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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Document preview page 1

Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 1

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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition

Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition helps you tackle difficult exercises with expert guidance.

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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 1 preview imagefnstructor'sSolutionsManualtoaccompanyMODERNCOMPRESSIBLEFLOWThirdEditionJohn D.Anderson,Jr.Curatorfor AerodynamicsNationalAirandSpaceMuseumandProfessorErneritusUniversity of Maryland
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 2 preview image
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 3 preview imagel.l.008121.2e:#:sffiP:mr.g/m-lpRT(10X1.01x105)(1.38xLO''11ZZO1pr= p_(10)(1.01x105)RrpRT(11.0)(8314)(320)87xI0'"lm0.,0345kg-molekg/n:1.3Fromthedefinitionofenthalpy,h:e*pv:e*RTEoracalorically perfectgas,thisbebomescpT:c"T+RT,or6JFForathermally perfectgzrs,EQ.(A1)isfirstdifferentiateddh:de+ Rdtof,codT:c"dT:Rdtof,(Al)
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 4 preview imaget.4sz-sr:c,tn?-.^f(a)R:1716ft-lb/slug"R"r=h= W=6oo6ftJb/slug.Rs2-sr:(6006)tn(1.687)-(t716)tn4.5(b)R:287joulelkg.K"r=h:W:too4.5jouletkg"Ksz-sr= (1004.5)tn(1.657)_287n@.1)52-sl=59.9ftJb/slue"s2-sl=3'6ioul/1.5Pr-PrW:ppz824.3=--r'RT,(1716X400)t.4=1800(400/500)0.400121.6Volume of room:(20X15XB)=2400ffStandardsealeveldensity:0.002377slug/ft3Massofair:(0.002377)e400)=IS.ZOslud
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 5 preview image1.7(a)dp=-pVdVanddp:ptdp,orCombining:dp,_L=_pVdVprdp:-"p2vdvdp4=-tpVdVpdP=-rpv2dVVp(b),,:1=-17P(1'4[l^ol"1d):7'07x1o-6m2/NY:r,py2+= -(7.07iiro{y1r.23x10)2(0.01)+=FT?.ro=1p(c)Here,&*iffbelargerbytherati,(tooo\'p"[roly:(:8.7xrof(-*9'Comment:Byincreasingthevelocityofa factorof100,the fractionalchangeindensityisincreasedbyfactorof104.This isjustanotherindicationofwhyhigh-speedflowsmustbedp:dPpr
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 6 preview imageCHAPTER22-lConsideratwo-dimensionalbodyinaflow,assketchedinFigureA.Acontrol volumeisdrawnaroundthisbody, as giveninthedashedlinesinFigureA.Thecontrol volumeisboundedby:l.Theupper andlowerstreamlinesfaraboveand belowthe body (abandhi,respectively.)2.Lines perpendiculartotheflowvelocity faraheadofandbehindthe body (aiandbh, respectively)..3.Acut that surroundsandwraps the surfaceofthebody(cdefg)The entire control volumeisabcdefiria.Thelyidthofthe control volumeinthezdirection(perpendiculartothepage)isunity.StationsIiand2arcinflowandoutflowstations.respectively.Assumethatthecontour abhiisfarenoughfromthebodysuchthat thepressure isinflowvelocity urisuniformacrossai(asitwouldbeinafreestream,oratest sectionofawindtunnel.)Theoutflow velocityu2is notuniformacrossbh,becausethepresenceofthebodyhascreatedu*ukattheoutflow station.However,assumethat bothu1andu28rointhexdirection:hence,u1=coostdfltandu2:f(y).Considerthesurfaceforcesonthe controlvolumeshowninFigrreA.Theystemfromtwocontributions:l.Thepressuredistributionover the surface, abhi,-lJoutabhi
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 7 preview imagee__6FigureAThesr:rface shear stressonab andhihas beennegfected.Also,notethatin Figwe A.thecuts cdandfgaretakenadjacenttoeachotherihence anyshe6rstess orpressuredistributiononone is'equal and oppositetothatontheother; i.e.,thesurfaceforcesoncdandfgcancel each other.Also,note thatthesurfaceon defistheewaland opposite reactiontotheshear shessandi-*pressuredistributioncreatedby theflowoverthesurfaceofthebody.Toseethis more clearly,examineFigrueB.Ontheleftisshowntheflowover thebody.The movingfluidexertspressureandshearstressdistibutionsoverthebodysurfacewhichcreatearesultantaerodynamic forceperunitspanR'on thebody.Inturn, by Newton'sttrirdlaw,thebodyexertsequal and oppositepressureandshear stressdistributions on theflow,i.e., onthepartoftheconftolsurfaceboundedbydef.Hence,the body'exertsaforce-Ronthecontolsurface,asshown on theriofFigureB.Withtheaboveinmind,thetotalsurfaceforceontheentirecontrol volumeisSurfaceforce:-lJabhi
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 8 preview imageMorgover, this isthetotalforceonthecontrol volumeshownin FigureAbecausethevolumetricbodyforceisnegligible.Consider theintegralform ofthe momentum equationasgivenby,Equation(2.1l)inthetext.Theright-handsideof thisequation isphysicallythe foree on thefluidmovingthroughthecontrolvolume.For thecontolvolumeinFigureA,this forceissimplythe expression givenbyEquation(1).Hence,using Equation(2.11),withtheright-handsidegivenbyEquation (1), wehaveovlldv+'VQ)flS(pv'ds)v:-lJnns-n'abhiFlowexertspandron the surfaceofthebody,givingaresultantaerodynamicforce RFigureBAssumingsteadyflow,Equation(2)becomesR':-flrou'ds)v-lJnusa surfaceforceonthesection of.thecontrolvolumedelthatequals-R(3)
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 9 preview imageEquation(3) isavector equation.consideragainthe confrol volumeinFigureA.TakethexcomponentofEquation (3), nothingthattheinflowandoutflowvelocitiesu1and u2 dreinthexdirectionandthexcomponentofR'isthe aerodynamic drag perunitspanD,:D':-#(pv'ds)'-lf(pds).Sabhi(4)InEquation(4), the last term is thecomponentofthepressureforceinthexdirection.[TheexpressionGdS),.is thexcomponentofthepressureforceexertedonthe elementalareadSofthecontrolsurface.]Recall that the boundariesofthecontrol volumeabhiare chosenfarenoughfromthebodysuchth"t pis constantalong theseboundaries. Foraconstant pressure.llro dS)*=oabhibecause,lookingalong thexdirection inFigureA,the pressureforceonabhipushingtowardthe./rightexacflybalancesthepressureforce pushingtowardtheleft.This istrue no matterwhattheshapeofabhi is,aslongasp isconstant along thesurface.Therefore,substitutingEquation(5)into(4), weobtainItlD':-(flfnV.dS)un\rsEvaluatingthe surfaceintegral inEquation (6), wenotefromFigureAthat:l.Thesections ab,hianddefaresteamlinesoftheflow.SincebydefinitionV'isparallel to thestreamlines anddSis perpendiculartothe controlsurface, alongthesesectionsVanddSareperpendicular vectors, and henceV'dS=0.Asaresult.thecontributions ofab,hiand defto theintegral inEquation(6)are zero.(s)(6)
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 10 preview image2'The cuts cd andfgare adjacenttoeachother.Themassfluxoutofoneisidenticallythemassfluxintotheother.Hence,the contributionsofcdandfg tothe integralinEquation (6)canceleach other.Asaresult' theonlycontributions to the integralinEquation(6)comefromsectionsaiandbh.These sections areorientedinthey direction.Also,the control volumehasunitdepthinthe zdirection(perpendiculartothepage).Hence,forthese sections,dS:dy(l).The integralinEquation(6)becomesflrou'ds)u:-I,pru2rdy.JJnfzdySNotethatthe minusin frontofthefirst tennontheright-handsideofEquation(7) isduetovanddsbeinginoppositedirectionsalongai (stationIisaninflowboundary);incontrast,vanddsareinthesamedirectionoverhb(station2isanuoutflow boundary),and hencethesecondtennhasapositivesign.BeforegoingfiftherwithEquation(7),considertheintegralformofthecontinuityequationforsteadyflow.Applied tothecontrol volume in FigureA,thisbecomesla1b-J,prur dy+Jnp2u2dy:0or.J,"p,u,at:JrtpzwdyMultiplyingEquation (8) byu1,whichisaconstant,weobtainJ'"o'u"dv:IJPzwurdYSubstitutingEquation (9) intoEquation (7), we havefrou'ds)u: -IJpzvuray*J,'nizdy(7)(8)(e)
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 11 preview imageflfon'ds)u=-IJpzuz(ur-uz)dySSubstitutingEquation (10) intoEquation (6)yieldso':Jrtpzuz(ur-uz)dy(10)(l1)(r2)Equation(l1)is the desired resultof thissection;itexpressesthedragofabody intermsoftheknownfreestreamvelocityu1 andtheflow-fieldpropertiesp2andu2,&croSSa verticalstation downstreamofthebody.Thesedownstearnpropertiescan bemeasuredinawindtunnel,andthedragperunitspanofthebodyD'canbe obtainedbyevaluating the integratinEquation(ll)numerically,using themeasured dataforp2andu2asafunction of y.Examine Equation (11)moreclosely.The quantrtyur-uzis thevelocitydecrement atagiveny location.Thatis,becauseofthe dragon thebody, thereisawake thattrailsdownstreamofthebody.In thiswake, thereisalossin flowvelocifur-uz.Thequantiy mwissimplythemassflux;whenmultiplied byot-u2,itgivesthedecrementinmomentum.Therefore,theintegralinEquation(11) is physicallythedecrementinmomentumflowthatexistsacrossthe--W6ke,andfrom Equation(11),this #akemomentumdecrementisequalto the dragonthe body.Forincompressibleflow,p:constantandisknown.For thiscase,Equation(11)becomes.D'=oJJuz(ur-uz)dyEquation(12)is theanswertothequestions posedatthebeginning ofthis section.Itshowshowameasurementof thevelocity distributionacrossthewakeofabodycanyieldthedrag.Thesevelocity distributionsareconventionallymeaswedwithaPitotrake:
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 12 preview image''.t.tlJ.::'rit'--!i,l:.l;il::11..--,,..::,r_..il":.iUpTerf"("\2.2ia-ILo,uet-Wa//f2(r)Denote thepressuredistributionsonthe upperandjowerwallsbypr(")and.p,(x)respectively.Thewallsarecloseenoughtothemodblsuchthatpudrdpz tr€not necessarilyequaltop-.Assumethatfacesaiandbharefarenough upstreamanddownstreamofthemodelsuchthatP:P-andv:0andaiandbh.eIIIcl?,,IIiiuttt.y-component ofEq.(2.11)inthetext:L:-#rpiat;v-lJrnaSlvsabhi++Thefustintegral=0overallsurfaces,eitherbecauseV'ds =0 orbecausev =0.Hence.bhL':-JJtnaSlv:-tJPudx-Jn,d*1abhiaiMinussignbecausey-component isindownwarddirection.Note:Intheabove,the integrals over iaandbhcancelbecausep =p.onbothfaces.HenceL':hbc.'fJp,o*-Jpuoxi-a
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 13 preview imageCHAPTER33.1FromTableA.1,forM:0.7;pJp=l.3BTandTo/T:1.09g.Hence.r^\Po=pl?J:o.e (1.01xto-5y11.387y:Notethat P*/pocanbe obtainedfromTableA.lasthe valueofp/poforM: l.Hence,fromTableA.lforM=1;po/p*:1.893andTolT*:1.2.Thus.o*:n[&-)fa'.l:0.e(r.0r*ro)tr.s;7)/rr8e3:-'\p/(p"/r*:rt+)[+l:250(t.os8)/1.2:ET&m\1./\lolu*=.'ffi:^l@:bog2'A;d1.26x10"N/m'(or1.248 atrnx10"N/m'(or0.659 atm3.2po_1.5x106_",.,p5x104FromTable4.1'Fod@nearinterpolation)Also fromTableA.1:Hence,T,T:2.643(linear interpolation)/r\ro=r[+l:2oo(2.64T=ltt6E\ t./
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 14 preview image3.3' a:JAT:rftr.4xr7r6x500) :1096ff/secM:yla=3000/1096=(2.74i,From Eq. (3.37)(M*;z:y+l2.4:3.63.4FromTableA.2,forMr:3;pzlpr:10.33,pzlpr:3.857,po,/po,=0.3283,andW4.4:.s2.rhus,pz:pr@zlp):(lxt.0txtO511tO.:3):1.043x106N/m2(or10.33atm)Thus,//f.\Pz=PrIgI:(1)(l.olxto)1t0.:3):\p,/./^\_Pz:pr |,,| :1.23(3.lsT=wqUffi\pr)12:o:=#:#:F6m-pzR(4.744)(287)u:Jtar,=,[(L4)Q87)Q66)=554.8 m/secw-a2M2=(554.9)(0.4752)=EOI.OrnAidFrom TableA.l,forMr:3,Po,/p1:36.73a1dT",lTr:2.8.Hence:(P"\tiu,n.nnurn.Ao^\po,:pltijfr:(1.01xrc\Q6.73X0.32s3):h218 "' 10fr-/*-1.i''(-Lr)'*'-'M-:Tq
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 15 preview image3.5,1.22x10'(a)pJp=ffii#:7.2r.Fromrablee.t:F<s3,7222(b)pJp: ffi:3.413.FromTableA.l;wefindthis wouldcorrespondtoM-=1.45.However,sincethisis supersonic,anormalshock sitsin front ofthePitottube.Hence,Poisnowthetotalpressurebehindanormalshockwave. Thus wehavetouseTableA.2.1)))p^-/p,=t-z-!-'-:3.412.FromTableA.2'M;= l.jl^"?2116'l3tg:n.85.FromTableA.2'M;=Til(c)Po,/Pr=.,OZO3.6For theshockcompression, useMriISisentropiccompression,usepzlpt:(v1lv1)-7:NORMALSHOCK COMPRESSION"(pzpr)-tMrpzlil=vzlYrPzlqraparameter,alongwithTableA.2.FortheISENTOPIC COMPRESSIONvzlvrwlPr1.0t.21.5t.72.02.53.0I1.341.862.202.67J.JJ3.8610.7450.5370.4550.3750.300.2591l.5l2.463.214.s07.1310.3I0.7s0.500.400.300.2011.502.64.3.615.49.51
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 16 preview imageNotethat:t(a)Foragivendecreaseinvolume,a shockwaveyieldsahigherpressureandthereforemany beconstruedtobemore"effective"thananisentropiccompression.(b)Ontheotherhand,becarisethe entropyincreasesacrosstheshock,ashockwaveislessefficientthanan isentropiccompression.Toobtainagivenpzlptitrequiresmorework viaashock.(c) Note thatfor vzlvrnearunity,theshockandisentropic curvesareessentiallythesame.This isareflectionthat theentropyincreaseacrossweakshocks(Mr<1.3) isnegligible.3.7FromTableA.1foM-::g,fJT=289.9;To:(289.8)(270):78246"4Thistemperatureis almost8times thesurfacetemperatureofthesun.Longbeforethistemperature is evenreached, the airwilldissociateadiorize,andyis no longer constant,nor
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 17 preview imageanywherecloseto1.4.Theabove answer istotallyinaccurate!Sincedissociationandionizationrequire energy, the actual temperature is considerablyless,morelike11,000"K.The propercalculation of thisanswer mustbe madewithtechniquesdiscussedinChapter14.3.8(a)FromTableA.3 forM1:2.0:fr:0.363,+:os2us,+:0.7s34FromTableA.1 forMr:2.0:ft:,.*-T.Hence:(r*t/m\ro*=I+-It+lr,=f+^.)(r.8X288)=653.4oK(Tq/\T,/\0.793ry'/Thisis thetotaltemperatureattheexit,sinceforchokedconditionsattheexit,M2=IandhenceTo,:To*,p2:p*, etc. Attheinlet,(1,:1.8Tr:(1.8X288):518.4"K.g:cp(t,-T",):1005(653.4-581.4):1.357xIpz=p*:p1/0.3636=#H=WdTz:T*=T110.5289:^^:F@0.5289(b)FromTableA.3,forM1:0.2:prlp*:2.273,TrlT* :0.2066,1,/To*:0.7736FrcimTableA.l,forM1:0.2:
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 18 preview image1,:1.00gT,.n*_ft"-)fT",\( t\ro*=t-r"JtiJt':Iorrru1(l'008)(288):r672K1,:1.008T1:1.008(2SA;:290.3"K9:cp(f., -1,)=1005(1672-290.3)=1.389xl0oioulPz:P* =p110.273=Iatm2.273Tz=T*:T1/o.2o6u:#=h39aB..3.9Islugair+0.06 slugfuel:1.06slugof mixture= 424loR0.044.5xlos\n:#:2.547.x107ft-lbper slugof mixturec.=A-Q.4)(1716):6006ftlb'y-l0.4slugoR.n."q-2.547xl07'or-to,-%-6006FromTableA.1:AtM1:0.2,4,= Tr:1.0081,=1.008(1000):l008oR1,=4241+To,:4241+1008:5249"RFromTableA.3:ForM1:0.2,prlp*:2.273,TrlTo*:0.2066,T",flon=g,l
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 19 preview imageTo*:T",10.1736:1008/0.1736:5S06Rl,[o*:5249/5806= 0.904From TableA.3:For t",/To*:0.904,(by interpolation)N{r:0594,pz/p*:1.433,T21T*=0.9895n,=;?fpr:(1*r(#)(10):tu0;dt,:-ftT11=(o.e8er,(#J(rooo)=F7sed3.10 4,=.To*:5806'R(fromprob. 3.9)e:cp(T",-T",):6006(5306- 1003)=2.88xlOiftlb/slugLetF:fueVair ratiobymass.o:lA(4'5xlot):2.88xro7-(1+FA)FA=2.88x107i4.5x108-2.88x10?)=0.06840.03(4.5x108)3.11q::1.311x107ftlbper slugof mixture1.311x10?6006:2182oR1.03qco4,-1,=q,_T",T*q*However,_2182T*^o1,:.To*forchokedflow
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 20 preview imageAlso,H= l*r:l(sincetr4z:1)T22To*:l2Tz:1.2(4800):5760"R.,'L, -2182T"*5760:0.621T",q*FromTableA.3:For+:0.6212,3.12FromTableA.4:For M2:0.5,ry=\.069, pzlp* =2.l3gandT2lT*:1.143.4L,*=4L=,***=1.069+4(0.q0-t(40)!or.o,DDDFromTableA.4:For4L'*D=41.069,m=0.12iTrlT*:l.l96,prlp*:8.675n,=fr #,,:(8.67s)(#3J(rarn)=Eooad3.13FromTableA.4:=2.63V,AgTTFForMr:2.5.'iLtJrD:0.432,L=0.2921.T'-P*'T'F= 0.5333,po,/po*FromTableA.l:ForM1:2:5,!"'-:17.0gpr
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 21 preview image4Lrn-T*flo=0.432*4ff=0.132FromTableA.4:ForY:0.132,M;=l-4q,pzlp*:0.6125,T21T*:0.8314,po,/ po*-r.169*:;i#p,:(0.6t2r,(#)(os):m48d,,=hft,=(0.8314)(#)(5zo;=Fro.FEpo, =#Hfn'=rr.rur>ffi(r7.0e)(0.s):ltrs8;d3.14-L=10.Also,p1=100atm.Hence,p2:10ugn.Iftheexitischoked,the4p*=p2=Pz/10atm.Thus,ptlp*:100/10:10.FromTableA.4, forp1lp*=10,4L,*I:55.83DD=4inches:0.333ft.55.83Da.l-=-='4f,ss.83(0.333)=FgoE4(0.00t3.15StartingwithEq.(3.95),repeatedbelow:dp+pudu:-(ll2p*,Tandnoting that(pr'\t'pu2:(rp)|\YP)a-(3.es)
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 22 preview imageEq. (3.95)becomesdp*Mdu_-A,I'flgl)pu2(D/Fromp:pRTdp dodTppTFrompu:constdp:-dupuCoining (A2)and(A3)dp_dT-dupTuSubstitute(Aa)into(Al).rdr*(yMz_l)!* -tM'Tu2From the adiabatic energy equationfr+{=const2dh+udu:codT*udu:A-dT*udu=0y-l-l-g.4du Idr*14,$=oy-I T/RTuy-lTu(Al)(42)(A3)(As)t:.f:...iof,dr'')M'jgT:-(T-r'uFrom thedefinitionof Machnumber
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 23 preview image,]:a2M2:M2yRTThus,dr:M*1dTuM2T(A7)Substitute(.4.6)into (A7)du-dM -1rn-ttM2iuuM2"u^duJOIVeIOr_.u+:#u*+(y-r)rvr't-'Substitute(,4.6)and(A8) into (A5)-(y-r)M'#U*+(y-r)att-,+(TM,-t,gtl*:(y-r)rvr,t-,tM'z-|-Tr,/P+Mrl#tt*)<r-rlrl-'=ry(atr.)M--t)#u*+(y-r)M,r-'=-ry(9or,finally+#(1-M'?)#tt*jtr-r)rur't''whichis Eq. (3.96).(A8)g=cp (To,z-To,r)
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 24 preview imageAtM=2.5,fromTablea.3,lU-=0.7101.'Tt€SincecpTo,tisthetotalenthalpyofthegasenteringthe duct,and qisgivenas30%oofthis totalenthalpy, we haveq=0.3toI,,Hencek:(l.3xo.7ror):0.e23T*FlomTableA.3,for+:O.g23,wehavetot,,0.3:l''T*;
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 25 preview imagevlz4.1Ia1:1ffi:{14rc(520t&r:1118fl/secM1=v1la1:335511118:3.0Mo,:M1 sinP:3.0sin(35)"=1.72FromTableA.l.Pz/pr:3.285Tzttr=1.473:Mn, :0.6355?P2:3.285 (2000):@OtAlfrT2:1.473(5201:@M,,:M1cosp:3cos(35o1:2.457w1:2.457(1I 18)=2747fVsec:w2a2:JIRT2=JQW=1357.3fflsec.u2:Mo,a2=0.6355(1357.3):862.6ff/secTan(F -0):wlwz:862.612747= 0.314p-0=17.4,,,..e=p- 17.4:35-17.4=
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 26 preview imagevr=rli]**r'=@:Es79m;d4.2Mn=2Fromthe O-B-Mdiagram:F:39.3oMn,:MrsinB=2sin39.3"= I.2668FromTableA.2:po, lpo,:FW4.3F1o-the O-B-M diagram,at M1:3,0r*:34.1oandp= 66o.Hence,the wedgecan benolarger thanahalf-angleof34.1e.:iMn,=MrsinF= (3.0)sin66-o=2.74FromTableA.2:pzlpr:8.592pr=1.01x10sN/mz (standifdsealevelpressure)pz=bp:8.592(1.01x10s1=8.678xlgs+prm'Thisis themaximum pressure.Anyhigherpreslure wouldconespondtoawedgehalf-angle>0,or.,theshockwouldbe detached.4.4P'-5.FromTableA.2,PrMn,2.2sinP=:j'': =,p=38.94o'Mr3.5Yn,=2.2,Mor:M1sinp,Thus,From the O:B-M diagram,fo{M1'.-3,5
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 27 preview image4.5Fromthe O-B-M diagram,lB:3S.1Mn,:M1 sinB:(3.0)sin3go=l.g5FromTableA.2,forMn,=1.85;pz/pr =3.g26,T2/Tr:l.56gand M".:0.6057P2:3.826p1:3.826efi6)=18096h/f,Tz=1.569Tr:1.569(5te;=@v,:ffi=ffi:lr-td4.60-___:l5z=2ooM,=9.6Cr;,QFromthe0-p-Mdiagram,gt:34Mo,:Mrsin34o=(3.6X0.559)=2.01FromTableA.2:M,.:0.5752M2=.y:"'::= '0'5752:2.378x2.38-sn(p-0)sin(34-20)F.q-* e-B-Mdiagram,forIvIz=2.38and02:20",p1-=,j'\,.
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Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 28 preview image4.7From theO-p-Mdiagram:ForMr= 2.g and 0:30o,0r=Ilo.Mn,Mrsin30o:2.9 (0.5)_1.4FromTableA.2: Mn,:1.4:pz/pr=2.12,T2/Tr=1.255,po,/po,:0.95g2,Mn,:0.7297M-n.ra^11'Itdz::;:"':. --0"739-7:2.272sin(p _0)sin(30_Il)Frome-p-Mdiagram:Fortr42=2.272and02:Ilo,pz= 35.5o.Forthereflected shock.designatethe upstreamnormalMach number asMn,,.Mn,':I\42sinp2:2.272sin(35.5")=1.319FromTableA'2'forMn-'=r.3r9;ptlpz:1.g66,Tzfiz= r.204,po,rpo,=0.97Sg,Mo.:0.776vr,:ffi=*ffi:lr.8,o,:33pr:(1.866)e.r2)(r)=F.e56;dPzPtTTt,=ifrr:(t.2o4xl.25s)(3oo):F53JE'2rl
Page 29 of 31
Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 29 preview imageFromTableA.lforMr=2.g,po,/p1=27.14Po'-Po'Po'Po'-Po,p;p,:(o'9758)(o'ss82)Q7.l4xl)wherefotM:= l.g7t0",=30,=(6.398)(3.956)=25.ta,,rr-p:-Clean enoughwithin roundofferrors.Po,Es3-sudAsacheck,po,Po,/Pscan becalculated moredirectlyfromTableA.l=6.3984.8(a)Irrsr.s.PrFromTableA.2for]Wn,:2.57:Po,=0.4715*OMo, =0.5065.porFrom theO-p-Mdiagram,forMr=4and,p=40:0=26.3oyr:-r,.--0.5065-Sin(F-e)Si{A-26,,=2.139FromTableA.2,forMr=4.0,*,=0.13gg.FromTableA.l,forMr:4.0,Hence,Po,=L fo,=(o.l38go(rsl.8Xl)Mn,=MrsinF:4Sin40o:2.57Po,(b)
Page 30 of 31
Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 30 preview imageFromTableA.2,forNh:2.139,no':0.6557po.p",:+I9nt:(0.6557x0.47r5xr5r.sxr)-Po,Po,PrNote thatthetotalpressureincase(b)ishigher thancase(a),indicatingamoreefficientshockcompressiontosubsonicflow forcase(b).Theupstreamtotalpressureis po,:(l5l.8xl)=151.8 atrn.Forcase (a)the loss is totalpressure=151.8-2r.07atrn:130.7 atut.Forcase(b)the lossin totalpresstre:151.8-46.93=104.9atm.Hence,case(b)experiences asmallerlos!inpo, ?ndtherefore is moreefficient.Thisis,fgenerallytrue.Obliqueshockinletsonjetenginesare moreeffectivedevices thantheoldernormalshock inlets.4.9From the O-B-M diagram,andFigure4.22;For0z=20?andMr:3,g=37.8o.Mn,=M1:sinP=(3) sin(37.8)=1.839;P'-3.783PrMo, :0.607g;Mr:,,Y,"', :o='90.t*-== l.ggS:u;r(6-qSin(37.s-20)For0g:l5oandMr:3,F:32.2Mn,:MrsinB:(3) sin(32.2):1.60;b:2.82PrMo,=0.6684;M3:Mo'slun//-e)_0.6684=2_26Sin(322-It.:..',l'..,'j',.,..i..:'.-,:;,';;:--':-:..:.':.'
Page 31 of 31
Solution Manual For Modern Compressible Flow: With Historical Perspective, 3rd Edition - Page 31 preview imageFortheupstreamflowrepresentedfollowingcalculations:byregion2,plota press're deflectiondiagramfromthe0t'pM""=M2sinpPt'Pz9=Pa'Pzo:02-oa,PrPzPt30JJ.J37.24t.646.753.5FortheupstrOamflowfollowingcalculations:I1.091.201.32t.451.60M."=lrzl3sinp3.7834.6r5.727.068.6510.67&-=&-&-o:oa_03PrPrPr20t6l2840048t2t620I1.2r91.5131.8662.2862.820represefltedbyregion3,plotapresswedeflectiondiagramfromthe0+pP+Pz2629JJ36.84t.546.853.7I1.096r.231.351.501.651.82Ir.231.598t.962.4583.013.6982.823.474.515.536.938.4910.43-15-l I-7a-JI59048t2t62024
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Mechanical Engineering
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Modern Compressible Flow: With Hist...

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