Aerospace Engineering - Compressible Fluid Dynamics

Exercises of the course

Complete course

Compressible fluid dynamics exercises b)Eix. y)= 2×1-1×+41I •S'MPLTCONNECTEDDOMAIN+IRROTATIONAL=CONSERVATIVEFIELDiI K•I=nF= det7×24Iz=IFY-IF×'≤=-11=0NOTCONSERVATIVE--Ix34FxFyFza)-0 =20I +70I +70E=zxz>e#+"" I + gzz @✗ 2+44I +zz@✗ 2+44 ±JX04OZb)I= -0 +2e✗2+44e-°Il=/22-2 @✗'+44+ 4×22-2 e#+44)+1g ZZ @✗'+44≠0notSOLENOIDALii K•_^I= detIx24Iz=Tm z-Junè +Dux-SuzI +Tiny-JMXI =74IZJZ2xIX74Mxmylez=/gz@×"""_gz@✗'+"" )è + / c.✗ze✗"¥44_a✗ zè""") I + (g✗ZZe✗'+""- 8×2-2 @✗'+""I±=IRROTATIONAL al◦FIRSTOFall,LET'SCHECKTHATTHEFIS1220TAT I O N A L,SOTHAT:'RTDOTATIONAL+S'MPLYCONNECTED=CONSERVATIVEWEcan-DEFINEAPOTENTIALFUNCTIONiI K•^I= detIx24Iz=7ft-JFYè +3Fx-JFZI +71=4-JFx± =-34IZ77IxIX74Fx Fy FZ=Gxy-^-4×4+1=02✗2✗∅= |6×2-2×4 >+4d.✗= 2×3 -✗ 242 +y✗+C2✗• E =_00= | - 2×24+4 +✗ dy =-✗ 242 -144+c•∅= 2×3 -✗ 242 +y✗+44+C •Il/×,4,t)=/0,3✗+244)è+|1,64+1,2) I • g = f /ti• Flo)=1,2kg n}•MASSCONSERVATION:79+I°-S+ f -°Il=☐set•-°Il=0,3t1,6=1,9§ +1,99=o i 9101 =1,2•g=gol" j =foie"go/ ✗+1,9)e"=o✗=-1,99 =goe"◦ |g =1,2e- ""ᵗ1,2= IogIo)=1,2 •MASSConservation:79+ f _°Il+Il° § =Osit•_°Il=0,8+0,6=1,4 § +1,49=o i 9101 =1,6.g=gol" j =foie" soli +1,4)e"=o✗=-1.c, 9 =goe"◦ |1,4= Sog =1,4e- "◦tgIo)=1,6 • |219ª) dv + | ,Sartorids= | , Iodv+ /l - PE + E) ◦Idsatss• 2191)+o/Saa) =-98-_P+◦IT--=st.sisa)+≈◦ 1911 + 1gal °_I=- SI _-p+-◦ Iat◦a79+ g 71+I_ 019A)+ giroI=- PItatMass• SI °I=- P -----_7mIn06sx04•,Il==JUJo 6/4+2)-614+2)DX74✗2------__'spIx-suInnIntvIn0+GV--sxdi 74 =- g ''•=- f -nv.◦°"=- g -3P 74 -IVIVinTVtVIVn614+2)- Goofy +2)DX74Ix74✗2--__--•DP=51,84PamDX10,310,4).DP=-729,216Pam0410,3,0,4) a)dh=Td s t r r d ph=e+PV•Td s=de+Pdr+Udp-vdpTd s=de+Pdrda=1de+PduTT•D=ste.ir) DI =DAde+IsdosesuVe1TPT-1-1◦T=DA=C EREF ^=ETEEEREFCV b)c=S-SREFT"=E'=10≥kCaInE1EREFc) -1.T=CEREF1=2E.EREF2EEREF°•Eau,Librium:Ta=TBi2E?EREF=E"EBEREF=Ea≥ EB =C-a≥=2,5Eacc4METHOD1(✓MONOATOMICGASPOAT O M I CPOL>AT O M I CLINEARGASnon-LinearGAS°A=3NTr a n s l a t i o nal3r5r3R22NROTATIONAL(v=9r2NTr a n s l a t i o nal3r3ar3ar✗=112229naaa,.ua,|,,=µ,NVibrationalA:NUMBEROFA:NUMBEROF2AT O M SAT O M SMETHOD2•✗=☐OF+2=3'3+2=11☐OF3.39• È = gè = g e+1 Ù = gCut+12V2 T=1 Et _ v2 =3522,3K2Cv f 2.r=9314=188,955JCv= 850,2975 Jzmc+MoKook1cgK •c=✗RT=901,9m5 a)htot=h+^ v2To=Tttu≥2ZCP b)✗=CP=Cvtr=1+ECv=rCvCvCv8-1◦e=CvT=RT=PV6-16-1T=6-1eP=6-1erti•h=e+ po =e+6-1lo= te = 62T O6-12[2CO+ v2 ◦ho=h+ Ùg-1=f-12222+6-1 v2 [ È =1+f-1M≥To•Co=C,=1+8-1M≥2[2221-8•'soentrzopicFlow:P V6 =costPrt✗=costP^-✗T✗=costT P °=☐p % Po =1+8-1 M2p2c)To=Ta o1+8-1 nè 2 ◦Al=0,15 è un=240m5TI=300KPt=70'000PaU2=120M5• 91 =Poi= 0,8139kgn>Rtlmi=ha+12n }Tz=Tu+il } - MÌ =321,5K°ADIABATICi"¥1+1=2ZCP'Cz=✗rtz=359,22n5•Mz=U2= 0,334CZ•P9-°=costSrtf-✗=costT} "-✗=costg 1-" % =cost1^11-01-f1-6• 91 Ti= fa TZ fa = 91Ti=0,968kgm}TZ' 91 Allen= 92^-2 U2AZ= Sent A1=0,252n≥92hr2 Se52-z-z2h~,P,n/Z)=llmax1-ZZ✗h--DM:NUMBEROFPHYSICAlvariaBLES◦TI-THEOREM:M-✗=2✗:nunTSEROFFundamentalUnitsa:NUMBEROFDIMENSIONIESSGroups•n=5~,P,h,D,n.✗=3L,M,T•a=2°un=M,P=MLtz/h=mc-è•☒e=SC> 502è =1029,11Wm22•=1010810①e=150,125dB①eref • I =1atmÌ=293,15K=60dBV=200HZ• 5 = Ag cos / kx-wt)a)•g-=Ò=1,204kgn≥RT• [ =✗RF =343,202MSt◦ § =E | s dt S/t)=1 d}=ZITVASSin/kx-mt)=-Sosin/wt)c-dtEHP:✗☐=0So•=1010810①e∅e=10"°①ref=. 10-6 W m2①erefeOle= èjc >s'dt= è g-c> SÌsirflwt ) dt tti t : TCOSZX= COSZX -SIR✗=1-251IXSIM?=1-COSZX2 |Siriexdx=✗-1^ ( X-sinxcosx)2 | COSZX=✗_1Slnzx=2224COSZX= COSZX -SIR✗= COSÌ -1cos≥✗=1-1COSZX2 |COSZXDX=✗+12 | coszx=✗+1sinzx=^ ( ✗+sinxcosx)2242 --•de=9-E> SEè|; e-cos/zwt)dt=9-E> SÉè T-1sin/ut)=T2T22W21T-_= f-E> SEè 2•FORMULAFORSINUSO/☐alWaves:e=SC>S }è50=2Ef.è>=2,03. 10-7 2 b)p'=g-c-2s=Pc-2sp'=c-2rtèrt°•naxp'=C-2So=2,8. 10-7Panaxp'= 0,028Paèrtc)r=2078,SJkgK•✗=☐OF+2=3T2=53☐OF3ec-=✗RT=1007,73ns •So=g-. 10-6 •✗=1,8M•Ì=304,15Kè=1atmal• § =Ò=1,161kgn}RF•E=✗rt=349,58m5•p'=g-c- 25P' Max= §è 2So=0,165Pa•ne'=ESn'Max=Esco=4,059.10'"nSTb)+1✗≤0SoS=So SS = | ✗-1✗>0•e= è g-[ 35dt=9-c-3 è / t èdt=9-E> sè / t S2dt = JÈSÓè =6,69. 10-5 W ma tti TT1o0•=1010810①e=78,25dB①eref tt• Siti =E | ,sic)di =E|SoSde =c- solo • § Max=EsoT=ESo✗2=1,0.10-°m2c-=50,12T di ta|.Sourcea•[a=2km=5,7215E•t=✗=5,149.10'>SE RIGHTRUNNING•S=A f I✗io)al• § =è=1,1768kgm≥E=62T=347,1887mSrt-9'= Isf' Max= j a.1=1,1769- 10-4kgn}◦p'=g-c-25P' Max= f-EZA.1=14,1851Pa•µ'=ESu'Max=[a.1=3,4719-10'≥m5 b)de=g-c-352 è TT.s'2=1 | , 52DI=^A≥ DI =a≥TT / ◦•de= jè>A≥ è =0,4925Wm2•=1010810①e=116,9241dB①erefttEs de =c-At cig = | .n'di = | ,• { Max= §(t2)=c-a✗=a✗= 5.10-5 ~E ◦ d)torssc-ruc.rs =5km=14,4014Sc-[sound=✗=2,980.10'≥c-,✗0p'g-c-2A | PI-X,o)=g-EZA--•S=-A1+ZXPRIGHTrunningWAV E✗✗---g-c-2APIO,0)=-g-da•P=96Pa✗=15,24MSTANDARDSEALEVELCondition: Ì =15°C= 288,15 KÒ=1atm.E=✗rt=340,2626mS• § =Ò=1,2252kgm}rt. P =2g-EZAA=P=3,384.10'"n2g-c-2Tz-RIGHTrunning,Tz-_. Smax = | , n'dt= | "≥c-Sdt= | "≥-EA1+zxdt=- capo1+21×0 -it)dt✗✗00------.-=-Eat-zET2=-Ea✗-ZE✗×1=-c-at=a>=1,289. 10-3 m2✗8zè✗c-✗8ai4-__-'e=g-c-352è •52=1 /è5dt=1, /[ A≥' 1+21×0 -Et)'≥di= /[ 1+4c-222-GEede=T✗✗2✗--=A2Tt4[2T}-2ET≥=A≥TtGT-2T=A≥T3✗2✗T33 •e= fè>a≥=1,8424Wm23.=1010810①e=122,6538dB①eref-co=411,36n5Po=101.480Pa-..-•ne/×,t)=Co0,41136 ( ✗ -411,3Gt)--0,017885(✗+ 411,3Gt)"canonicaform:n/✗,ti=[f-(✗-It)- gli -Et)-coCo-- f/✗-Cot)81✗+cut)•(o=OrtoTO=(o≥=421,1475Kor• Io =Po=0,8396kgm>rto----• 9 '=90- f /×-Cot)+gI✗+Cat).=So0,41136 / ✗ -411,3Gt)+-0,017885(✗+ 411,3Gt)"coCo-- f/✗-Cot)81✗+cut)----•p'= giocò - f /✗-Cot)+gI✗+Cot).=So0,41136 ( ✗ -411,3Gt)+-0,017885(✗+ 411,3Gt)"coCo-- f/✗-Cot)81✗+cut)• f + I =-0,05248(1,110,06)°9'/1,1,0,061=-0,044061cgn}°P' / 1,1,0,06)=-74 5 6,076 8Pa -◦P/✗it)=107-000+ 149,8/✗+ 438,25T )"=P'=g-c-2149,8/ ✗+ 438,25T )"2' F èg-c-2--p' g/✗+Et)•E= ORÌÌ=E'=478,0066K te • È =Ò=0,7800kgn}rt•µ'=-Eg/ ✗+cit)m'/6,0,0,7)=-7, 7 5 0 2ns•g'=g-ng(×+Et)9 '16,0,0,7)=0,013794kgn> a)7(n)+I12in)- 3×2 =OCONSERVATIVEDX04INtZXIN- 3×2=0 QUASILINEARDX24b)a=1,b.=2x,[=- 3×2 •Di=B=ZXDXA•TU=-C= 3×2 sxa(✗I✗di= | zx dxY/×)="¥e+✗=-✗ È il 0YFXFWENEEDTORETRIEUETHESTARTINGPOINT•Y(XP)=YFtXP≥4F=G- 22 =0THESTARTINGPOINTFORTHECHARACTER'STICLINETHATPASSESTHROUGHP(2,4)isF(0,0)•Character'STICline:Y(X)=✗2 di | "" ( ×3• du = 3×2dxU/×)=ne+×}-✗F0UF✗F•nelxp,yp)=UF=10•COMPATIBILITYElevation:U(X)=×}+10e)MI2. 4) =V12)=18 •A=1,B=ZX,C=-SYZp/0,YP)Startingpointll(2, 4) •di=ZXdx•du= SYZdx{ "'×'וzxdxY/×)=yp+✗2-× }y,d>= / ×,o•YI✗e)=yp+✗ èlyp=G- 22 =oP(0,0)StartingPoint•CharacterISTICEdiY/✗)=✗=' | "" du = ( × 542dx= | " 572dis/✗2)=dxulx)=up+✗s-✗ ps =L : upxpxp•up=il/0,0)=S'0+10=10•V1×)=✗ St 10•U/✗e)=mixa,ya)= 25+10=42 •A=1,B=-ZY,E=-342IN/TIALPOINTP•di=-ZYdi•du=342dxYlx)-_•-1, | ^di= | "dx.1_InY- lnyp _=✗-xpIn✗= lnyp -2x✗249XP-ZX•Y=YpE✗• | """du= | " 342dx= / ×3/ypè " Ìdx =3.yp' | e-"×dxucx)=up.3yp≥ è ""_ e- " % 41UPXPXPxp-up=SYP+10ayp≥ /e- וU/✗I=SYE"+10-3gy>e"" /e- "×- 1) = SYZE >×+10.3✗2+3✗2 ehi 44 •a=1,B= 3×2 ,C=YPSTARTINGPOINTll(2,9)•TY= 3×2 sxeDU=-YDXYlx)• | di= | "3×2DX✗il=yp+✗3-xp}0ypxp3•Y/✗a)=✗ è -Xp✗= 23 -9=-1xp=-1•CHARACTER'STICElevation:Y/✗)=✗}-11•up=nel1,0)=-2✗✗Ulx)• | du = | -Y dx = | -1-✗}dxulx)=up-✗+xp-✗"+xp"=-11-×-×"4444UPXPXpav12)=MI2,9)=-354 •A=✗,B=ZY,C=-SU≥•di=24di✗•du=S v2 DX✗. { "'✗I11dxInY=Inypt2In×_ ztnxp Y=yp✗2 dy =2 ) ; y✗ypeY/✗e)=YP✗ èyp=109•Y=10✗29•up=64p=203. | """1,du=5 | "1 dx -1+1=SIn×-5InxpU=1upUxp×°"P1-Strixup◦v13)=nel3,10)=-0,1872 +1,3✗=o•2,4t≥7m+sut1,3×=0Sixat'A=2,41-=,B=1,C=1,3✗•di=1DX2,4T2du =13✗DX24T2t""1-2di=1 | × dx T}= tè +1✗-1xpT>=1,25×-1,25XP• i 2,4332,42,4tpxp-0,3648•Xp=1,25✗e-T≥/✗e)=1,25-0,85760,1464-Up=0,8090aT=°1,25✗-1,25XPU/✗I✗du= | 13✗↓✗ANNOYINCINTEGRAL•(n,24✗P/1,25×-1,25xp)≥>-In+2,4t?In+1,3×=ositsx•A=1,B=2,4t≥,C=1,3× DX=2,4 Èdtdu=-1,3✗dt✗"" dx = /È ,=, Gt2dt✗ ltl=xp+0,8-13- 0,8T} ' / ×.-0,3648•xp=✗Ita)-0,8tè =-0,85760,1464-up=1,1✗ è =018090. / "×'✗It)-1,3✗di= / t-1,3Xp- 1,0Gt > dtUlx)=up-1,3✗ptt1,3✗ptp- 0,2Gt "+0,26tp" du = | ,uptpeU(1,1)=ne/0,7,1,11=0,2874U/1,3)=u(0,9,1,3)=0,5158 ✗It--01'levasiLinearform:ImtnIN=Osit2xIX=uatCHARACTER'STICSTRAICHTLINES f. =.se✗It)to dx = | Udt✗It)=✗o+ ult - to ) / ×.to✗It)=✗otnot• | "du= / ᵗ0 dt u=no+tonoto1)✗It)=✗o✗o≤-11234Ult)=OIt2)✗It)=✗o+/✗◦+ 1)t-11 alrenerot=65,a=-1ns≥,ta=SSy✗•µIta)=ata=-SMS•a=constilLinear✗Parabolicit2[=6s-ta◦"¥=0✗•JÓ=JEmo-2CO=ma-2CzCz=(o+f-1MZ= 339,294 ns6-1f-12no=0 ( n..,#=....To=Po=288,2031KCo=ORTO=340,2940nsforCHECK:DETACHMENTOFTHEFLUID~ [ °""/¨ACCELERATEDTHEN(1MUSTDECADETO0.BE'NGCi=(RT,Cyn,,=Ortuin=0OK•no-2CO=U2-2C2llzun=-2Co=-1701,47ns6-10-16-1>SINCE[=00SURELYu>NLLIMTHEFLOW☐C-TAC H E S,HENGEWECan'TPROCEEDcittta.1