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B. E./B "Tech. DEGREE EXAMINATI0N, NOVEMBER/DE CEMBER z0 1 0

Fifth Semester

Aeronautical Engineering

AE 2303 -

AERODYNAMICS -

II(R'egulation 2008)

Time : Three hours Maximum : 100 Marks(Gas Tables permitted)Answer ALL questions

PARTA-(tOx 2=2}Marks)Differentiate perfect gas from real gases.

Why do you need a convergent-divergent nozzle to accelerate the flow fromsubsonic to supersonic speed?

Define characteristic Mach number and give its maximum value for air.

Define shock strength and express it in terms of Mach number for a normal

!t1te the limiting values of shock wave angle in a supersonic flow for zero flowoellectlon.

Explain Mach reflection.

1.

,

3.

4.

5.

What are Riemann invariants?

\Yrite down the Prandtl-Glauertexplain.

Reg- No. ,

similarity rule for pressure coefficient and

6.

7.

8.

r) lII-L^r:- -

-ir: I It t r.i. yrrruL ls crlttctll i.vlacn numDer/

10. Describe boundary layer fence and state its purpose.

(a)11 (1)

PARTB-(5x1G=B0Marks)Derive the one dimensionai adiabatic steady state enerp- equatio,and deduce the isentropic relations for a peject gas. (g)Air flow is discharged to atrnosphere at sea level through u .o.ri.nozzle' If the air storage at the reservoir is 40 x 10s N/m2,determine the pressure, temperature and density at the exit of thenozzle- Assume that the reservoir air is at

=uu lurll te*fe.atrre. (g)

(ii)

Or(b) (i) Explain the phenomenon of choking in a nozzlethe mass flow of a choked norrle can

. P A*m = 0.6847 jg

J RTo

(ii) A storage charnber of a compressor is maintained at l.g atmosphereabsolute and 20 degree c. If the surrounding -jr"rrr." i,1 atmosphere, calculate the velocity with which air fl"ow takes placefrom the chamber to outside through a unit area hole. A-lso,calculate the mass flow per unit area. Assume air as a perfect gas.(8)

12. (a) (i) Derive the Rayreigh supersonic pitot formula. (g)(ii) A Pitot tube is inserted in to an air flow of Mach 2 where the staticpressure is I atmosphere- Calculate the total pressure measured bythe tube and. the losi of total press,r."

"*p".ienced. (B)"" ,.t \

. u_ ;'-',.,.. tt" 'r I Ot

\q'(b) (i) -- Derive a relation between flow turning angle, shock angre andFree-stream Mach number r"i"uriq"".io.t *Lru... (g)(ii) consider a Mach 2.5 supersonic flow over a compression cornerwith a defl-ection angle-of 14 degree. Calculate the increase in shocksrrength if rhe deflection angre is doubled to 2B;;;;;-""d giveyour comments on the shock *ur".hu*.teristics. (g)

(a) (i) Explain the concept of prandtl-Meyer expansion aroundcorner and represent it in Hodograpi, pta"",

(ii) A supersonic flow at Mr = 1.5g and pr = 1 atmosphere expandsaround a sharp corner.-If the pressurulo*rrrt;;;; of the corner is0"1800 atm-, carculate the deflection arigle of thl co.ne." (g)

13.

flow and show thalbe expressed as

(B)

a convex(8)

Or

53013

(b) (i)(ir)

Explain the proceclure Lo obtarn supersonic nozzle contour 'br

ugrven Mach number using Illethori of characteristics (g)An incidenl shock rvave angre ,i 35 cieg:-ee impinges on a straigrriwall' If the upstream florv properties are M = 3, p = l atm. andT = 300 K, calculate Lhe reflected shock wave angiewith respect tothe wall and the flow propert,ies M, P and T downstrearn of thereflected shock wave. (B)state-the assumptions made in small-perturbation potential theoryand show that the linearized pressure coefficient is a function of theperturbation velocity in the main florv direction

";ly. --

14. (a) (i)

(b) (i)

15. (a) (i)

Describe Prandtl-Glauertover airfoils and highiight

affrne transformationits significance.

(10)

lor subsonic flow(6)

2n(-,- = -=-' v_'

(ii)

Or

Derive suitable expressions forpiate airfoil at small angles offlow fheory.

A flat plate of 1 m x 0-2 m size is kept in an air stream of velocity1800 kmph at an angle of attack of 5 degree. carcurate the lift usingsupersonic linear theory- Assume that the static p."".,r." ,rrdtemperature of the free-stream air are 2 x 10s N/m2 and 2gg Krespectively. --'- """(;;'Describe the transonic flow regime with suitabre sketches of flowpattern over a two-dimensional airfoil" (6)Explain how large drag increase takes place at transonic flow. Whatare the control measures adopted at the design stage? (6)what are the flows detrimentar effects experienced in sweep backwings?

G)

lift and drag coefficients of a flatattack using linearized supersonic

(8)(ii )

(ii )

(iii)

(b) (i)Or

Write a note on transonic area rule"

Discuss the major differences between(8)

Rayleigh and Fanno flows.(g)(ii)

53013

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-' ,y Degree . I].8 /B.Tech /Brarrch : B-E_Acronarrtical Errginecring (Senre-ster : 5

Code No. ISubject : AD z3a3/AERODyNAMTCS _ II U- & Nov_lOe c ZOIAtleE'ff':iass-t"ht,'lis 4 B*r tt*- s[.d*^tvYsw

respective ly aild arc called Riernanrr Ilivariants.(8,) Write down the Prandtl-ClaLrerl similarity rule for pr-essLtre coefflcient and explain.

Answer: . Prandtl-Clauert sinrilarity rule lor oressure coefficient is C ,

' For lvi-:0, Cp:Cpo (the incompressible Cp).

: For l\vl- r0, Cp increasesasymptotically with the increase in free-stream N4achnumber and it gives the compressibility correction up to the transonic regirnebeyond which.rhe expression is invalii

6 *,,,,, is critical Mach nr.,,,',lr.r?\_-/Tnswer: Criticai Mach nuntber is detlneci as the flee-strerrn Mach nuntber for wiriclrthe local N4;tch nuntber M at some point ovel tlre body surface reaclies unit',,.

(lSDescribe a boundary layer f'ence and stare its purpose.Answer: ' These are flat plates kept perpendicular to the plan-form on the ripper sirrt?rce

of the wing at specitied span-wise locations., lt acts as a barrier to tip-rvard flow on swept back wings

Key for pART-B

I I (a) (i) Describe the one dimensional adiabatic steady stare energy equariqn arrd deduce theisentropic relations for a perfect gas.

o

Answer: . Energy conservation equation across a contr-ol volume, Introduction of errthalpy and simplification lbr q :0. Deduction of adiabatic energy equation

h+ t/rt)2 : const.. Deduction of temperature ratio for a perfect gas; p!-essure

' (2nlarks)ttl rn;r rk s)

(8)(2 rnarks)

(3rnarks)and density ratios

''i,

\(3marks):lt.

I\U-1e

rusing isentropic relation

l=r*, "t

M'T2' , * =lr*r-)M'li

,

*=Lrnt-|u,t'

P.p,,=fr=48.4kglnrjT'/To = 0,833

= l-':239.9K

(ii) Air flow is discharged to atnrosphere at sea level through a sonic no-rz.le- tf the airstorage at the reservoir is 40x 105 N/ m2, dete nnine the pr*rrrr., ternperature anddensity at the exit of the nozzle" Assunre that the reservoir air is at sea leveltemperature. (8)

Answer:

2/ta

rf ,o'1i)p'lpo: Olrr, :- J)' - 2t.lx lO5 N/nr2p'/ps:0.6338

=) p : lO.l Xg,ni "ll (b) (i)Explain the phenornenon orcrroking in a nozzreflow and show

ola choked nozzle can be expr_essed as

ryL)

(2 nr irrks)(2marks)

tlrat the ntass flow

/ (8)III

a

i = 0.6847 ryJ nr,, (5rnarks)

.fi'

"r+ (ii) A storage chamber of a compressor is maintained at l.g atmosphere absolure andq6#'Y 20 degree c" lf the surrounding pressure is I atmosphere, calculate the velocity with

-

" which air flow takes place fiorn the chamber to outside through a unit area hole. Also,

Answer:

"ul"ulut. th" *u* A. Assurne air as a perlect gas.Answer: . Calculation of po/p and M

yn = 0.6841 14'

"' . Jnr,, (8)

:F?t aco1$Eert.nozzlc connected to a reservoir of high pressrrre (po), if-rhe!::I pressure(Tb) is decreased berow, ihe reservoir pressure to a vaiue orPbJ!9=0.5283. the llow ar the Dinr!!!!rrt qry:a_of.cross secrion (;;;r;;;.n.,a speed of Mach r.0 and the nrasi-ffow iEactei trre maximurn varue. Anylurther reduction i, back pressure doer ,g!3Ei,h" ri.* allribution arread'of rhroat and does not increase tt'" qagf]g@-qqy-rr4h.. Trris conditionis called choking. " -''-'

.

(3rnarks)*= P A a

+

"o*=1 p,,.tqnr- A.po

Introduction of gas relations & further simplicaiion

;=^E#Er*l-Substituting the value of y=1.4

Po/P: 1.8; M:0.96 (2marks)'Calculation of 'a' Frotn temperature obtained tiom isentropic rclatio. arrd V

T/To=0.844

=T:247.3Kq

=.lyRT =3l5.2ntl.vV= J02.6 rn/s (3marks)

1l rn

le). Calculation of p and then yn I Ap:P/RT : 1.427 kg/nrrpV : L427x 302.6:43 | .8 kgis (3marks)

' 12 (a),(i) Derive the Rayleigh:supersonic Pitbt formLrla. (B)"Answer:

. Pitot pressure measuied [s the stagnBtion pressure behind the norrnai shcck

P,, P,, Pi,,,-=P, Po, P,

.,From first iaw of iherrlodynalnics. change in entropy can [:e deterrnined andis dr = C:,,+- R!! (2mar.rrs)UI'T P. After integration and simplification, the pressure loss across rhe norrnalshock can be obtained and is

l_t

Po, -,1r-*)tuti t'' , lP,,='*7;;;,rl A M,, -r -,,)/,-, ({r,arks)

' f+l f+ln

" Substituting this value ," #, Rayteigh supersonic Pitot ferrnula can be. It

obtained and is

D 1t*t 7r z,rl1-,I 02 _

,; tvtl ) (2marks)P, tJ-, M,, -r4l\-,. r+t /+t

(ii) A Pitot tube is inserted. in to an air t'low ol' Mach 2 u.here the static pressure is Iattnosphere. Calculate the lotal pressure measured by tlre tube and tire loss oltotalpressure experiencpd. (8)

lpAnswer: . Tqtalpr{ssure rneasured Po, =4P,= 5.64 atm. (3rnerrks),ri P, I

' Fr".-rtr.arn total pressure O, = ? n, = 7.824 atm- (3nrar.ks)

. Loss of total pressure :7.824-5"64 = 2.184 atrn. (2marks)

lZ (b) (i) Derive a relation between flow turrring angle. shock angle and free-streanr Maclr

t.- e '\'$) (3)nurr,ber lor oblique shock rr,aves.4t l0

)@f z'^(T

" skdtch ot'upstrearr & downstreanr rlow veloc.ity cornponerts ac'oss ano (2rnarks); From geometry tan B = u1lrv ; tan(fl-0; - tr2;/w

tan(R-0) _!] _(r-t)M,'sinz p+2

'rqnF ut 0+l)lvt,2sin2 p-

B= 35.87 deg.' Mnr: Mr sin p = I .465

From normal shock table p2lp,:2.335Shbckstrength {=i.3:S

Pl. Deflection angle 0:28 deg

B: 56.33 O.g; { = 3.881Pl

. From these results. it is clear tt,at tf,{sno.k@P.j"fl"rtion ungt. o.t *u*,f, i. ulJ-linear function of tlow deflection angle. (2nrarks)

l3 (a) (i) Explain the concept of Prandtt- Meyerexpansion around a convex cor.ner andrepresent it in Hodograph plane. (S)

(2rnarks)

(2rnarks)

(2rnirrks)

II

Ia

' Schematic sketch ofl prandtr-Meyer expansio, corner and description of trreflofit'ounalh-'e extransion fan. rJrvrr evsrrs;r crru ucscl

' Prandtl-M.y., furl"iion corresponding to rhe upsrream frow is,(*('D]';::tlparticularexpansion angle 0; Prandtl-M1yer fupction v(M2) for ttre florvdo-wnstream of expansion = r,(Mr) + 0. Knowing the prandtl-Meyer firnctiorrv(Mz), M2 can be determined.'The lociof the florv verocity downsrrearn ortrre. expansion,"va,,es !::.,lyt'traced in a Hodograph prane rvhich gives rh.

"-;;;;r-H"o"jr,,oh. (3rnarks)

)2-u)()A nsrver

(3nrarks)

'Expanding tan(p-0) and simprirying, rhe foilowing0-p M r.erarion can beobtained-

tan9 =2cot p M,1 sin' p -lM

,t (y + cos2p) + 2 (3 mrrrks)

I Z U (ii) Consider a Mach 2.5 supersonic flow ovcr a conrpression corner wirh a detlecrio,angle of l4 degree. Calculate the increase in shock strength if thedetlection angle isdoubled to 28 degree and give your comments on the slrock wave clraracreristics. (g)A.swer: ' Knowing M1=2.5& e:r4, B can be deterrnined rrorn 0-p-M chart

Answer:

ta\> \1A \rl3 A)iii) A supersonic flori'at N{ l: I .5E rnd I'i=

tne Ilrcssure dorvnstreanr oIthe corner is 0o t' the corner

A ns r,r,e i':

h Pt. Pot P2: 4.l27xlx l/0. I 306 : 1.6

M2:2.r1 frorn norrral shock tlrblev2= 47.79

#Pressuredorvnstrearn p- =

4-P., -

o rnt P, P, rl

-()'rJTem perature downstreanr Tl:5 5 5 K

nrial rhe.ryff":*i[]'the linearized pressttre coellrcient is a lunction oIthe pcrrurbarion ,"lo.ity ;n rt,. nu*drrectron cinly.

Cp--2u/y.Answer: . AssLrmptions

Answer:

( l0)

. t\,>\

,--${.,/r#'

'I The perturbation.verocity co,rponents rr,v,rv are smr, cornpared to' the mean velocity U ' -"-r-.uvtt)'' i'o, r({t

* Flow is two-dimensional and isentronic Vetn.ir., ^^..

' L,, = -. . , [-:-- l]vM'-P

' From energy equation, temperature ratio can be simplified toT

_, r-1.Y2-y 2T =t -;-(-----, -,,'a L (l

. l n trod uc i n g pe4!&alia!-yg!g91!les a n d ded.u c i,, g, p/!]s u s i n g i se n rropicconditions, p/p* in tenniofpertirbat;on velo"ir,"*rbor"rir" (2rnarks). Using Binomial expansion rra,i*piifyirePt , )/-1 ,, r.2u u2+vzL-' *r-** ,C*-U ) i2rnarks)

. Substituting it in Cp and simplifying2uV- (2marks)

1,? Tr:r,ne prandrr-Gtauerr afrne rranstbrnration for subsonic nf or",, airfbirs and

h igh l/ght its rign in.rniJ.---.-....-----.---._'__-

(2nrlr-ks)

(6), .,1ffjj,,:1,,:1".:,:::sfbrmation

retares rhe corrrpressibte florv Gver an airfbitin physical plane ro the incomprerriir" ii.* iln transfornred spacc over thesame airfoil.Transflornrationis(=x

. The governing esuatio, a. rl,;rlJc compressrbb liJ^:frtlr arr airroir isgiven by11"-M_216,,+Q,r,=o

The transfonned equation can be shown to take the form

r{ibJ'ri ?:, , ,\ ,, ,\1 L. q)a i) which governs the incornpressible flow over tlre same airioil (2nrarks)

. The significance of this transformation is that it relates the Cp over an airfbilin compressible florv to Cp overthe same airloil in incr:rnpr-essible florv

CP run,, =

(',t(2rnarks)

l+ (b) (i) Derive suitable expressions fir.tin 4 drag coelilcients of a llat Plate airfoil at' snra{l angles of attack using liriearized supersonic florv,theoiy. (tt)

Answer: ' From linearised theory

( r;)

2aLPr= r , (compression)

(expansion) (2marks)Cn 2a

.Cn=

288K respectively.

Answer: . Velocity: 1800 knrph = 500 nr/so = JynT =34Aml s

M: 500/340=1.47. c,

-3:= 0.32'" JM' -l

L'lqr,uo

: ! fcr,c

,u,.

'Cp,,)clx --

-+-,rM-. _l

(2rnarks)

.Ca-

('P,,)dY = 0 since dY =0 (2rnarks)

Cr: Cn since cos o = IC6: C, a since sin a = crTherelore C', =

C,I7

.+G=- (2marks)M-'-l

(ii)Aflatplateof lmx0"?msizeiskeptinanairstrearnofvelocity lS00kmphatanangle of attack of 5 degree. Calculate the lift using supersonic linear theory. Assunrethat the static pressure and temperature of the freestrearn airdre 2x l0s N/rn2 and

II;

(8)

(2rnarks)(Siiiarks)

(3nrarks)

M,

r Lift = % yHrt2PS Cr. = 19160 N

B/t0

l-i (a) (i) Describe thetransonic fllrrv regirne rvith sLr itabre sketches t-rlflo,,i,,aLre, overatwo-d imensional airtb i I

A rrsv'er: (6)

' Transonic flow regiine is between rower and upper criticar Mach numbers.Definition of lower critical Mach number ' I rrqv' I I rur I ruc;r 5-Definitibn of upper crirical Maclr number

- Flotv pattern over air[oil in tnirrsonic reginre

hock{ur),tshlr-k

(4x l:4rnarks)

Transonic Area Rule:' The area rule is a sirnple sta{enlent that the cross-sectiorral area of the bocl-v should har.ea srnooth yariation with longitudinaldistance along thebody;rhere shoLrld l;c no rupiclordisco,tinuous changes in the cross-sectionar are. cristribi,tion.'Application of this rule to a conventional wing-body.o,nt inution willalso lrave ro bed iscussed.'lllu.stration of the rule with sketchesof air-craft nl2p .-,ipq..7q anrr ^-^^^ - ..Var.iations y,u,yr dlu Lruss-sccuonal arca

(lr)

c)/ I (t

rc

. Aerodynarnic ad.rantages oltlre area-nrled aircraft rvillalso havc to be broLrglrt orrt hyconrparing the Cd Vs Mach ntrrttber plots olarea-r'Lr led and non area-rr.rled cases o['ana ircraft. (4x2:Srnarks)

16 . I' tiii Discuss the major differe nces between Rayleigh and Fanno florvs. (8):Auswer:

. Plot of tlre therrnodylarnic properties flor.one-dimensional flow with heat lddirion in'aMollier diagram, g?]l:d Rayleigh.cLrrve which describes the Rayleigh Flow.describes Fanno a[ffit as' similar pfot for flow with. friction' called Fanno crtrve ,vh ic'

' - l-leat.aaaition lfwJri'd.'r., the Mach nunrber tolvards l. decelei-ating il supersonic florvarrd accelerating a subsonic florr.

Where as, in a lranno tlorv, liiction drives the Mach nurnber lorvlrdd l,clccelerating a supersonic llorv and accelerating l subsonic flow.. Rayleigh curve can be traversed f,rom the top and bottom. -E Such things are not possible in Fanno flow..lrr Rayleigh flows, the total pressrlre decreases and the total temperature increases.

Where as, the total pressure decreases and the totaltempeiaturg rernainsconstant.in Fanno fl ows.

(4x2:8marks)

((

Il.I!a

r0/r0