The first practical supersonic wind tunnel, built by A. Busemann in Germany in the mid-1930s.

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2 Introduction

3 Th first rcticl sursonic wind tunnl, built by. Busmnn in Grmny in th mid-93s.

4 lrg hyrsonic wind tunnl t th U.S. ir Forc Wright ronuticl Lbortory, Dyton, Ohio.

5 Comrssibl Chnnl Flow Qusi--D Flow qusi-on-dimnsionl flow is on in which ll vribls vry rimrily long on dirction, sy x. flow in duct with slowly-vrying r (x is th cs of intrst hr. In rctic this mns tht th slo of th duct wlls is smll. lso, th x-vlocity comonnt u domints th y nd z-comonnts v nd w.

6 Continuity qutions liction of th intgrl mss continuity qution to sgmnt of th duct boundd by ny two x loctions givs Th qusi--d roximtion is invokd in th scond lin, with u nd ρ ssumd constnt on ch cross-sctionl r, so thy cn b tkn out of th r intgrl. Sinc sttions or cn b lcd t ny rbitrry loction x, w cn dfin th duct mss flow which is constnt ll long th duct, nd rlts th dnsity, vlocity, nd r. ( m ( x u( x ( x constnt

7 omntum Eqution for qusi-on dimnsionl flow. Th ur nd lowr surfcs of th control volum in Fig. r strmlins; hnc, VdS = long ths surfcs. whr (ds x dnots th x comonnt of th rssur forc.

8 Th intgrl on th lft of Eq. (.3 bcoms - u + u Th rssur intgrl on th right of Eq. (., vlutd ovr th fcs nd of th control volum, bcoms -(-P +P. Evlutd ovr th ur nd lowr surfc of th control volum, th rssur intgrl cn b xrssd s momntum qution for stdy, qusi-on-dimnsionl flow.

9 Enrgy Eqution for stdy, dibtic, inviscid qusi-on dimnsionl flow.

10 Diffrntil xrssions for th Continuity qutions Diffrntil form of th momntum qution Eulr's qution momntum qution for stdy, inviscid, qusi-on-dimnsionl flow

11 Diffrntil form of th Enrgy qution

12 r-vlocity rltion. For < (subsonic flow n incrs in vlocity (ositiv du is ssocitd with dcrs in r (ngtiv d. Likwis, dcrs in vlocity (ngtiv du is ssocitd with n incrs in r (ositiv d. d du du ( ~ u u For subsonic comrssibl flow, to incrs th vlocity, w must hv convrgnt duct, nd to dcrs th vlocity, w must hv divrgnt duct. lso, ths rsults r similr to th fmilir trnds for incomrssibl flow. Onc gin w s tht subsonic comrssibl flow is qulittivly (but not quntittivly similr to incomrssibl flow.

13 . For > (sursonic flow n incrs in vlocity (ositiv du is ssocitd with n incrs in r (ositiv d. Likwis, dcrs in vlocity (ngtiv du is ssocitd with dcrs in r (ngtiv d. d du ( ~ u For sursonic flow, to incrs th vlocity, w must hv divrgnt duct, nd to dcrs th vlocity, w must hv convrgnt duct; thy r th dirct oosit of th trnds for subsonic flow. du u

3. For = (sonic flow, It shows tht d =vn though finit du xists. thmticlly, this corrsonds to locl mximum or minimum in th r distribution. Physiclly, it corrsonds to minimum r, s discussd blow.

s soon s sonic conditions r chivd, w must furthr xnd th gs to sursonic sds by divrging th duct. Hnc, nozzl dsignd to chiv sursonic flow t its xit is convrgnt divrgnt duct.

14 3. For = (sonic flow, It shows tht d =vn though finit du xists. thmticlly, this corrsonds to locl mximum or minimum in th r distribution. Physiclly, it corrsonds to minimum r, s discussd blow. = t th throt d ( du u ( du u If w wnt to tk gs t rst nd isntroiclly xnd it to sursonic sds. W must first cclrt th gs subsoniclly in convrgnt duct. s soon s sonic conditions r chivd, w must furthr xnd th gs to sursonic sds by divrging th duct. Hnc, nozzl dsignd to chiv sursonic flow t its xit is convrgnt divrgnt duct. If w wish to tk sursonic flow nd slow it down isntroiclly to subsonic sds, w must first dclrt th gs in convrgnt duct, nd thn s soon s sonic flow is obtind, w must furthr dclrt it to subsonic sds in divrgnt duct. Hr, th convrgnt divrgnt duct is orting s diffusr.

15 Sonic Conditions (I In th dvlomnt bov, th stgntion conditions ρ o nd o wr usd to normliz th vrious quntitis. For comrssibl duct flows, it is vry convnint to lso dfin sonic conditions which cn srv s ltrntiv normlizing quntitis. Ths r dfind by hyothticl rocss whr th flow is snt through duct of rogrssivly rducd r until = is rchd, shown in th figur long with th fmilir stgntion rocss.

16 Th rsulting quntitis t th hyothticl sonic throt r dnotd by ( surscrit. Th dvntg of th sonic-flow rocss is tht it roducs wlldfind sonic throt r, whil for th stgntion rocss tnds to infinity, nd cnnot b usd for normliztion. Th rtios btwn th stgntion nd sonic conditions r rdily obtind from th usul isntroic rltions, with = luggd in. Numricl vlus r lso givn for γ =.4. ( T T ( (.6339 (.99 ( ( T T.583 ( = Sonic Conditions (II u u

17 Sonic Throt r Th sonic flow r cn b obtind from th constnt mss flow qution (. For th sonic-flow rocss w hv u u u m ( nd w lso not tht u= sinc = t th sonic throt. Thrfor, u u Using th rviously-dfind xrssions roducs ( ( ( ] ( [ ( ( ( ( ( ( T T u ( ( ] ( [ (5

18 r-ch numbr Rltion This is th r-ch rltion, which is lottd in th figur blow for γ =.4, nd is lso vilbl in tbultd form in ndix. It uniquly rlts th locl ch numbr to th r rtio /, nd cn b usd to solv comrssibl duct flow roblms. If th duct gomtry (x is givn, nd is dfind from th known duct mss flow nd stgntion quntitis, thn (x cn b dtrmind using th grhicl tchniqu shown in th figur, or using th quivlnt numricl tbl. Givn (x, m,,, T ( [ ( ] ( ( ( m Not tht for ny givn r (x, two solutions r ossibl for th givn mss flow: subsonic solution with <, nd sursonic solution with >. Which solution corrsonds to th ctul flow dnds on whthr th flow ustrm of tht x loction is subsonic or sursonic. ( [ ( ] Solv for (x, ( x /

19 (x = f((x/; i,., th ch numbr t ny loction in th duct is function of th rtio of th locl duct r to th sonic throt r. must b grtr thn or t lst qul to ; th cs whr < is hysiclly not ossibl in n isntroic flow Thus, / yilds two solutions for t givn / - subsonic vlu nd sursonic vlu. Th rsults for / s function of r tbultd in ndix.. For subsonic <, s, / ; i. th duct convrgs. t =, / =. For sursonic >, s, / ; i.., th duct divrgs. For xml / =, w hv ithr =.3 or =.. < > ( [ ( ] / =

20 Lvl Nozzl Flows (I =.58P o.833t o,6 P,6 T,6 Th convrgnt-divrgnt nozzl ws dvlod by Swdish invntor Gustf d Lvl in 888. ssum tht th flow t th inlt is fd from lrg gs rsrvoir (stgntion condition whr th gs is ssntilly sttionry. Th rsrvoir nd inlt rssur nd tmrtur r P o nd T o. Th r distribution of th nozzl, = (x, is scifid t vry sttion long th nozzl. Th r of th throt is dnotd by t, nd th xit r is dnotd by. Th ch numbr nd sttic rssur t th xit r dnotd by nd P. ssum n isntroic xnsion of th gs through this nozzl to sursonic ch numbr = 6 t th xit Th corrsonding xit rssur is P 6 For this xnsion, th flow is sonic t th throt, = nd t = t th throt.

21 Lvl Nozzl Flows (II =.58P o.833t o,6 P,6 T,6 Th flow rortis through th nozzl r function of th locl r rtio / nd r obtind s follows: Th locl ch numbr s function of x is obtind from Eq. (.3, or from th tbultd vlus in.. ( [ ( ] t For th scifid = (x, w know th corrsonding / = f(x. Thn rd th rltd subsonic ch numbrs in th convrgnt ortion of th nozzl from th first rt of. (for < nd th rltd sursonic ch numbrs in th divrgnt ortion of th nozzl from th scond rt of. (for >. Th ch numbr distribution through th comlt nozzl is thus obtind nd is sktchd in Fig. b.

22 Lvl Nozzl Flows (III Onc th ch numbr distribution is known, thn th corrsonding vrition of tmrtur, rssur, nd dnsity cn b found from isntroic rltions, rsctivly, or mor dirctly from.. Th distributions of P/P o nd T/T o r sktchd in Fig. c nd d, rsctivly.

23 Lvl Nozzl Flows (IV =.58P o.833t o,6 P,6 T,6 For th isntroic xnsion of gs through convrgnt-divrgnt nozzl, th ch numbr monotoniclly incrss from nr t th inlt to = t th throt, nd to th sursonic vlu,6 t th xit. Th rssur monotoniclly dcrss from P o t th inlt to.58p o t th throtndtothlowrvlup,6 t th xit. Similrly, th tmrtur monotoniclly dcrss from T o t th inlt to.833 T o t th throt nd to th lowr vlu T,6 t th xit. For th isntroic flow, th distribution of, nd hnc th rsulting distributions of P nd T, through th nozzl dnds only on th locl r rtio /.

24 Vribl Exit Prssur P Lt us xmin th ty of nozzl flows tht occur whn P /P o is not qul to th rcis isntroic vlu for,6, i.., whn P /P o P,6 / P o P /P o =P,6 / P o P /P o P,6 / P o

25 Subsonic Nozzl Flow t Exit Prssur P / P or P / P P /P o P,6 / P o If P /P o =, no rssur diffrnc xists, nd no flow occurs insid th nozzl. If P = P or P = P, it will roduc low-sd subsonic flow insid th nozzl. Th locl ch numbr will incrs through th convrgnt ortion, rching mximum vlu t th throt, s shown by curv & in Fig.(b. This ch numbr t th throt will not b sonic; rthr, it will b subsonic. Downstrm of th throt, th locl ch numbr will dcrs in th divrgnt sction, rching finit vlu = or =, t th xit. Th rssur in th convrgnt sction will grdully dcrs from P o t th inlt to minimum vlu t th throt, nd thn will grdully incrs to th vlu P, or P t th xit. This vrition is shown s curv& in Fig. c. Pls not tht bcus th flow is not sonic t th throt in this cs, t is not qul to nd t > for bck rssur conditions t P or P

26 Sonic Flow t Nozzl Throt for Exit Prssur P 3 / P Now, lt us rduc P = P 3, such tht th flow just rchs sonic conditions t th throt. This is shown by curv 3 in Fig(b. Th throt ch numbr is, nd th throt rssur is.58p. Th flow downstrm of th throt is subsonic. For givn nozzl sh, thr is only on llowbl isntroic flow solution for th sursonic cs. In contrst, thr r n infinit numbr of ossibl isntroic subsonic solutions, ch on corrsonding to vlu btwn P > P >P 3. Hnc, th ky fctors for th nlysis of urly subsonic flow in convrgnt-divrgnt nozzl r both / nd P /P.

27 For givn nozzl sh, thr is only on llowbl isntroic flow solution for th sursonic cs. In contrst, thr r n infinit numbr of ossibl isntroic subsonic solutions corrsonding to vlu btwn P > P >P 3. P /P o =P,6 / P o P /P o >P,3 / P o >P,6 / P o

28 Nozzl Flow t Exit Prssur P 4 / P <P 3 / P Now, S w rduc P = P 4, th flow rortis rmin fixd t th conditions shown by curv 3 in th convrgnt sction of th duct, tht is th flow rchs sonic conditions = nd th throt rssur is.58p. Howvr, lot hns in th divrgnt sction of th duct. rgion of sursonic flow rs downstrm of th throt. Howvr, th xit rssur is too high to llow n isntroic sursonic flow throughout th ntir divrgnt sction. For P 4 <P,3 but P 4 > P,6 (Fig. c, norml shock wv is formd downstrm of th throt. Btwn th throt nd th norml shock wv, th flow is givn by th sursonic isntroic solution, Bhind th shock wv, th flow is subsonic. This subsonic flow isntroiclly slows down furthr s it movs to th xit, Corrsondingly, th rssur xrincs discontinuous incrs cross th shock wv nd thn is furthr incrsd s th flow slows down towrd th xit, Th flow on both th lft nd right sids of th shock wv is isntroic; howvr, th ntroy incrss cross th shock wv,

29 Loction of Shock Wv t P 4 / P or P 5 / P Th loction of th shock wv insid th nozzl, is dtrmind by th rquirmnt tht th incrs in sttic rssur cross th wv lus tht in th divrgnt ortion of th subsonic flow bhind th shock b just right to chiv P 4 t th xit, s P is furthr rducd from P 4, th norml shock wv movs downstrm, closr to th nozzl xit, t crtin vlu of xit rssur, P =P 5, th norml shock stnds rcisly t th xit.

30 Bck Prssur P B Effcts P 6 <P B <P 5 (ovrxndd nozzl Th bck rssur is still bov th isntroic rssur t th nozzl xit. Hnc, th jt of gs from th nozzl must b comrssd such tht its rssur is comtibl with P B. This comrssion tks lc cross obliqu shock wvs ttchd to th xit. P B =P 6 (mtchd nozzl Thr is no mismtch of th xit rssur nd th bck rssur; th nozzl jt xhusts smoothly into th surroundings without ssing through ny wvs. P B <P 6 (undrxndd nozzl Th jt of gs from th nozzl must xnd furthr in ordr to mtch th lowr bck rssur. This xnsion tks lc cross cntrd xnsion wvs ttchd to th xit

31 (ovrxndd nozzl (undrxndd nozzl

32 If w ssum tht th flow in th duct is isntroic, th stgntion dnsity ρ o nd stgntion sd of sound o r both constnt. This llows th normlizd ρ nd u to b givn in trms of th ch numbr lon. m ss Flux of Isntroic Nozzl Flow ( x u( x ( x constnt Th figur shows ths vribls, long with th normlizd mss flux, or ρu roduct, ll lottd vrsus ch numbr. Th significnc of ρu is tht it rrsnts th invrs of th duct r, or ( (3 (4

33 x ss Flux t Throt It is vidnt tht th mximum ossibl mss flux occurs t loction whr loclly =. This cn b rovn by comuting d d u ( t mximum which is clrly zro t =. minimum Thrfor, th duct must hv locl minimum, or throt, whrvr =.

34 Considr isntroic subsonic nozzl flow with throt, connctd t its inlt to vry lrg still ir rsrvoir with totl rssur nd nthly P r,h r. Th duct xit is now subjctd to n djustbl xit sttic rssur, somtims lso clld th bck rssur. s P is grdully rducd from P r, ir will flow from th rsrvoir to th xit with mss flow. W first not tht th stgntion conditions r known from th rsrvoir vlus ll long th duct. ( / / ( ( T T ( ( ( ( h P ] [( h h P u m / / / ( ( ( ( / / / / ( ( ( h RT RT h m ( ( ( ( ( ( ( ss Flow Rt of Isntroic Nozzl

35 ] [( Subsonic nozzl flow nd Chocking s is rducd, mss flow rt will first incrs, but t som oint it will lvl off nd rmin constnt vn if is rducd ll th wy to zro (vcuum. Whn mss flow rt no longr incrss with rduction in, th duct is sid to b chokd. ( ] ( [ throt throt throt throt u u t u m RT m ( ( ] [(

36 ,,, ( ( ( T f RT m Givn (x, / ( x m T P,, ( ( ( ] ( [ Solv for (x, Subsonic nozzl flow bfor Chocking ( Solv for / ] [(

37 ximum ss Flow Rt ftr Chockd Condition Th onst of choking coincids with th throt rching =. Thislso corrsonds to th mss flux ρu t th throt rching its mximum ossibl vlu ρ, which is givn by Thrfor, th only wy to chng th mss flow of chokd duct is to chng th rsrvoir s totl rortis nd T. For givn sonic nozzl, idl chockd mss flow rt is roortionl to. t t t u m ( (,, ( ( ( ( ( ( t t t T P f RT h P m ( h ] [( h Bfor chockd flow condition Chockd flow condition,,, ( ( ( T f RT m,, ( ( ( t t T P f RT m ] [(

42 Idl (isntroic v.s. ctul sursonic diffusr diffusr is duct dsignd to slow n incoming gs flow to lowr vlocity t th xit of th diffusr with s smll loss in totl rssur s ossibl. m m ctul whr RT C C d d ( ( f ( P, T, ( RT ( : disch rg cofficint t t t ~ T Th rt of diffusr dsign is to obtin s smll totl rssur loss s ossibl, i.., to dsign th convrgnt, divrgnt, nd constnt-r throt sctions so tht P /P is s clos to unity s ossibl. It is xtrmly difficult to slow sursonic flow without gnrting shock wvs in th rocss. Not tht th sursonic flow in convrgnt sction will inhrntly gnrt obliqu shock wvs, which will dstroy th isntroic ntur of th flow.

43 ( ( ( ( whr C RT C C RT C m t d t d ctul ctul ss Flow Rt of Sonic Nozzl GRD-R-3

44 Sonic Nozzl ss Flow Rt Clibrtion by Grvimtric Systm htt://

45 Sonic Nozzls for Gs ss Flow tr m ctul whr C C D D ( RT ( : disch rg cofficint t ~ T

46 Clibrtd Sonic Nozzls for Gs ss Flow tr Clibrtion

47 SUPERSONIC WIND TUNNEL DESIGN (I If you wnt to crt ch.5 uniform flow in lbortory for th uros of tsting modl of sursonic vhicl, sy, con, How do you do it? dsign ( W nd convrgnt-divrgnt nozzl with n r rtio / =.637 ( orovr, w nd to stblish rssur rtio, P o /P = 7.9, cross th nozzl in ordr to obtin shock-fr xnsion to =.5 t th xit. (3 In this dsign, th high-rssur ir suly is t 7.9 tm, You nd n ir comrssor or bnk of high-rssur ir bottls-both of which cn b xnsiv. It rquirs work, hnc mony, to crt rsrvoirs of high rssur ir-th highr th rssur, th mor th cost.

48 SUPERSONIC WIND TUNNEL DESIGN (II Cn you ccomlish your objctiv in mor fficint wy, t lss cost? Th nswr is ys, s follows, imgin tht you hv long constnt-r sction downstrm of th nozzl xit, with norml shock wv stnding t th nd of th constnt-r sction; this is shown in Fig. blow. Th rssur downstrm of th norml shock wv is P = P B =tm.t =.5, th sttic rssur rtio cross th norml shock is P /P = 7.5. Hnc, th rssur ustrm of th norml shock is.4 tm. Sinc th flow is uniform in th constnt-r sction, this rssur is lso qul to th nozzl xit rssur; i.., P =,4 tm. P = 7.9P =.4 tm dsign

49 Howvr, th "norml shock diffusr" sktchd in dsign hs svrl roblms:. norml shock is th strongst ossibl shock, hnc crting th lrgst totl rssur loss. If w could rlc th norml shock in dsign with wkr shock, th totl rssur loss would b lss, nd th rquird rsrvoir rssur Po would b lss thn.4 tm.. It is xtrmly difficult to hold norml shock wv sttionry t th duct xit in rl lif, flow unstdinss nd instbilitis would cus th shock to mov somwhr ls nd to fluctut constntly in osition. Thus, w could nvr b crtin bout th qulity of th flow in th constnt-r duct. 3. s soon s tst modl is introducd into th constnt-r sction, th obliqu wvs from th modl would rogt downstrm, cusing th flow to bcom two- or thr-dimnsionl. Th norml shock sktchd in dsign could not xist in such flow. dsign

50 SUPERSONIC WIND TUNNEL DESIGN (III Th min sourc of totl rssur loss dsign 3 ( convrgnt-divrgnt nozzl rovids uniform sursonic flow into th constnt-r duct, which is clld th tst sction. ( This flow is subsquntly slowd to low subsonic sd by mns of diffusr. (3 This rrngmnt-nmly, convrgnt-divrgnt nozzl, tst sction, nd convrgnt-divrgnt diffusr is sursonic wind tunnl. (4 tst modl, th con in dsign 3, is lcd in th tst sction, whr rodynmic msurmnts such s lift, drg, nd rssur distribution r md. Th wv systm from th modl rogts downstrm nd intrcts with th multi-rflctd shocks in th diffusr.

51 t t u m t t m Lt us ssum tht sonic flow occurs t both sttions nd t t Sinc is constnt for n dibtic flow ( = dibtic, not isntroic dibtic, isntroic / / RT RT t t Sinc T is constnt for n dibtic flow (T = T ( ( t t t t

53 ROCKET ENGINE NOZZLE DESIGN (I Th thrust of rockt ngin nozzl cn b dfind s F mv ( F ( m[ V ] m m V q T [( ] V / ( RT V Th scific imuls (I s is th rtio of th thrust roducd to th wight flow of th rollnts. I s F mg V q g whr F = Thrust q = Prollnt mss flow rt V = Vlocity of xhust gss P = Prssur t nozzl xit P = mbint rssur = r of nozzl xit

54 ROCKET ENGINE NOZZLE DESIGN (II nozzl xit Undrxndd mbint (idl Ovrxndd If nozzl is undr or ovrxndd, thn loss of fficincy occurs rltiv to n idl nozzl.

55 Ovrxndd If norml shock r insid th nozzl, thn loss of fficincy occurs rltiv to n idl nozzl.

56 Rockt Nozzl Dsign: Otimizing Exnsion for ximum Thrust

TOPIC 5: INTEGRATION

TOPIC 5: INTEGRATION. Th indfinit intgrl In mny rspcts, th oprtion of intgrtion tht w r studying hr is th invrs oprtion of drivtion. Dfinition.. Th function F is n ntidrivtiv (or primitiv) of th function