.7681.3764).2873.4253.8265.5347.2657.5740.6849.5034.0341.4127.2633.I.323.8028.8420.2824.6416.6620.2212.8615.588.66610.496.1777.4680 0

.

.- —-... .—— — —.—

52 l!rAcAmT3’p2.

.

—

ii a ue

*

ii a

f

,y, v

ue

+

--x, u

(a)Withrespecttowall.

Expansionwave

.

y, v

-iue

,,U

shockwave EkpSIISfonWave -

,(h)Withrespecttowave.

Figure1.- Coordinatesystemsusedtostudyboundary_ behindawavetivandngitioa 6tatiouarymid.

--.,- .’

.“

—. -- — .— -.. .— — .—..-— .

, ,

.

t

y, vEdgeof thermalboundarylayer

3959 ‘

. Mge of velocity boundary layer

u*

e

2’

%

Figure2. - Temperatureandvelocitydistributionsbehinda shock wave.

x, u

I1

I