Aerospace Engineering - Compressible Fluid Dynamics

Completed notes of the course

Complete course

Compressible fluid dynamics Legend New paragraph / new theorem Demonstration Definition Important ConservationLawsMassConservation(IntegralForm)edm=o 79da + | giroIds =odta.s | ,atsMassConservation(DIFFERENTIALLAW)• | , 79da _° 1911da =o+ | ,st. | Ildr+-o 191 )da=o 79 +_°'911=0DI+ f-ore=ositsitDta•IFWEPASSFROMTHEINTEGRALTOTHEDIFFERENTIALLAWWEcannoLONGER☐C-ALWITHDISCONTINUITIESLIKESHOCKWAV E SCONTACTSURFACESLIPLINESDISCONTINUITIESARENOTDIFFERENTIABLECOMPRESSIBCEincompressiBLEFLOWS-d= | -o1DITHEINTEGRALOFTHE☐IVETLCENCEOFVELOCITYISTHEVariationOFTHEVOLUMEOVERTIMEatmit)•inCOMPLESSIBLEFLOW:THEVOLUMEOFTHEFLUIDDOESN'TCHANCEOVERTIME-°I=0•THEVELOCITYFIELDOFANINCOMPLESSIB.LEFLOWISSOLENO/Dal•KEEPinmin☐THATincompressib.LEFLUIDSDON'TEX'ST:THEY'RETHERMODYNAMICALLYandPHYS/CALLYMEANINGLESSMOMENTUMBALANCE(Integralform)• dimmi = Id|finda + | SI10?ds= | T. s u?ds+ |ErdrdtdtnzSSv2•TSiSURFACESTRESSTENSOR•FI:VOLUMEForces-pressore:Tp=-PIGravity: [ g=- fly ==v'scuusstress.ES:IT=MOMENTUMBALANCE(DIFFERENTIALForm)•2191)+.° /97±+ PE )=°IT- SI -=at •in29+ gsia+-p+±_ 0191)+ giro±=_°it- SI -=atatMassConservation• g Dei+P=°- SI --☐tVISCOUSStressTENSOR•ti= Il9 , 97 , Et )=•HYPOTHESIS:'SOTROPICnewtonianiFLUIDTHEVISCOUSSTRESSTENSORISAL'NEARFUNCTIONOFTHEStrainRATETENSOR"I+(y)T- E =12.-µ=µ(T,P)DynamicviscusiT>COEFFICIENT.it= un -µ+(y/T'+✗/ o= ) ,aere | =---I=✗IT,p)volumeUISCOSITYCOEFFICIENT•BULKVISCOSITYCOEFFICIENT:New=23 µ +✗•STOKES'HYPOTHESIS:µU=0it'sValID☐ultForFEWMOLECULES|notValid,forEXAMPLE,forCO2 ) ENERGYBALANCE(INTEGRALForm ) C:InternalENERGYPERUNITMass2•To t a lC-nero>perun/tvolume: Et = f lt g Il22 g IliKINETICENERGYPERUNITMASS2•dEtda+ Etyoads =-|1Pa )oads+ |tuttoads- |9g oreda- | 90sds --at / a / s-SISPOWEROFPRESSOREForcesPOWEROFUISCOUSSTRESS.ESPOWEROFGRAVITYFORCESHEATTr a n s f e r•FourierH>POTHESIS: 9 =-KT,WITHK=K(T,P)THERMALCONDUCTluiTYCOEFFICIENT.dg, |Etde+ | IE '+ Pirroads= |laf) ☐ads+ | 1k_TI°Ids- / affari da rsss ENERGYBALANCE|DIFFERENTIALFORM)--• set +,oa/c-'-+PI=-◦ tuIl --◦9-- SI ◦Iset-_•2lget)+◦ IPI)+◦ Ietgir )=◦ inEl --◦9- SI ◦I----set.et>9+ gsè +◦ 1Pa )+ et ◦' 91)+ 910È =◦ inEl -_° ? - SI ◦Isetat-MASSCONSERVATION• g ☐ et +◦ 1Pa)=.◦ tutt)--◦ 9 _ SI ◦I=☐t-2eg☐e+a+◦ IPI)=◦ tutt)--◦ 9 - gg ◦I11)Dt2--=°LET'Scorsi☐C-aTITEMOMENTUMBALANCEEQUATIONANDMULTIPLYInunBOTHSIDES:--_-2Io9 ☐≈+P= Io ◦ I - SI9 ☐a+noP=Io/◦IT)- pagoI12) -----=DtDt2-_-_--_--P◦≈=ero/◦ E) +(I - Ion ◦ I ◦it)- 9£01 e11)-12): f Dl-_--_◦9- SI °I-I_=☐t-_--•g☐e=-P-o≈+(IT)°In--°9- de = SW - SL 1ˢᵗPRINCIPLEOFTHERMODYNAMICS=-☐t•ALSOFROMCONSERVATIONLAWSITCANBEFOUNDTHAT: g T DI = ✓ -◦9T da ≥ d9ClausiusincavatiTY--Dt• ✓ :DISSIPATIVEFUNCTIONALUVAYSProviDESAPOSITIVECONTRIBUTION THERMO DYNAMICSOFMOTION•THERMODYNAMICSYSTEMATEquilibriumCONDITION:tOBSERVATION[RELAXATIONS,E,uniFORM|P,T)°IFTHECONSIDEREDTIMEOFOBSERVATIONISMUCHBICCERTitanTHETIMETHATACERTAinVOLUMEOFFLUIDTA K E STORELAX(WHICHMEARSTOCHANCEITSTHERMODYNAMICSTATE)WEcantalkPROPERLYOFTHERMODYNAMICEQUILIBRIUMPz=/Py,Tz=/Ti•HYPOTHESIS:LOCALTHERMODYNAMICEQUILIBRIUMITMEANSTHATTHEEllulLibrium2Pt>TI^ConditionisUALIDLocali,SOWECANUSETHELAWSOFTHERMODYNAMICSTODESCRIBETHEFLOW,WHICHISMADEOFPARTICLESinTHERMODYNAMICEllulLibrium•THEDIFFERENCEBETWEENTHERMODYNAMICSandTHERMODYNAMICSOFMOTION'STHATinTHESECONDONETHETHERMODYNAMICEQUILIBRIUMISGUARANTEEDLOCALLYPOSTULATORYTHERMODYNAMICS•allPROPERTIESOFanISOLATED,mono-COMPONENT,SINGLE-PHASETHERMODYNAMICSYSTEMATEQUILIBRIUMAREDesertBEDBYASINGLEFUNDAMENTALRELATION:S= SI E,V,M)•E,V,MAREinCapitalLEITERSBE'noEXTENSIVEQUANTITIES•ENTROPY'SPROPERTIES:1)SISAHOMOCENEOUSFUNCTIONOFDEGREE☐NEWrtallINDIPENDENTvaria@LES:5/TE,tu,)M)=✗5/E/V,M)✗EIR2)SisaMONOTONEFUNCTIONWATTHEInternalENERGY:IFE1>EzS( £1,V1,M1)>S(EZ,V2,M2)andviceversa3)SisSUPERADDITIUEiS/E'/V1,Mi/tS/C-2,V2,M2 ) ≤ SI C-1+C-z,V1+V2,mi+Mz)°MI✗'NGTUOSYSTEMTHEENTROPYISHIGHERORATLEASTEllvalTOTHESunOFTHE'SOIATEDSYSTEMS G) NERNST'SPOSTULATI☐~:TEoS075V.M CONCAVITYOFSWRTE•✗, p E112✗+ p =1•SI✗En+ pez ,✗v1+ pvz ,✗ma+ pmz)7as/c-1,v1,Mi)+ PSIEZ ,V2,Mz)=✗Si+ psz S52/sldeni-PE.at.✗si+PS2.Seic-1✗C-e+pezEzEENERGYFUNCTION•S,BECAUSEOFTHEmonotonicit>WRTE,'SinvertiBLEiS=SIE,V,m)E=E(S,V,m ) •conSEINENTLT,EHASSOMEPROPERTIESTOOi1)E'SAHOMOCENEOUSFUNCTIONOFDEGREE☐NEWETallIndipendentVariables:E1)5,tu,✗m)=✗E(S,V,m)✗EIR2)EisamonotoneFunctionWatTHEInternalENERGY:IF51>52E (51,V1,M1)>E(52,V2,Mz)andviceversa3)Eissubadditive:E/51,V1,mi/tE/52,V2,Mz ) 7E(St+52,un+v2,ma+Mz)•MI✗INGTVUOSYSTEMTHEENERGYisLOWER02ATLEASTEllvalTOTHESUMOFTHE'SOLATEDSYSTEMSCONVEXISITYOFEWrtSe✗, p E112✗+po=1•EI✗51+ psz ,✗v1+ pvz ,✗ma+ pmz)E✗E/51,v1,Mi)t PE/52,V2,Mz)=✗Ei+ PEZ sC-2'dei+ PEZ.E/✗Set Pez )..C-1Si✗ Setpsz 52E SPECIFICENTROPY,ENERGY•From#OMOGENEITY:SIEM,VM,1)=S/E,V,M)se=a/E.V)e=e/si,v)MSTATEPRINCIPLE°LOCALTHERMODYNAMICSTATEISDEFINEDBYNINDEPENDENTTHERMODYNAMICINTENSIVEVA R I A B L E S / ACCORDINGTOTHEMATERIALVOLUMEMODEL)•GIBBS'aule:N=NUMBEROFREUERSIBLEWORKMODESt1°FORSINGLE-COMPONENT,SINGLE-PHASE,ISOLATEDSYSTEMS:N=2 SL =p dv EQUATION'SOFSTATE•anEQUATIONOFSTATEISTHEDERIVATIVEOFAFUNDAMENTALQUANTITY•BECAUSEOFTHEMONOTONICITYOFIwrtE:I=A/E.V)E=e(zio)1T=Il2sede= Td s - pdvde =seda +se dv vDIIV2VIP=.IlTosTHERMODYNAMICPOTENTIALS•THERMODYNAMICPOTENTIALSAREFUNDAMENTALRELATIONSWHICH✗l'0WTODESCRIBECOMPLETELYATHERMODinamica(SYSTEMe=e/sir)c-nero>de= Td a -pdrT=Ilp=-Jlastovss=Ileiv)Entropy tds -de+Pdr^=2sP=2sTIlv.TTve h = hII ,P)c-ntitalpr dh =1- da +vdpT=7hV=2hDAJPPsa=a / T,V)HELMUT-2Function da =-Ddt-PIUI=_DaP=_Da3TosuT I = 8/Tip)GibbsFunction dlg =- Ddt +VDPA=_DIrt= 283TJPPTshpe g VaT3 •THEMirrorsOFAPOSITIVEDEFINITEMatrixarePOSITIVE: 1)22C >oJTossaasvv•USEFULRELATIONS: 1)dh - de =Td d t r t d p-1-da+pdr=d/Pv)h=etpro>o>o2)de - da = sdt + Td s =dist)a=e-STMAXWELLRELATIONS Je =2se=-JP•enero: | »»→»»v2TJPgy=-asJe =2Il=2Tavarrastoastos•LET's☐C-TERMINETHESANERELATIONUSINGTHEMAXWELLSQUARE:2Dentµtip"°""1.CHOOSEaTHERMODINAMICPOTENTIAL^2.noveCLOCIC-WISE:THEADJACENTVariableISTHEDENOFTHEFIRSTDERIVATIUE.THENUMOFC & THESAMEDERIVATIUEISTHECOEFFICIENTAFTERTHEPREUIOUSONE.3.THEOTHERDerivatiUEis☐BTA/NEDMolino'~THEOPPOSITEVUAYG.IFTHEDENOMINATORIAREBOTHONTHEARROWHEAD☐aGOTHOPPOSITETOIT,THENTHEDENZzrTNUMZaCOEFFICIENTIareconcordant3•c-NTALPY:2T=IVspasapahp•Helmholtz:ID=IPe & ToITTVvaTeCribbs:IV=-INJTJpPT THERMODYNAMICSTABILITY•E(I,TT)CONVE✗ It (l),WHICHISTHEHESSIANMATRIX,15POSITIVEDEFINITE=-- aresiessaJoss• It/e)== JeJe Isto avea --•THEMirrorsOFAPOSITIVEDEFINITEMATRIXAREPOSITIVE: 1)22C >oJT0JAZsavv221èeèeIl >o Ìl >oJPAT O M I C(VLINEARGASnon-LinearGas.Cv=E=RTn1=RNTZT2NTr a n s l a t i o nal3e5r3R22NROTATIONALNTr a n s l a t i o nal3E3ar3ar222NROTATIONALa:NUMBEROFa:NUMBEROFNVibrationalAT O M SAT O M S•REMark:CONSIDERINGNTr a n s l a t i o nalandNrotationalASACTIVEMODESWEsai☐WEcanASSUMEN=5+SWEOBTA/~THESANERESULTS ISOCHORICSPECIFICHEAT•✗=(p=Cutr=r+1Cv=RCvCvCvf-1ENERGY•e/T)=(v(T-To)tCoifWEASSUMETHEFOLLOWINGREFERENCECondition:To=0Klo=oJ KI WEOBTAIN:e(t)=CvT=rte(T)=rt8-1O-1C-NTHALPY•h/TI=et PV =e+rt=✗rt+hoh/t)=XRT+hof-1f-1ENTROPY1)1- da =de+pdr=Cvdt+et duda =Cvdi+rdvsi/T. t l =RInTE+CvInT2VTTV1Te21h=e+podh=deirrdp+pdrr=1- da +vdp1- da = dh - Vdp =Cpdt-rt dpp•da=CPdt-Rdpse/T,-0)=cpInta-rInPzTPTipe3)LET'Sstartfrom1) VP •Tz= Pzttz ea/T,v)=rInV2+CuInPzrtza/T,V)=CvInPz+cpInV2TirparteV1partypoiV1SPEEDOFSOUND2•c=v3PT-JP=Vr=Tf-1+RT=v✗rtc=JETIT✓CvIVVev2v2TFUNDAMENTALRELATIONForPOLYTROPICCASES-- exp8-1(s-so)r•EliminatiNGTFromA/T,V)andC(T,-0)cives;e/A,V)=--lovot-1V vanDERWA A L SFLUIDSi•THEORYINTRODUCEDTOSTUDYMULTI-PHASEFLOWS1/•vanDERWA A L SPressoreElevationiPIT,VI=RT-av-bv2 .•a=27 è TC?b=1rtc...64Pc8Pce' atrn-CTIVET-OZC-C.SI eiBETWEENMOLECULES°2ivanDERWA A L SForces v2 • b :co-VOLUME:IT'STHEVOLUMEOCCUPIEDBYAMOLEOFPARTICLESIFUtoTHENPOPIT,VI=rt_a v2 v-b f ..ae/T, v1 = ¢/TI+ | Tdp-pdv = ott )= 01T)+ | ] , dr =Io(t)-aITyVOVOFROMRECIPROCITYRELATION•Cv=Il=0'IT)2TvSUPERCRITICALREGION ÷eiSPEEDOFSOUNDz- è = v2 'JPT-JPJT✓CvJV-T- { """="-a v2 no-b(T)=e+a=nrtcu=0'/T)=N § V2e=nrt-a2V --•(2= V2 'r2TZ-_rt+2a= V2 'rt2+1-2a=V-bnrIV-b)2 v3 .lo-b)2NV3---2=Vat2+1-2av-bnv2CZVDW=C ? ☐EAL+ ÉREPULSIVE tCATIRACTIVE222◦ / g,,a,>,greg,,,,>,ganga,,,y,o-REPULSIVEFORCESDEPENDOnTHECO-VOLUME:THEcareerISTHECO-VOLUMETHEMOREIMPORTANTTHEREPULSIVEFORCESBECOME.LINKBETWEENTHERMODYNAMICSANDCOMPRESSIBLEFLUIDDYNAMICSMOMENTUMBALANCE(DIFFERENTIALFORM)•3197)+_° /91≈+ PE )=°IT- SI -=setpressoreNEEDSTOBECOMPLETEDFROMTHETMDMODEL•UNKNOWN'SOFFLUIDDYNAMICSARECALIEDCONSERVATIVEVA R I A B L ES:WEHAVETOWRITEPRESSUREWITTHISVA R I A B L E'Sp=p/} , fa , Et )•THERMODYNAMICMODEL:p=PIT,VIP=p/e, f) =P Et -^a≥, g92 | e=e/t.ro'Et = ge +1µ≥2•EXAMPLE:POLYTROPICIDEALGAS PIT ,v)=RTVPll, f) =/6-1)se =(6-1)fEt -^≈≥92 | e,,,,a,o-1 SIMILARI TYTHEORYITTHEOREMM:NUMBEROFPHYSICAlvariaBLES•M-✗=2✗:NUMBEROFFundamentalUnitsa:NUMBEROFDIMENSIONIESSGROUPSEXAMPLE:NEWTON'SLAWOFMotionF=mDVdt•m=GIF.n,-0,t)2=1◦✗=3 / MASSM,LENCHTL,TIMET ) F= FOÈ n= morì ~e0=SCALINGVA LU E.=SCALEDVA LU E | ,.ro ,t = tot DIMENSIONLESSCROUP•FoF-=morìvoduÈ =mortoindir todtFoto di •REMAREi☐0NOTFORCETOSCALEINITIALANDBOUNDARYCONDITIONSASWEIL•REMARE:IFSCALINGVA LU E SARECHOSENSOTHATDIMENSIONE55GROUPSAREUNITY,THEEXPRESSIONOFTHEPit>51calLAWisu-CHANGEDinTHENEWREFERENCE:RELATIONSEXISTAMONGSCALINGVA LU E SNAVIERSTOKESElevati0ns79+-◦ 197)=0set {s''≈'+◦ 'sea-+ PE' =◦'Tremare:volume-oacesareneaectei--=at-- set +,oµ/c-'-+PI=-◦ tutt --◦9-=set-_◦UNDERTHEASSUMPTIONOFanisotropicanewtonianiFluidi Ttf , SI , Et)= un '≈+( 1) t'+✗/ o≈ ) --. MASSCONSERVATION~• 9=9051= nomit= tot ✗=LoÌ =1----Lon=4 I9,1 ,t,±)•IT-THEOREM✗=3 ( L,T,M)a=1 SÌ ~• So +Sono_°/giù)=0tosi Lo•SCALINOSTROUHALNUMBER:Sto=LO2=1tono•Sto2} +°(§Ù )=OSCALEDMASSConservationElevation-se°REMark:IFno=LoSto=1:THESCALEDEQUATION,ASWeltASTHEBOUNDARYCONDITIONSAND'NITIALDATA,toAREC-✗PRESSEDUSINGTHESAMEMATHEMATICALEXPRESSIONSASITSUrSCALEDCOUNTERPartMOMENTUMBALANCEet- tot±=LoÈf = poÌI=no ÈP=Poèµ = Mori ,✗- puoi n=6 I9,1 ,t,±,P, µ ✗)-IT-THEOREM✗=3 ( L,T,M)a=3•3191)+,° /91I+ PE )=°IT,TI=µ '≈+I1) t'+ il ◦≈ ) -=-at-.◦ Sonoa/II)+Sonèi ◦(fuie) + POIè =uomo io µ - Iè+1IE) '-'+ I "◦ è} to se loLO LE --•MULTIPLYEACHSIDEBYLO:Sono≥--~.•Lo215in)+.◦(suiE) +Po' è =uno◦ {èIè+1IE) T+ Ì "◦ è} --tonoseSono ≥Sonolo__ °SCALINGStroutalnumber:Sto=Lotono•SCALINGEulernumber:[no=Po2=3 Sono '•scalinoREYNOLDSnumber:llo= SonoLo ✓o•SCALE☐MOMENTUMBALANCEC-levati☐nsi~.Sto Silja)+o /gèE) +Eno" è =1_• {ju ' Iè + IIè/t'+ Ì 'o è} -- setreo._•REMArk:sto=1,PO=1,Reo=1THESCALEDMOMENTUMElevati0nsPRESERVETHEMATHEMATICALFORMOFTHEUNSCAIED☐NE forzo >•☐NLYM-a=3SCALINGVA LU E ScanBECHOSENarbitrari(Y.THEremaining3ARECOMPUTEDFromTHEunitVA LU EOFDimension(ESSGROUPSEXAMPLE2""""p.So,NO,LoareCHOSENNO,PO,LoareCHOSENstoC-no=Pogo=Popo=Ponno=→ È90nè90µFv02POZLOZpolo>reo=90noloreo= SonoLo =uno nè -oNo= pomolo no=Momo=Polo→o SolopolounoTOTALENERGYBALANCE•t- tot±=LoÈg = poÌa=no ÈP=Po̵ - Mori ,✗= puoi K=KoÈT= TOÌEt =poÈt TOTALENERGY2• È = glt = Met = f E+1 È :TOTALENERGYPERUNITVOLUME:JIl~,=N.m=N✓2n}M2ENERGYperunitmassn=7 / ±,-1,1,PEt ,-✗,K,T ) •IT-THEOREM✗=3 ( L,T,M,@)a=3 --_• Jet +,oµ/c-'-+PI=-o tuIl +oKT,TI=µ±+I1) t'+✗| ,o =) --set--_.--z-•PoDE '+Pomo-o è/Èttpl =Uomoo luiÈ )-koto [ o è - Ì -.-to at Lo-_ Lollol °MULTIPLIEACHSIDEof(oiPorro- set -o è/Èttpl '_o luiÈ )-koto'o è - Ì Lo+,=unonomotoset --Polo-polonio--sto2unoSonoSono?Koto=Sono?mokoto Sono LoPoPoSono>LoPoSonolomorrà1Reo1Reo1Pro2°SCALINGStroutalnumber: Pro =unoCPOWITH[po=noKoTo•SCALINGEulernumber:[no=PoSono '•scalinoREYNOLDSnumber:Rlo= SonoLo ✓o•SCALEDTOTALENERGYBALANCEEQUATION:---sto set -o è/ÈT +Ó)+,=11o luiÈ )-111~o è _ Iset -_reoEno'=reoEnoPro--REFERENCEVERSUSSCALINGDIMENSIONLESSGroupse'~SimilariTYTHEORY,ITISSTANDARDPRACTISETOREFERTODimensionLESSGroupsELALLATEDinREFERENCECONDITIONSRTHATAREREPRESENTATIVEOFTHEPROBLEMATHAND.REFERENCEStroutalnumber:str=LatrnrREFERENCEREYNOLDSnumber.. Rcr = SeMrLe uneREFERENCEPrandtlnumber:Pra=MrCprKRREFERENCEMACHnumber:Mar=MrCRremark:inGeneral,REFERENCEVA LU E SARENOTUSEDASSCALINGVA LU E S •EXAMPLE:REFERETZENCESTATEAT00,MeasureinGM'MPERTURBEDFLOWCONDITIONS fa = SoÌoreco=Scorronolauso= Morto ChaoReo= Sono lomenoreo=reorèa ( •=,.ae,rèao = StoriaiosiaLa=Lo Ìco ◦War ning:GOVERNINGEQUATIONSANDBOUNDARYANDINIT/ALDATAMUSTBECONSISTENTGEOMETRICSCALINGOFTHEEULERC-QUATI0ns39+,◦ 197)=◦sig+◦ 1971 =◦set-set | s'sa)+◦ 'se÷+ PE'=◦it--=setDI97)+◦ /91I+ PE )=0-_-atset +,oa/Et+PI=-◦ intl-_°9-pe,_aeac,eaua,,◦,t= tot ,≤=LoE / """""""""""~Lo75 +◦ Ifrit =0toset-losiÌI )+'◦ IjetIn + FE )=0-to set -- set -lo+,o Ie/ÈT + ÒI =0to at -_•EULEREQUATIONSAREinvariantWRTTHESPACIALSCALEinTHESTEADYCASE02IFto=(oinTHEUNSTEADYCASE ACRITERIONFORCOMPRESSIBillTY~-z-•.° In = Cè -me?ino- È - Ì . È +celeMr III -Sr25 ' se +(22Ecrtr.2 It9-GRAVITY--EFFECTSWNSTEADINESSCOMPRESSIBLEEFFECTS mi '- è ☐ i| ' nè ~' zt '- Sacriè o_ È ?+Uv+^_° È)-+1Trza-I+1 gcz 3nrPrRer.Ti Rer -ITpCprtrI\SHEARSTRESSESISOTROPICCHANCEOFVOLUMETHERMALHEATFLUXESExpansionVISCOSITYTERMSTHECONCLUSIONISTHATITWOULDTA K EAMIRACLETOHAVEUNCOMPRESSIB/LITYiEACHOFTHESEFactor'sisRESPONSIBLEforCOMPRESSIB.'l'TYEFFECTS BOUNDARYLAYERGOVERNINGElevati0ns°HYPOTHESIS:STEADYFLOW•NAVIERSTOKESEILAT10nsi39+,o 191)=Oset {sise)+.. /se /¨+ PEI =•it-=set-- set +,oµ/Et+P)=-o intl-,o9-=set--BOUNDARYConditions•VECTOROFconservativeVA R I A B L E S:W=(f , Gry ,[£)tzioÀ=o-1)Woo=WooNeumann'sConditionsToin=O-CONSERVATIVEVA R I A B L ESOFTHEFLUIDEVALLATEDATTHEWA L LAREC-cavalTOTHECONSERVATIVEUARIABLESOnTHEWall2)Il=UNNOSLIPCONDITION:THEVELOCITYOFTHEFLU/DATTHEWallOFTHEBODYMWSTBEEllwalTOTHEVELOCITYOFTHEBODYvu- 3)Tw =TworToÀ w=-9W-KDOUBLE-DECKTHEORYOUTERREGION•Tr i c kOFMiltonvan-DICKiUSEDIFFERENCESCALINGVA LU E STOMATCHTHETWOREGIONSinTERMSOFFLUIDavantiTIES •DIMENSIONLESSSCALINGGROUPS:Seo=Lo=1Io=LoEconomoEvo=Po=1Po= format 2SONOAlla= $0 MMA(0=Alla=RMMr(rIFWEconsideraTHEREFERENCEConditionASTHEONEATINFINITY:amochr~ Pro =→☐no≥= Prr =→rur≥reso=reorèarèoo =1 Pro = ProProPico =1KOTOkrtr•THISISTHESMARTESTCHOICE,BECAUSEITALLONSTOSTOPFOCUSINGONTHEBCATINFINITYWEALREADYKNOWTHAT ÓE =1ForEVERTFLOWANDEVERLYFLIGHTCONDITIONATINFINITYANDALSOTHEVA R I A B L ESWHICHAPPEARINTHEREYNOLDSNUMBERAREUNITARYTOO.SCALEDNAVIERSTOKESEQUATIONS-◦ IÌÌ)=0-Reo=Rea=Reco'. /fuiin + IE/=e'◦ ÌT --=reso Pro = Prr = Pro -- i ◦ ù/È' -+ ÒI =1'◦ in:-) +1 Io E ' Ì ----RenRenfroSCALINGVA LU E S°THEDIMENSION(ESSSCALINGGROUPSTHATWEHAVECHOSENALLONSUSTOWRITE:reo=Reoèe = ÌaoùaoLà =1 fà , nè > LÀ,µÒ= 1 mioPro= ProÈ =miaouèIotè→•/ ùa ,KÓO, Tè =efa= Sofà90=90mio=no MÌ mo=MIO22Pao=PoPÌOPo =Po=SooMao~2=SooMao pàSoociaoMao- morìauno=MaoKoo=KoÉOOKo=KooTa o= TOTÒTo=Ta o BOUNDARYCONDITIONSHP è =In=oWMI•Wall: | , w, ,y ,yw, g , aaaaa ,,,Ta o fino =1 né =MIO=1. NÈ versarOFTHEVELOCITYMI•00: {g. =, TÒ =1METHODOFASYMPTOTICC-✗PANSIONS00INTERFACE✗INNERREGIONyOUTERREGIONXINNERProblem:THINinYDIRECTION°THE✗DIRECTIONIScomparabileinTHETWOSCALES,VUHILETHE4DIRECTIONISNOTOUTERPROBLEM:THICKIN4DIRECTION•WEASSUMEa☐IFFETZENTYINTHEINNERPROBLEMTOMAKETHESCALESCOMPARABLEALSOinTHISDIRECTIONOUTERPROBLEM• Se = Se /×,y) Il = Il/×,y) Pe =P@(✗,y)WITH✗ZETHEINNERPROBLEM:✗✗yYi4={Y ,E=1Reo•7=1774Esi◦DROPPINGTHE"I"FORsimpliciter:I/9th)+1I/f V)=osxEsi◦LET'SAPPLYTHEASYMPTOTICE✗PANSIONS:si' (fot{91 +[ 292 +...) / no+Eun+ Elena +.../'+17' (fot{91 +[ 292 +...) / V0+EV1+ EZVZ +.../'=o2XETY.-----.-si fono +E/fino + font /+...+2E-^ /foto )+ /fatto + forte) +E/Sorta + fatto + fiori)=0DXTY-_--◦LET'SGROUPTHETERMS: |9000 =constaconoy È2(porto )=0SINCE go >0:V0=074Bc:V0(✗,4=0)=0 È2(sono/ +3 (furto + forte )=o7(Sono/ +3 (forse )=O2XSYIXsi è2Ifont + fino/+7(four + fatto + ferri) =o2Ifont + fino/+7(fovztferri) =0sx74sxsi°REMARKiTHISISACASCADEPROCESSWHEREEVERYRESULTISUSEFULFORTHENEXTEQUATION MOMENTUMBALANCE-InnerFLOW✗-DIRECTION È2/gonovo)=027È2Igono '+Po)+2/gomorra)+7lSavona) +2(govone)=2zy10Tu oDX272737ZYy-DIRECTION È a |gov? +Po ) =o3Po=osisi- È a/portono) +2(governo + govoni + 91V02 +Poi)=2' /zero +to)Ivo2Pa=osxOYOY.27.TY•remare: pl ✗,Y)=Po+EH+ ÉPZJP= Tipo +E3Pa+ È2Pa aisetItatTOTALENERGYBALANCE-InnerFLOW•HP:POLYTROPICIDEALGAS:P=grte=RT(p=2h=frf-1JTp8-12{°Cp go mosto+V1sto=no3Po+✓☐Inot2KoItoa✗rgy2x2>2727 BOUNDARYLATERElevationsV0=0e'B.CiPOY00WEWEEDTOKNOWtoATTHEINTERFACE3Po=0IT3lfono) +2/forti)=0DXTYsi/Son} +Po)+2 /fononi) =2DXsi,y,1°>no{°si2Pa=0è2 go CPnoTTOtV1ITO=no2Pa+ µTu o+7KoITODX7TDX77DX37To/✗,0)=Tw•Bc:no|✗,)=☐,V1|✗,0)=☐WEWEEDtoSPECIFY6BC fo ,no,Pe,-10YcoMASSCONSERVATION-OUTERFLOW• selfie)+ altri)=o sisi e2-(go +E 911/ no+Eni)'+2' /go + Cfr ) / V0+Eun ) '=oDX--74.. Èzigomo)+a/Sotto)=0sx74E'a/four + fino)+si /9001+ Siro)=0DX74 MOMENTUMBALANCE-OUTERFLOWg-èro _ In +' Ì =e'☐ È ---=reco✗-DIRECTION:{° Sono 2no+ Sotto2notTipo=o2xseySixTo t a lENERGYBALANCE-OUTERFLOW- È2no /Eot +Po/ '+2'vo/Eottpo) '=osx..74.-EULEREQUATIONS°SETOFEQUATIONSFORTHEOUTERPROBLEMINWHICH{°2/Sono)+a/foto)=osx74•REMARK:THESEEQUATIONSareVA LIDFORaVISCOUS,2(fono ?+Po)t2/fottuto)=0CONDUCT/NCFLUIDFLOW.sxsuIT'SNOTTRUETHATTHEY'LEvali☐Foran/NUISCIDFLUID:SIMPLY,ATE°,THEWISCOWSTERMSDOnotCONTRIBUTE2/gomito)+2(gov} + Po)=oSexsu•REMARKiVISCOWSandConducttwittTe r m sARENEGLICIBLEup---2no /Eot +Po/ '+2V0/Eot +Po) =otoE'DX..74.. INTERFACECONDITIONS•LET'SANALYZETHEINTERFACECONDITIONSFORTHEC-✗EMPLARYQUANTITYM: NÉ INTERFACE= NÉ INTERFACE----eee•. mio/✗,Y)+ Emis/✗,Y)+E> NÉ/✗,Y)+...-interface=-no/✗, 4) +En1/×,Y)+E>M2/✗,Y)+...-Interface-_--eee•line. mio/✗,Y)+ Enit/✗,Y)+E> nè/✗,Y)+...-interface= live.no/X,EY)+Em1lX,EY)tE2n2lX,EY)+...-Interface0000INNERPROBLEMOUTERPropter=•THEEXTERNALSOLUTIONISEXPANDEDUSINGATAY L O RSERIESaround4=0i { " " """""="≤"""°"""° " "" """ " " (y-0)34'✗'°'2 rgyz ix.012 ne µ?/✗,{Y)Em ? I✗,0)tEYI12 EZY ≥IM ? It'sanExpansionOFanExpansionrgyIX.01+^ rgyz IX.01 neèy ≥ simeNE/×,EY)= MEIX. 0)+EY7a+1a☐y(✗io)z☐ 42 IX.01•REMARK:ne= ME + Enel t Eauze -.-•line_ mio/✗,Y)+ Emis/×,Y)+E> nè/✗,Y)+...-Interface=lineMEIX. 0)+EY7 EZY ≥ ÒME oargy" { ix.01" I rgyz ix.01+-22-+1+12 EZY ≥7 ne +En ?IX. 0)+EY7n ?Ei ≥7n ? + ÈNE '✗io)+E>I2yy,2ix.◦,sey"°☐y(✗io)z yyz IX.01IX.01-INTERFACE-.-•line_ mio/✗,Y)+ Emis/×,Y)+E> nè/✗,Y)+...-Interface=lineNE '✗io)+EY7 ME ++ix.0,m ?IX. 0)••74-2-1,+ MEIX. 0)+ è ◦+ Y 7 ne +... YZ I Ù (✗io)2 74 2rgyIX.01-INTERFACE Ècinuè/×,Y)= MEIX. 0)00E"cinuè/✗,Y)= lingYI (×,◦,+ UE (✗,0)INTERFACECONDITIONSseymi 00212Y≥3 neèlignina /×,-1)= ein ◦+ Y 7n ? + miix. 0)342 1×101ryyIX.01 SOLUTIONPROCEDURE 1)[ ^ Voi/✗,Y)=0nsEas.OFTHEInnerproblem-- 2) {° liz _ Vol/✗,0)- VÒ l✗,Y)-=0 NÉ/✗,0)=0zeroNormalVELOCITYConditionSLIPConditionnoPENETRATIONTA N G E N T IALVELOCITYcanBEANYBUTNORMALVELOCITYONTHEINTERFACEISNUKWEOET: gol , NÉ , Vol , Pol3)saidpoe =WEGETTHEBCFOR go ,MOTO,POATTHEINTERFACESOLVETHEInternalFLOW:WEGETasasxDXSOLUTION Vai {° 4) V1canBEUSEDTOSETTHEBCATTHEINTERFACEASWEBIDATPoint2)5)ITERATETOTHEOLDERWEDESIRE:THEHIGHC-2ISTHEOlderTHEMOREPRECISEISTHEAPPROXIMattonTHERMODYNAMICSCONSISTENCYg/×,y)= gol ×,y)+E fa (✗ 141 +{ 292(×,y)+... { """""""""+"""""""""""""WEHAVEMADETHESEEXparsi0nsTI✗,y)= To/✗,y)+E-11(✗ 141 +E> Tz(×,y)+...ISTHEREST'llconsistente>b.C-TWEENTHESEFormulati0ns?1- IX. y)= TIP /×,y), 9/×,y))•T=T/Po+{Poi+E>Pa, go +E91+E> fa)= TIPO , go)+2T(p _po)+2T(g-go)= OP299Pago PPo .joTIPO , go)+IT(po+Eri+ ÉPZ -Po)+IT(go + Egr + E'92 - go)= OP299Pago P Pago =TIPO, fo)+E |2T.poi+2T. 91| +E' |2T.pz+2T.92)apsigapasto 9Pago P Pago9Pago PPo .joTiTz ACOUSTICSGOVERNINGEQUATIONSEULEREQUATION'SFORSMOOTHFLOWSNOSHOCKWAV E SMASSConservation 29 +no_ 9 + 9 °1=o--atMOMENTUMCONSERVATION f Dei+p=ott- gg su+Io±+1g-P=o--=☐tatTOTALENERGYConservation+Il0-e+Pg☐e=-P-o=+1 ? -lore-_° ?seg _°Il=o☐tat•LET'SINTRODUCEaPERTURBATIVIAPPROACHi gix.ti= f +g' lx.tl| ±ix.t,=±+±,/×,+,e ix.ti=e+e'/×,ti•EULEREQUATIONSBECOMEi79'+ le +±')o.9'+ /f-+9 ')-°±'=oset(sei + le +±''._±'+-e=.otg-+9'de'+1è+='/°_e'+P-°E'=oset9-+9'•HYPOTHESIS:1)REFERENCEFLOWISSTILL: È =02)SMALLPerturbati0ns:f' 5 •p'(e',g')= 18 -1)/ g-e'+ s'è) 29'+g-◦ È =o129'+◦ È =o--atssetsui +^P'-◦ { 0ª'+^P'-◦sit 5 -sitg--• /µ ,p.. , ,,se'+P-◦ È =0atIatI •INTHE1ˢᵗand3ʳᵈEQS.WEHAVEOBTAINEDTHEDIUETZCENCEOFTHEPERTURZBEDVELOCITYWHICHISATLELELANTPARAMETER.LET'STAYTO☐BTAINTHEGAMEWITHTHE2ndEQUATION:-_-° DÀ +1P'=o2I_◦≈')+ =p '=o11)sitg--se-S-_""""°"°"°""""€"""€""/¨arbitrarifunctionh(t)canBEADDEDTOQWITHoutMODIF>INGE'•LET'SDERIVATETHEElevation:72+È=29/=Ostag-set•FROMMASSGoverningElevation:129"t0 È =o- Sst . 50 +c-2_1amg---'=o 50 -c-2 20=0stag-sta-0AllIwanTITIES } ",,P"SATISFYTHESAMEWAV EEQUATION ☐NEDIMENSIONALWAV EElevation Ó -C-= Ò LinearPartialDIFFERENTIALEQUATION•y✗z=°staF/5):IRIR, §/✗it)=✗-Et•LET'SDEFINE:0(×,t)=F/✗-Et)tG(✗+c-t),with:GIR):IRIR,2/×,t)=✗+Et°remare:gothFANDGarescalareFunction'SOFaSINCLEScalareINDIPENDENTVA R I A B L E0/✗it)=F/glxit ))+o /Il✗it))•20=F'+G'g,=F'/ g)75+0'/2)dasxsx. 50 =F"/ g)35+o"/2)DX=f-"+o"ziasxDXa20=F'/ g)35+0'/2)DX=-EF'+cio'atnotata 50 =-EF"/g)35+E0"/✗132=E>F"+c- 26 "at'atat•Finale>,WEGET:[=F'"tÈ>G"-C-=(F''+G")=0Identity•0(×,t)=F/✗-Et)+6(✗+c-t)•µ'=_0=F'/✗-Et)+g'/✗+cit)•STARTINGFROMTHEMOMENTUMEQUATION:-g-'•30+c-= gi =og'=- 5 20= è/EF'- cio ')=e.F'✗-ct-0'✗+ct.segc-2sec-a--•Ia'+_P'=O_'20+P''=o20+P'=o P' =-g-70=g-c-.F'✗-ct-0'✗+ct.sitJ -setè .Itè set•byIntroducing: f/✗-Et)=F'/✗-cit),6/✗+Et)=-G'/✗+cit)we0,THENM>OINTHEPOSITIVEDIRECTIONTHEFLOWISPUSHEDINTHERIGHTDIRECTIONCASE2: f =0•LEFT-RUNNINGWAV E S " """""= "" """|" ""=↓"""="°" ( n pi = gczg 'g= casi|g '=_g- è .""""""=" " """c-P'lx.tl=pc'g ✗+ctIi =-c9'p'=_giu'Sp'=- scià •P"=( 29 ':IFIREDUCETHEVOLUME(9"),THENP"iP'=-g(µ':IFP">0,THEN NICO THEFLOWISPUSHEDINTHELEFTDIRECTION ACOUSTICIMPEDANCEP"=f(µ"THEACOUSTICIMPEDANCE [[ISTHECONSTANTOFProportioneCITYLINKINGVELOCITYANDPRESSUREPERTURBATION•LET'SCONS/DERARIGHT-runningWAV E:--_--pa.siFLUIDS-KIM}-C-MS-SC-M-controlvolumeHC0,16641007167,6Il= È +Il"Air1,205343,3413,7P=F+P'=f-+f-c-µ'CO21,839266,2489,6/iCHG0,594446,1265,0H2o/LIQUID)998,41484,1148- 10º n=In=o Hg /LIQUID)13-600145019,7.10°P= è •il"=P":AFTERAPTZESSUREPerturbati☐n,THEHIGHEEISTHE'MPEDENCETHELOWERISTHEConsellVENTVELOCITYperturbatiUn9-c-•example: §èlHe)< §È(Air)il"He> À AIRIFWEINHALEHELIUMWECETanHICHERFREQUENCYTONEOFVOICERIGHT-RUNNINGSIMPLEWAV E SP"'INITIALSolutionat[=0:P"(✗,0)=Po'(X)tcit-_ È •n'ix. 01= mio lx)=Po'/✗I--9-egtx✗°•9'/✗io)=So'/✗)=Po/×)c-2gtx×"gyxt ~gy.it >•P'/x.tl-gcafx-ct-gcaflslx.tl)✗0✗,✗1✗2✗3✗a✗• P' =constIF5/✗it)=const✗-cit=constconst•LET'sconsidera✗It>0)=✗☐:✗=ct+✗oP"/✗it)=Po'(§/×,-1))✗-Ct--'SEt°THEDISTANCETHATaPOINTOFTHEWAV EDOESin✗-DIRECTIONinTHEIntervalt=_OiÈ' ✗[☐iacramsiRIGHT-runningSIMPLEWAV E St°THEX-tDiagramisbyDefinitionWTZONG,B.ECAUSETHISISINVERTED"EÉ '✗=✗(t)✗ISTHEDEPENDENTVariable,WHILEtISTHEINDEPENDENTONE,+E¥ +[BUTinTHISGRAPHWEHAVEINVERTEDTHEAXISUSINGXasTHEINDIPENDENTVA R I A B L E+ È "+[=✗THEHIGHERISCTHELOWERISTHESLOPEC•THESELINESareCALLEDACOUSTICLINES'_..ABC✗1stPROBLEM:SolutionATAGIVEN È FORall✗tEÉ '1)pressoreDistributionatt=0MUSTBEKnown+EÈ +Ei +è2)LET'SFIXt=ÈinWHICHWEWar tTOEVACUATETHEPRESSURE+[=ÈPa'.Pà.Pi. 3)p'/x.tn)=Po'(gli,È))✗-cepo'(✗alEXAMPLE:p'l✗A, EI=Po'/✗a)'_..ABC✗ 2nd PROBLEM:SolutionataGiverÈFORalltt+=+=×"+canBENOTSYNC#20miZED,NEWETLTHELESSTHE>30THOBEYTOTHEWAV EEQUATIONSINUSO/DALWAV E S✗iWAV E L E N G H TSo:AMPLITUDEP- f/✗-cit)=-So.sin21T ( ✗-Et/'✗✗---eP'= g (2 f/✗-c-t)=-g-c-2Sosin21T ( ✗-it)'✗----•µ'=Ef/✗-cit)=-ESosin21T ( ✗-It)✗--ttt--•Gp/t)=E|flx -Eclde=E | f/ ✗PIO)-c.c)di=-Eso / sin21T ( xp/0)- ETIde=✗to_-to=plotcto=plotc-__-t-=-ESOt._1-cos21T ( Xp/0)-II)=Sot/ 1-cos21T / xp/0)-cit)' } 21TE✗21T✗----to=plo)c--|iDISPLACEMENTAMPLITUDEMAXDISPLACEMENT3×Il21321 QuaeTa o,talti=c- | , f/pit)-c- e)di + fde}9maxMax II. QMAX2799Max=9(t-Tz)Imax¥9LET'SSUPPOSE✗☐=Xp/☐)-•9Max=q/T2)= Aq{ 1-cos21T ( Xp/0)-ET) [ | 1-cost | 9Max=2^-9a= Aq| c-T-•CONDENSATION:S=9-9=9"= f.(✗-cit)S=9-9=9"= g/✗-cit)ssssRIGHT-RUNNINGWAV ELEFT-runningWAV E•FORACOMPLETEWave:S= &/✗-cit)+ g/✗-c-t) ACOUSTICAPPROX/NATION✗:WAV E L E N C H T•LET'SCons/☐EraSIMPLEWAV EWITH:SoiWAV EAMPLITUDE"WEWar ttoveriFYIFTHEPROPERTIESOFTHESimdLEWAV E SRESPECTTHENSGOVERNINGElevati0ns'THEN,THEFOLLOWINGMAGNITUDEEST/NATIONSHOLDS: Sg ~9-Sog'ix.ti= gf ✗-etSP - gczso P'/x.tl= gcznf ✗-CtSunc-Soµ'/x.tl=c f ✗-ctSx~aSt~✗CcontinuiTTElevationIS+goil+µo- g =omonodimensionali29+gIn+µ39=o-atatSexSex•9SoC+9-E502 +9-c- 502 =o✗a1MOMENTUMBalanceElevationIll+µo_µ+1-P=4→'Inmono""""""In+µIn+1Jp=4→'Inset9 3SSexatSex9sx3SSex•WITH:4Ne'=4 31 + Nero 3•c-250+c-2502+c-250=→'c-So✗×✗g-✗a•ASSUMPTIONS:1)So~ IÓC ForordinariAUDIBLESOUNDS SÉ CANBENEGLECTED2)FORDILUITEGASES:un"IN~}C--MOLECULARnearFREEPATHiMEANDISTANCECOVEREDPotaPARTICLEFROMACOLLISIONTOTHENEXT 9SoC+9-E502 +9-E502 =o9SoC=o✗a1X I c-250+[2502+c-2509[1Eso1+1=IHavedeviDEDB>È>So7,ATEACHsi☐E✗=g-✗✗✗2 | .a.☐se.nu..ee:"=^microscopiCSCALE✗macroscopicoSCALE•AlainSTANDARDConditions:1=6,9.10-&MTOGETKn=1:I=C=IV=C=5.10"HZVknIFREQUENCY1C-✗CESSIVELYHIGHTOBEREACHED'~STANDARDCONDITIONS•THISMEALSTHATINSTANDARDCASESVISCOSITY'SNEGLIGIPOLEiSOUNDWAVESPROPAGATESWITHNODISSIPATI☐~noCHANCEinFREQUENCYOVERTIMESPHERICALWAV E S•THEY'REEXACTLYTHEWAV E SOFOURVOICE•LET'SWRITETHEWAV EEQUATIONWITTHEvelocityperturbati☐~/~SpiteriCalcoordinateS:I>①-è2==OstaSPHETZICALCOORDINATES. 50 -c-=17v230=o 2201 -c-2zr20+i. 2320 =osta1-2serJrsta+2Jrsera•I>¢_ è [ s' 'dtT / ◦T9T0 SinusoidaleWAV E•S=SoSinnotin=21TWT=21TT,T 52 =1 | ,52dt=1 | " SÌsin'wtdtTTOCOSZX= COSZX -SIMZX=1-251IXsin?=1-COSZX2 |SiriexDX=✗-1 | coszx=✗-1sinzx=^ ( X-sinxcosx)222GZ-T-1• È =S ?| ; siiwtdt= SÌ/è 1- costante)dt = SÌ/ ,aw-sin/wt)-sin/w-0) :| = SÌ TT21T22•e=9C> 502è 2DECIBELSCALE•REFERENCEVA LU ETHRESOLDOFHEARING:①e,.gg=10'"≥Wm2•WEBER'SLAWOFSensation:HEARINGISProportione-1LTOTHECOCARITHOFTHEINTENSITY•=10/0810①eSOUNDINTENSITYINDECIBEL①lref0dB:THRESOLDOFHEARING | yg,gg,ga,☐anag, NONSIMPLEWAV E SIn-[=In=oGenericvaria@LEW(✗it)sett2×2WIX,t=0)=wo(X)2°OlderElevation2BCv/✗,[=0)=Vo/✗Izo=duedt | ........>•=.I+c2I=ose/×,-1)=SI✗-c-t)atSexatSexatSexse/×,ti•REMARK:I+c)W=OW= &/✗-[£)remare:apureTr a n s p o r tElevationHASTHEFOLLOWINGSHAPE:atSexI+K)WHETLEKISTHETr a n s p o r tVELOCITYatIxJ-c2w=ow=g/✗+It)atSexIn_con=S/✗-cit)atSexsourceTe r m☐wI✗,t)=W"/✗it)+nel✗it)HOMOGENEOWSPROBLEM•THEHOMOGENEOUSPROBLEMisaSIMPLEADJECT/OnElevation,WHOSESOLUTIONISKnown:DW-C2W=☐un"(✗,t)=W/✗+[[)atSexFORCEDPROBLEM✗+cte3W.caw=SI✗-cit)WP(✗it)=K |SIG )d} WHY?BECAUSEISTHEONETHATWORKSBETIERsetsx✗-Ct----• zur =K.SI✗+c-t)-s/✗-c-t).2W'=KE.SI✗+c-ti+s/✗-c-t).Isexat----•KE.SI✗+c-ti+s/✗-c-tl.-Ek.SI✗+c-t)-s/✗-c-t).=SI✗-it)•ZKÈs(✗-Et)=s(×-cit)K=1ZE ✗+ct•w IX. t)=W/✗+c-t)+1 |515 ) di GeneralSolution2C✗-Ct | """""""""""""""°""nel✗,t=0)=wo(✗)W/✗)=wo(✗)--rt/×,[=0)V0/✗1ÈW'/✗I+1.SI✗+c-t)+s/✗-cit).=V0c-wo"(x)+SIX)=V0/✗I2S/✗I=V0/✗I-C- Wil ✗)✗+ct--✗+ct✗+ct•w/ti=no/✗+c-t)+^ Volg)-c-wo' 151dg =no/✗+It)-1_Eno/5)-×-ci+^VOIG) =ai|aizè| ✗-Ct✗-Ct✗+ct=no/✗+cit)-1 un' ✗+it'+'"◦'×-it ; ""'s' =22ZE| ✗-Ct--✗+ct•W/✗,t)=12-wo(✗-It)tWol✗+Et).+1 | V0 (5)DS SOLUTIONOFTHENorSIMPLEWAV ECASE✗-ctSIMPLEWAV EnonSIMPLECONTRIUTEWIX,t)=F/✗-cit)+o/✗+c-t)• | E/✗-c-ti=eno/✗-it)-1-ò/✗-it)22Cwitered-'o=V0/✗Idxo/✗+Et)=1no/✗+cit)+1-io/✗+Et)22Ct.E, EiTHEFIRETERMOFTHESOLUTIONISanAV E R AG EOFTHECONTRIBUTIONSOF0-ELEFTANDOWERIGHTSIMPLEWAV E S+ ix """"+:on0nFINDIPENDENTVA R I A B L E3)ACHARACTER/STICCURVEISTHEINTEGRALCURVEOFTHEVECTORFIELDofDIRECTIONSal☐noWHICHTHEDERIVATIVEOF if isUNDEFINED •THROUGHaPointPASSESINFINITEDIRECTIONS.LET'SCONSIDERAPOINTANDDEFINEAVECTORFIELDCONTAININGEUERYDERIVATIVEDIRECTION.•ACCORDINGTOHP}ThereareSOMEDIRECTIONWHERE & ISNOTDerivabile-anINTEGRALCURVEISacurve(☐CALLYTA N G E N TTOTHEVECTORFIELDinTHISCASETHECHARACTER/STICCURVEISTHEINTEGRALCURVEOFTHEVECTORFIELDofDIRECTIONSALONGWHICHTHEDERIVATIVEOF if isUNDEFIREDG)THECAUCHYPROBLEMFORTHEQUASILINEARPDEISILLPOSEDIFINIT'ALDATAAREGivenaLONGACHARACTER'STICCURVEy"°"",aa.a.www.aa.o.aa.a,•✗= fila),y= fy/DI ✗•HP:se✗y=Y/×) f = f/✗,y)F/✗)==p ( X,Y/✗))nonWEHaveascalarefunctionFFunctionOFONEScalareINDIPENDENTVA R I A B L EXal✗,4,f)a / ×,Y/×),Flx))=a/×,F)b/✗,4,f)B/ ×,Y/×),Flx))=B/×,F)CI✗,4,f)[/ ×,Y/×),Flx))=C(×,F)•a/×,y, f)If +b/✗•y, f) > f +c'✗ 14,7)=o✗,>(×)DX×,Ylx)24×,Ylx)•al✗,F) If +B/×,F)If tc/×,F)=osxsryX,Ylx)×,Ylx) •ASSUMETHATTHESLOPEOFTHEcharacter,STICcurveis:DX=B(✗,F)dxaIX.F)THISCHOICEISCorsiSTENTWatTHEEXAMPLEOFSIMPLEWAV E S30+E00=o't✗l'×4'a/×,y, f)If+b/×,y, f)If +c/×,y, f) =ob=Esetsxasxè34oatc✗Ylx)•df=df/×,y)= If +ofdi= Of +Bix,F)ofdxDXIxsudisxa/×,F)rsy✗,Ylx)×,YIX)×,Ylx)×,YIX)•(C-T'SDivi☐EBYA/✗,t)ATEACHSIDEOFTHEGOVERNINGEQUATION:(WEcanDOitBECAUSEOFHPt)If +b/×,F)IL +c/×,F)=oIXa/✗,F)74a(✗,F)✗,Ylx)×,Ylx)d.Fdxd.Y=B/✗/F)CHARACTER/SticElevationIT'ScallEDCHARACTER'STICEQUATIONBECAUSETHEOUTCOMEOFINTEGRATINGTHISDXA/✗,F)☐DEISTHEDEFINITIONOFTHECHARACTERISTICCURVE4=✗(×) |g,=.a,,,,,compa,.ae,,,eaa.io,DXa/✗,F)HPZiALONGACharactersSTICCURVE,THElevasiLinearPDECanBEWrittenasaSYSTEMOF2☐☐EyY/xp)=ypJr.we.ee,iosx.ae,w. u a,contano,ioALONETHECHARACTER'STICLINES•LET'SADDTHEINIT/ALCONDITIONS: d = èdtplt)=rtt+xptltp= a) =Xp=0* èÈ× .dt{,no,.ua, +"(treno)= 001Io))=◦(xp) t.E.EE × STEADYZDMASSCONSERVATIONn29+v79=on-1-0sx04|g,,,,,,ay,,n,,,aong,•FirSTLYWEHAVETOCHECKTHATTHEINIT'ALSOLUTIONISCONTINUOUSANDDerivabile°CHARACTER/STICCURVES:se✗y=Y/×)se✗y=(×) f = f/×>41F/✗I= f/ ×,Ylx)) 9=9/×>y)R/✗/= g/ ✗, / ×))°CHARACTER/STICEQUATION"di=blx.FId =b / ×,r)=~diAix/F)dxna / ×,r)•COMPATIBILITYEQUATIONidf=-clx.FIdr =C / ×,r)=odxa/×/F)d.✗A / ×,r)"LET'SSOLVETHEODESiIx) i dy=o diY/✗I=v / ✗-xp)+yp=v✗+yp / yp^nn✗p | ""dr=☐rlx)= fp =zyp+3 Jp " { """"""""" più "×" in ✗YP. C-✗AMLE i In+zxIn- 542 =0.012,41{ susu?ilre/2,4)✗no/y)=ne/0,y)=54+10°CHARACTER/STICCURVES:se✗y=(×)u=ulx>y)U(✗/=un / ✗, / ×))°CHARACTER/STICEQUATION"DY=B/✗sul=2.✗DXA/×,v)•COMPATIBILITYEQUATION:du=-C/✗sul= 5×21×1 DXA/✗,U)•LET'SSOLVETHE☐☐ES:✗ | "" da =zxdxY/×)=✗2-✗ È +yp=✗2+yp/ ×.ypUlx)✗✗z 5×21×1| ""du=du= | 5/✗'+Up)dite , / xpupXP--✗U/✗I=up+5✗s+4} ✗+2✗>yp=up+✗s+ SYPZX +10✗>yp533--XP0'LET'SCOMPUTEninPointll/2,4):1)Y/✗)=✗=+ypyp=Y(✗a)-✗ è =G- [ =oSTARTINGPoint:Xp=yp=02)no/Up)=up=10ss3)U/✗)=up+✗S+ SYPZX +10✗>yp=up+✗SU/✗a)=upt✗a=10+2=423 •GRAPHIC-111YTHEPROCEDUREHASBeer:1)WEKNEWTHATXP=0ALWATS42)WEUSEDTHECHARACTER'STICEQUATIONTORETRIEVETHEStartingPOINTRELATEDTOll2.ll/2,41 ! 4YP=0✗3)ONCEHAVINGDEFINEDTHEin/TIALPointP,WEFOUNDTHEIN/TIALDATAMpG)WEUSEDTHECOMPATIBILITYEQUATIONTOFINDU(✗e)BURGER'SElevationNEGLICIBLE'nu'SCID◦BURGERWA N T E DTOUNDERSTANDTHEnonLinearTe r minNSEas:In+mIn=-^IP+U 32m ItDXSSix2×2•CONSERVATIVEFORM:In+^I RE =☐sit2sx"itcanALSOBEWrittenas:In+I &(m)=0,WHERE & /m)= À ISCAILEDFLUXFunctionsetsx2'ADVECTIONForm:IN+µIN=☐setsx◦WEcanUSETHEADUECTIONFORMTOEXPRESSTHEBURGER'SElevationasaDE'~Quasi-L'NEARFORM:alt,✗, f)77+blt,×, f)If +Clt,×, -91=0 1Itn>×o✗= It )°CHARACTERISTIECURVEi |Ult)= ulIt ),t)td U(t)inITIALCONDITION:XP=(tp )CHARACTERISTICEQUATIONfp / = didu =Oinit,alCondition:up=U(TP)COMPATIBILITYEQUATIONp.dt✗ •LET'SSaveTHEODES:e plt)t d -Up/t)dtplt)=xp+ | Upltldt % - |tptp t Wp(t)=up [ UisConstant|IT'SVA LU EISTHEOWEATTHEFOOT)ALONETHECHARACTER/STICCURVEtt•p/t)=Xp+ | Up/t)dt=xp+ | up dt p/t)=Xptupt-[pLinearBehaviourtotpt•CHARACTER/STICLINESARESTRAIGHTLINES fp o•ALONGEACHCHARACTERLSTICLINETHESOLUTIONISConstant1upp' la 1ma•DIFFERENTI-1FROMACOUSTICS,THEVELOCITYVA R I E SFROMPOINTTOPOINTASEFFECTèOFTHENorLinearTERM.ConsellVENTLYTHECHARACTER'STICLINESHAVEDIFFERENTINITIALDATALINESCOPESandcanINTERCEPT.THISTHEMECHANISMTHATALLONSSHOCKWAV E STOEXIST✗RAEEFact/ONWAV Eare+uIn=0atSex | no,,,,µ,,,o,no0✗E0✗•no(X)=☐ I =1>=-me,g.×=o fin)=n>2,ma=/1(nr+me)at3nr-ne2•REMARKiMODIFYINGCONSERVATIVELAWSLEADSTOWEOnceRANKINEHUGONIOTJUMPCONDITIONSTHEMISTAKEISTHATWESTARTEDFROMTHEDIFFERENTIALFormandNOTTHEINTEGRALONECOMPRESSIONWAV E Sare+uIn=0atSex | no,,,=,,,,o,no1✗E0☐El'UNKNOWNS: fr ,Mr,Ert57UNKNOWNSin3Elevati0ns4FREEPARAMETERS'EXAMPLE:icanpickASFREEPARAMETERSTHERIGHTSTATE(fr ,Mr,E È|+1MOREVA R I A B L EOFTHELEFTSTATE.THEREMAINING3Varia@LESWILLBETHEUNKNOWNSOFTHEPROBLEMSiSrÈ=LÌr =Senenrmi=me-S sta =nr-StttEcs EÈÈ ≠Ec ÈÉ ≠Erael=C- È -1 miera= EÌ -1 MIè =le èr =erSe2Se2Pc=P/Se.ec) Pr = Plfr >er) È =afr =PrLABORATORYREFERENCESHOCKREFERENCE"~" ^LABORATORYREFERENCE:FIXEDOBSERVERCOOKINGATTHESHOCK•SHOCKREFERENCE:OBSERVERMOVESONTHESHOCKWAV E(S=☐)MASSCONSERVATIONLABORATORYREFERENCE• Sana - Seme =5/Se - Se/SHOCKREFERENCE .gr/rir+sl-9clne +SI=SIfa-Sc)Srnir = Serie MOMENTUMBALANCELABORATORYREFERENCE22• gaur +Pr- Sene -PL=s /gaur - fine )SHOCKREFERENCE--• Salute+512 + Pr -Selui+s)≥-Pc= s.ge/uir +s)-Se/niet S)-22• gaur +faè + ZSRÙRS +Pe- Sent -SLS-2 Series -PL= fair s+faè - gentes -SLS≥2e farei + SaÙRS - fine - Series +Pr-Pc=o22• gaur + SISRNÌR - Sini )- Sene tPa-PL=O fruire -1Pr= Sini +PLMASSCONSERVATIONENERGYBALANCELABORATORYREFERENCEt•ma /Er+Pr)-ne/c- È +Pc)=s/EÌ - EÌ)SHOCKREFERENCETOTALENERGY-TOTALENERGYPERUN/TVOLUME: È = jet = net = f E+1 NE V2ENERGYPERUNITMASS - Et +p= g e+1 è +P=Se+P+1 gri = g e+P+1 fà = gIh+1 m2 )22g22"¥TOTh-_-_-• Et = g e+1lui+ SÌ '=ge+1n'2+ g 1 È +is222---_-- : •STARTINGFROMTHELABORATORYREFERENCE:ma /Er+Pr)-nel/C- È +Pl)=SIE È - Elt)--• tua + SIÈE + fr ^ È + ìrs +Pr- tue + SI ' ÈI + Se 1 È + Iris +Pe'=22-__---=s ÈE + gr ^ È + ìers - ÈI + Se ^ È + Iris 22----__. ìrÈE + fr 1 È +n'estPr+s ÈE + fr ^ È +n'estPr+22-_-_----- ù , Èt + Se 1 È + Iris +Pe-S Èt + Se ^ È + ìes +Pc=22-_-_--s ÈE + gr ^ È +n'es- ÈI + Se ^ È + ius 22--e sir/Èrt + Pal - in/È + Pel + noigr 1 È + ìers - mifa ^ È + ìecs +s/Pr-Pc)=022• sia/Èrt + Pal - in/È + Pel +^, è/Santa - Serie )+s /farti +Pr- giù? -Pc)=OMASSCONSERVATIONMOMENTUMBALANCE• sia/Ért + Pal = in/ÉÌ + Pel 'ᵗ= rischiHÌ = È • ÌrgrhrMASSCONSERVATION Srnir =SiÙLfruire tPr=IlÙÌ +PLRANKINE-HUGONIOTJUMPCONDITIONSinSHOCKREFERENCE vir/Ér '+ Pal = in/È + Pel[È = LÉ CONTACTDISCONTINUITY• MI =0 mia =0 SrriètPr= Sini +PLPr= Perir /Ért + Pal = in/ÉÌ + Pilfa≠Si| ."→1 M(g, f >(g)f-219)T>o1)M1:df>oa"(g)=f-' 219)-f"/f)f-19)dgf >(g) PHASEPLANESUBSONICEntri g7 SUPERSONICENTRY•THISGRAPHSHOWSISOLINESOF=PbeinG f =fu=constALONGanISOLINEIFfINCREASES,THENme☐ECREASES12SUPERSONICFLOWSUSONICFLOWM>1M>1M1:HYPETZBOLIC'.TWOSEPARATECHARACTERISTICSW=Il=v2+v2VIlre=wCOSO,V=WSin@0O=atanvnnM1MACHANGLE: le =astri^Mmm?-1.d±=u± C2M2 -1d±=tanto±µ )div2 -[2DXyet ÷ ✗e- COMPATIBILITYEQUATION.(v2 -C2)dud+(v2 -CZ)d=o d@±=±m2-1 dvvd@±=± m' -1dmdxdiDXWM1+(f-1)m2 °INTEGRATINCTHISEQUATIONFROMTHEFOOTOFTHECHARACTER'STIC Pt TOACERTAinpoi-TPWECETi--@±-@pt=±-U/M)-V(Mp±).COMPATIBILITYElevation•Prandtl-MEYERFunction:U(M)"◦=✗+1ifan-^6+1(m?1)-fan-^µ?16-1f-1•WEcanREWR'l'ETHECOMPATIBILITYElevationas:⊖±FUIM)= @ p±IU/mpt)i:'-(ti:-.-it)4④+UIMP'/=constOpt-Ulmpt)=const• | "+=+'°"" " " | "±="" | ◦±'"""±"=°"'°""""dk+=odk'=odidxK-=@è+UIMP-)KÉ=KP-@I+V/MI)=@p-tv/Mp-)e'.ptp-✗ | '"""±"=°"'°"""" | °'=°"'°""""+"""±""☐DI+VINI)=Op-+VIMP_IVINI)=Op-+vlmp-)-OI221i@±=Opt-Ulmpt)+Op-+Ulmp-)-OIOI=@pttop -Vlmpt)-VIMP')22•'2VINI)=Op-+vlmp-)-Opt-Ulmpt)+VINI)u/MI)=Vlmpt)+VIMP')- Opt - Oè 22 SIMPLEWAV EREGION•K-=@è+UIMP-)=constEVERYWHERE. do -=_ M' -1dmM1+(f-1/ MZ +" | =""+° " '"""""'"""""=""""°""e-et22e:e.eièU/MI)=Vlmpt)+VIMP')- Opt - OÌ =constAlona ègo.JO '%'K22/i"tpt..THEVA LU EOFÒANDUATTHEINTERCEPT10nsARECONSTANTALONGCK'p-✗.d±=tantoIN )characterISTICS@+ARESTRAIGHTLINESdxtanto+11 •Hp:↑>1•ifd@'OPRANDTL-MEYEREXPANSION"Hp:↑>1•ifd@'>0 dm 1•ifd@+70dm>OPRANDTL-MEYEREXPANSION"Hp:↑>1•ifd@+PICALOFRE-EntrinoSPACEVEHICLESinTHEAT M O S P H E R E,THATAREBUILTINTHISWAYTOSLOWDOWN≈•ASACONSEQUENCEWEWEEDTOCOATCAPSULEWITHHEATSHIELDSttTOTALENTHALPYREMAINSTHESANE:ho=hz. | ,,,gang,nn,go.gg,☐