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S A E C H N I C A LT R A N S L A T I O N

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T T F - 5 4 2

LOAN COPY: RETURN TOAFWL [WLOL-2)KtRTtANO AFB N MU(

N A M IC S A N D F L IG H T D Y N A M I C SF TURBOJET AIRCRAFT

Press, Moscozc ; 1967

L A E R O N A U T I C S A N D S PA CE A D M I N I S T R A T I O N W A S H I N G TO N , D . C . S E P T E M B E R 1 9 6 9

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TECH LIBRARY KAFB,NMIllllllllllllllllsllllllllllllllllllllllllI

AERODYNAMICS AND FLIGHT DYNAMICS OF TURBOJET AIRCRAFT

By T. I. Ligum

Translation of "Aerodinamika i Dinamika PoletaTurboreaktivnykh Samoletov"Transport Press, Moscow, 1967

NA TIONA L AERONA UTICS AND SPACE ADMlN ISTRATIONFor sale b y the Clear inghouse for Federal Scient i f ic and Technical Informat ion

Spr ingf ield, Virginia 22151 - CFSTl pr ice $3.00

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Table o f ContentsI n t r o d u c t i o n .Chapter 1 . The Phy sic al Bas is o f High-speed Aerodynamics .

5 1 . V a r i a t i o n s i n the Parameters o f A i r w i t h A l t i tu d e .The Standard Atmosphere .52. Compre s s i b i l i t y o f A i r .53. The Propagation o f Small Dis turbences i n A i rSound and Sound Waves .54. The Speed o f Sound as a Cr i t er io n f o r th e Co mp res s ib i l i t yo f Gases .55. The Mach Number and i t s Value i n F l i g h t Problems .56. F l i g h t Speed. Cor rec t ion s t o Instrument Readings Neces si tate dby Compress ib i l i t y .7. The Charac ter o f the Propagat ion o f Minor Per turbat io nsi n F l i g h t a t V ar io us A l t i t u d e s .58. Trans - o r Supersonic F low. o f A i r Around Bodies .59. Sonic "boom".510. Features o f th e Formation o f Compression Shock Du rin g FlowAround Various Shapes o f Bodies.

9 1 1 . C r i t i c a l Mach Number. The E f f ec t o f Com pres s ib i l i t y on theMo ti on o f A i r F l y i n g Around a Wing .912. The Dependence o f th e Speed o f th e Gas Flow on t he Shapeo f the Channel. The Lava1 Nozzle .13. Laminar and Turbul ent F low o f A i r .514. Pressure Di s t r i bu t i o n a t Sub- and Su pe rc r i t i ca l Mach NumbersChapter I I . Aerodynamic Ch ar ac te r i s t i cs o f the Wing and A i r c r a f t .The E f f ec t o f A i r Compres s i b i l i t y .

5 1 . The Dependence o f the Co e ff ic ie nt c on the Angle o f At tac k .Y92. The E f f e c t o f th e Mach Number on the Behavior o f th e Dependencec = f ( a ) .Y93. The Per mis sibl e C oe ff ic ie nt c per and i t s Dependence on theMach Number . Y54. Dependence o f the Co ef f i c i en t c on the Mach Number f o r F l i g h tYa t a Constan t Angle o f A t tac k .55. The Af fe c t o f the Mach Number o f the C oe f f i c ie nt cx .56. Wing Wave Drag .57. I n te r fe rence .58. The A i r c r a f t Pola r. The E ff e c t o f the Landing Gear and WingMechanization on the Polar .59. The Af f e c t o f the Mach Number on the A i r c r a f t Po lar .Chapter I l l . Some Features o f Wing Co ns tr uc ti on .I. Means o f Inc rea sing the C r i t i c a l Mach Number .

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52. Features o f Flow Around Swept Wings . . 4953, Wing Const ruc t io n i n Tur bo je t Passenger A i r ' c r a f t . * 5354. Drag Propagat ion Between Separate P ar ts o f A i r c r a f t . 59Chapter I V . Ch ar ac te r i s t i c s o f the Power System . . 615 1 . Two -Ci rcu it and Turbofan Engines . . 6152. Basic Ch ar ac te r i s t i c s o f Tur bo j e t Engines . . 6653. T h r o t t l e C h a r a c t e r i s t i c s . 674. High-speed Charac ter is t i cs . . 695. H i g h - A l t i t u d e C h a r a c t e r i s t i c s . . 7156. The E f f e c t o f A i r Temperature on Turbojet Engine Thrust . . 72S7. Thrust Horsepower . . 7398. P os i t i on i ng t he E ngines on t he A i r c r a f t . . 74

Chapter V . Takeof f . . 815 1 . T ax i i ng . . 8192. Stages o f Ta kof f . . 8153. Forces Act ing on the A i r c r a f t Dur ing the Takeof f Run and Takeoff 8454. Length o f Takeoff Run. L i f t - o f f Speed. . 8755. Methods o f Ta keoff . . 8856. Fa i l u re o f Eng ine During Takeo ff . . 907. Inf l uen ce o f Various Factor s on Takoff Run Length . . 9858. Methods o f Improving Takeof f Ch ar ac te r i s t i c s . . 100

Chapter V I . Cl imbing . . 1055 1 . F orc es A c t i ng on A i r c ra f t . . 1052. Determinat ion o f Yost Su i ta b le Cl imbing Speed . . 10753. Ve lo ci ty Regime o f Cl imb . . 11094. Noise Reduction Methods. . 1 1 1S5. Climb ing w i t h One Motor Not Oper ating . . 115

Chapter VI I . H o r i z o n t a l F1 i g h t . . 1165 1 . Diagram o f Forces Ac t i ng on A i r c r a f t . . 11652. Requi red Thrus t f o r Hor i zon ta l F l i g h t . . 11753. Two Ho riz on ta l F l i g h t Regimes . . 12054. I n f l uenc e o f E x te rna l A i r Temperature on Required Thrust . . 12155. Most Favorable Hor izon tal F1 ig h t Regimes. Inf l uen ce o fA1 t i ude and Speed . . 123$6. D e f i n i t i o n o f R eq uir ed Q u a n t i t y o f Fuel . . 12957. F1 i g h t a t the "Ce i l i ngs" . 13158. P e rm i ss ib l e F l y i n g A l t i t u d e s . I n f l u e n c e o f A i r c r a f t Weight . 13359. ' Eng ine Fa i l u re Dur ing Hor i zon ta l F1 i g h t . . 134510. Minimum Per miss ible Hor iz on ta l F l i g h t Speed. . 136

Chapter VIII. Descent . 1385 1 . General Statements. Forces Ac t i ng on A i r c r a f tDuring Descent . . 13852. Most Fa vo ra bl e Descent Regimes . * 13953. Pr ov i s io n o f Normal Condi t ions i n Cabin Dur ingH ig h A l t i t u d e F l y i n g . . 140

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54. Emer gency Descent .Chapt er I X . The Landi ng .51. Di agr ams of Landi ng Appr oach .52. F l i ght Af t er E nt r y i nt o Gl i de Pat h.Sel ec t i on of Gl i di ng Speed .53. St ages i n t he Landi ng .54. Lengt h of Post - l andi ng Run and Met hodso f S hor t eni ng i t .55. Lengt h of Landi ng Run As a Funct i on o f Var i ousOper at i onal Fact ors .56. Speci f i c Feat ur es o f Landi ng Runs on Dr y , I ce o rSnow Cover ed Runways .57. Landi ng wi t h S i de W nd58. The M ni mum Weat her f or Landi ngs and Takeof f s

59. Movi ng i nt o a Second Ci r c l eChapt er X . Corner i ng .51 . Di agr am of For ces Operat i ng Dur i ng Corner i ng .52. Cor ner i ng Par amet er s .Chapt er X I . St a bi l i t y and Cont r ol abi l i t y o f Ai r c r af t5 1. Gener a l Concept s on Ai r c r af t Equi l i br i um .52. St at i c and Dynam c St abi l i t y .53. Cont r ol l abi l i t y of an Ai r c r a f t .54. Cent er i ng of t he A r cr af t and Mean Aer odynam c Chor d55. Aer odynam c Cent er o f W ng and Ai r c r af t .Neut r al Cent er ng56. Longi t udi nal Equi l br i um .57. St at i c L ongi t udi na Over l oad St abi 1 i t y .58. Di agr ams o f Moment s .59. St at i c Longi t u di nal Vel oc i t y St a bi l i t y .510. Longi t udi nal Cont r ol l abi 1 t y .51 1 . Const r uc t i on of Bal anc i ng Cur ve f o r Def l ec t i ono f El evat or .512. Ver t i c al Gus t s . Per m s s i bl e M Number i nCr ui s i ng F1 i ght ,513. Per m ss i b l e Over l oads Dur i ng a Ver t i cal Maneuver514. Behavi or of Ai r c r af t a t L ar ge Angl es of At t ack .515. Aut omat i c Angl e of At t ack and Over l oad Devi ce .516. Lat er al St a bi l i t y .517. T r a nsv er se St a t i c St abi 1 i t y518. Di r ec t i onal St at i c St abi 1 i t y .519. Lat er al Dynam c St abi 1 i t y .520. Yaw Damper .521. Transver se Cont r ol 1 ab 1 i t y .522. Di r ec t i onal Cont r ol l abi l i t y . Rev er se Reac t i onf or Banki ng .923. I nvol unt ar y Banki ng ( ' l Val ezhka' l )

. 144

. 150. 150. 151. 154

. 158

. 163

. 164. 167. 168

. 171'. 173. 173. 174

. 177. 177. 178. 181. 184

. 185. 188. 190. 194. 195. 197

. 199

. 203. 205. 206. 212. 213. 214. 216. 2i 6. 218. 223

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124. I n f lu e n c e o f C o m p r e s s i b i l i t y o f A i r on Contro lSu r face E f fec t i veness . . 230525. Methods o f Decreas ing Forces on A i r c r a f t Cont ro l L e v e r s . . 231526. Ba lanc ing o f the A i r c r a f t Du r ing Takeo f f and Landing . 233Chapter X I I . I n f l u e n c e o f I c i n g on F l y i n g C h a r a c t e r f s t i c s . 236l.General Statements . . 23652. Types and Forms o f I ce Deposi t i o n . In t en si y o fI c i n g . . 237S3. I nf lu en ce o f I c i n g on S t a b i l i t y and C o n t r o l l a b i l i t yo f A i r c ra f t i n Pre - land ing Guide Regime . . 239

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NTRODUCTI ONJet-powered passenger a i r c r a f t have been adopted and introd uced i n t o / 3*g en e ra l u se i n c i v i l a v i a ti o n .The f i r s t t u r b o j e t p a s se n g er a i r c r a f t b u i l t i n t h e S o v i e t Union w a s t h eTu-104, and t h e f i r s t f o r e i g n t u r b o j e t s w ere t h e De Havi l land Comet, t heSud Av iat ion C ar av el le , t h e Boeing-707, th e Douglas DC-8, th e Convair 880 ando the r s . These a i r c r a f t have been g iven the name f i r s t -ge ner a t i on tu rbo je ta i rc ra f t .I n b u i l d i n g t h e f i r s t t u r b o j e t p as s en g er a i r c r a f t , t h e d e si g ne r s a t t em p te dt o ach ieve long f l ig h t range and to pe r f ec t t he h igh -speed p rop er t i e s o f t he

a i r c r a f t , thereby compensat ing fo r th e heavy f ue l consumpt ion requi r ed by thej e t eng ines . The de s i r e t o c r ea t e new a i r c r a f t capab le o f compe ting wi thth e o ld passenger a i r c r a f t which were equipped with highl y economic pi st one n gi ne s l e d t o a maximum inc reas e i n the l i f t i n g capac i ty , and f l i g h t d i s tance and speed. The r e a l i z a t i o n of t h e s e q u a l i t i e s became p o s s i b l e o n lybecause of th e appearance of j e t engines .E xp er ie nc e i n u s in g a i r c r a f t ha s shown t h a t t u r b o j e t p a ss e ng e r a i r c r a f tmay be economic no t only i n terms of long-r ange f l i g h t , bu t f o r medium- andeven s h o r t -r a n g e f l i g h t as wel l . As a r e s u l t , s e c on d -g e ne ra ti on t u r b o j e tpassenger a i r c r a f t have appeared: i n the Sovie t Union the re a re the Tu-124,th e Tu-134 and the Yak-40, wh ile abroad the re are- th e D e Havilland-121"Tr iden tf1 , t he Bak-1-11, th e Boeing-727, t h e DC-9'and o th er s. These air

c r a f t a r e s u b s t a n t i a l l y s m a l l e r i n d im en si on s and i n t en d e d f o r u s e on s h o r t -range nets . The high power and low un i t l oad on the wing pe rmi t f l i g h t sfrom a i r f i e ld s having r e l a t iv e l y sho r t t ake -o f f and l and ing runways.Turbojet engines surpass p i s t o n e ng in es i n r e l i a b i l i t y . With t h e i rsh or t time i n se r i e s p roduc tion and use , se rv i ce pe r iods o f 2 ,000 - 3,000hours between maintenance checks have been es ta bl is he d. This i s an importantf a c t i n i n c r ea s i n g t h e economy o f u s in g t u r b o j e t a i r c r a f t , b ec au se t h e c o s tof th es e engines su bs ta n t i a l l y exceeds th a t of p i s ton eng ines . I n the F iveYear Plan f o r the development of t h e Russian economy from 1966 t o 1970, thef u r t h e r development o f c i v i l a v i a t i o n i s an t i c i pa te d and the volume of a i r -4t r a v e l s h ou l d i n c r e a s e by a f a c t o r of 1.8. New pass enge r a i r c r a f t are going

i n t o s e r v i c e i n t h e a i r l i n e s .T u rb o je t p as s en g er a i r c r a f t ha ve f l i g h t c h a r a c t e r i s t i c s which d i f f e r fromthose of a i r c r a f t wi th p i s ton and tu rboprop engines i n seve ra l r e spe c t s .These f l i g h t f ea tu res r e s u l t from the un ique h igh- speed and h i gh- a l t i t u dec h a r a c t e r i s t i c s o f t h e e n g in e s, as w e l l as t h e f l i g h t c o n d i t i o n s a t t h e s eh igh speeds and a l t i t udes .

..* Numbers i n the marg in in d i ca t e pag ina t ion i n the fo re ign t e x t .vii

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With the appearance of j e t a v i a t i o n , t h e r e ha s be en a r e s u l t a n t i n c r ea s ei n th e impor tance of h igh-v eloci ty aerodynamics , i . e . , th e motion o f bodiesi n air v iewed i n terms of t h e e f f e c t of i t s c o m p r e s s i b i l i t y , i . e . , t h ep r o p e r t i e s t o c hange d e n s i t y w i th a change i n pressure . . 'The f i r s t t o i n di c at et h e n e c e s s i t y o f e s t i m a t i ng t h e e f f e c t of air compress ib i l i t y w a s the Russians c i e n t i s t S.A. Chaplygin, i n h i s work "On G a s Flows" publi she d i n 1902. I twas he who de vel ope d a method fo r the t he o r e t i c a l so lu t ion o f prob lems o f th emotion of gas with allowance made f o r i t s compress ib i l i t y .

The Sovi e t s c i e n t i s t s Academicians S.A. Khr is t ianovich , M.V. Keldysh,A.A. Doro dnits yn, Pro fe ss ors V.S. Pyshnov, F . I . Frankl ' , . V . Ostoslavskiy ,B.T. Goroshchenko, Y a . M . Serebr iysk iy , A . P . Mel'nikov and ot he rs , throught h e i r s t u d i e s i n the f i e l d o f h igh -ve loc i ty aerodynamics , contributed muchwhich w a s of g re a t va lue i n the des ign o f h igh -speed a i r c r a f t .

The Sov ie t t u rbo j e t passenger a i r c r a f t c r ea t ed by ae rona u t i c a l eng inee r sA.N. Tupolev, S.V. I l us hi n and A.S. Yakovlev, ta ke t h e i r pla ces i n th e rankso f t h e f i r s t - c l a s s a i rc ra f t .The su cc es sf ul use of new av ia t i on technology by*fl i g h t and engin eeringpersonnel i s unthinkable wi thout a deep understanding of th e laws of aero

dynamics .Ai rc ra f t aerodynamics, when thought of i n terms of the f l i g h t crew, i s

us ua l ly c a l le d p ra ct ic a l aerodynamics. The number of problems involved i naerodynamics i s q u i t e s u b s t a n t i a l . T hese i n c l u de s t u d y in g t h e laws of themotion of a i r a n d t h e i n t e r a c t i o n of a i r flows with bodie s moving i n them,t h e i n t e r a c t i o n o f s ho ck waves w i t h v a r i o u s p a r t s o f t he a i r c r a f t , a i r c r a f tf l ight dynamics as a f f e c t e d by t h e f o r c e s a p p li e d t o t h e a i r c r a f t ( in c lu d in gaerodynamic forces) , and a i r c r a f t s t a b i l i t y and ha nd in es s.

I t i s t h e o b j e c t of th is book t o examine these quest ions i n terms oft ur bo e t p as s e n g e r a i r c r a f t .

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N A S A TT F-542CHAPTER 1

THE PHYSICAL BASIS OF H I G H - S P E E D A E R O D Y N A M I C S

ABSTRACT. T h i s book presents t h e physical bases of h i g h -s p e e d aerodynamics, and t h e i n f luence o f a i r compress ib i l i tyon t h e ae rodynamic charac te r i s t i c s o f w i n g s a n d a i r c r a f t .P r imary a t t en t ion i s turned t o passenger j e t s . T h e fol lowinga r e a s a r e c ov er ed : t a ke o ff c h a r a c t e r i s t i c s of j e t s andmethods o f Improving them; b e s t c l i m b i n g modes; horizontalf l l g h t ; t h e descen t ; t h e landing approach; turns and corners;c o n t r o l l a b i l i t y a nd s t a b i l i t y ; i ci ng and i t s influence onf l y i n g c h a r a c t e r i s t i c s ; a n d t h e c h a r a c t e r i s t i c s o f modernj e t e n g i nes .1 . Var ia t ions i n the Parameters of Air w i t h A l t i t u d e . T h e StandardAtmosphere

The f l i g h t of a i r c r a f t , l i k e t h a t o f o t h e r f l i g h t v e h i c l e s , i s a f f e c t e d /5by th e condi t ion of the a tmosphere the s h e l l of a i r sur rounding the ear th .Therefore, i t i s qu i t e v i t a l t o know the p rocesses occur r ing i n th e abnosphere .Only the a tmosphere 's l ower boundary, t he ea r t h ' s s u r f ac e i t s e l f , i scl ea r l y del inea ted. The upper atmosphere i s more d i f f i c u l t t o e s t a b l i s hbecause the dens i ty o f air decreases co nstant ly wi th a l t i tu de and even a t ana l t i t u d e o f .lo0 km it measures approximately one mi l l io nth th a t on the ear th 'ssurf ace . Normally, th e upper l i m i t of the atmosphere i s considered thea l t i t u d e a t which the air densi ty approaches t h a t of t h e g as es f i l l i n g i n t e r p l ane ta ry space .Data from direct and indi rec t observat ions show that the a tmosphere hasa l a y e r e d s t r u c t u r e . In 1951 th e In te rn at io na l Geodesic and Geophysical Unionadopted the d iv i s i on o f the a tmosphere in to f iv e bas i c spheres o r l a y e r s :th e t roposphere , the s t r a t os ph er e , t he mesosphere, the thermosphere and t h e

exosphere.The Troposphere i s t h e lcwest la ye r of th e atmosphere, which i n th e middlel a t i t u d e s extends t o an a l t i t u de o f 10-12 km, i n t he t r op i c s t o an a l t i t u d eof 16-18 km, and i n the po la r regions t o an a l t i t ud e of 8-10 km Thisl a y e r i s o f t remendous p r a c t i c a l i n t e r e s t i n a v i a t i o n , b ec au se a l l the mostimpor tant phenomena encountered by the p i l o t occur ba si ca l l y i n th e t roposphere . I t i s here t h a t the format ion of c louds and fogs , th e f a l l ofpr ec ip i t a t io n , and the development of storms occur.

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The mos t s ig n i f i ca n t f e a t u r e o f t he t roposphere i s t h e d e cr ea s e i ntemperature wi th a r i s e i n a l t i t u d e ( av er ag in g 6. 5" p e r km o f a l t i t u d e ) . Thetroposphere i s t h e area of thermal turbulence r es ul t i ng from the unequalh e a t i n g o f l a y e r s o f air a t t h e e a r t h ' s s u r f a c e and a t v a r i o u s a l t i t u d e s , aswel l as t he dynamic tu rbu lence r es u l t ing from the f r i c t i on o f the air w i t ht h e e a r t h ' s s u r f a c e and i t s i n t e n s e v e r t i c a l d is pl ac em en t a t the boundar iesbetween cold and warm a i r masses of a tmospheri c f r on t s .

The t roposphere ends in th e lay er of the t ropopause . The th ickn ess ofthe t ropopause f luc tua tes f rom a f e w hundred meters t o s e v e r a l k i lo m e te r s .I t i s u s u a l l y a cont inuous l a ye r which su r rounds the ea r t h ' s sphere i t s e l f ,whi le i t s a l t i t u de and t emperatu re are func t ions o f the geograph ic l a t i t u de ,the t ime of ye ar and the a tmospheric processes developing. Over the equ atorand i t s neighbor ing are as , the t ropopause i s l oca ted a t an average a l t i t udeof 16-18 km ( I n d i a ) , w h i l e i n t h e m id dl e l a t i t u d e s i t i s l oca ted a t ana l t i t u de o f 10-12 km, and i n t h e p o l a r r e g io n s i t has an a l t i t u d e of 8-10 km,while over the pole it may drop t o 5-6 km. J e t a i r c r a f t n o m a l l y f l y c l o set o t he l i m i t of the t ropopause, a ch a r ac te r i s t i c f e a tu re o f which i s t h eexis te nce of cy cl ic bumps beneath th e t ropopause i t s e l f .

The s t r a tosphere i s loc ate d above the tropopaus e and extends t o approximate ly an a l t i t ud e o f 35-40 km. Cons tan t t empera tu re wi th a l t i t ude isc h a r a c t e r i s t i c o f i t s lower layers . The in s i gn i f ic an t content of water vapori n t h e s t r a t o s p h e r e r e s u l t s i n t h e l ac k of c lo ud s from which p r e c i p i t a t i o nwould f a l l . According t o d at a from p i l o t s who have f lown a t a l t i t u d e s o f12-16 km, i n t he lo we r s t r a t o s p h e r e i t i s most f re qu en t ly c loud less . The a i ri s s t a b l e and v e r t i c a l m o tio n i s s l i g h t . T hi s a i d s i n smooth f l i g h t . T he rei s seldom bumpiness, and only then clos e t o the tropopause.

The mesosphere runs from the upper boundary o f th e s tr at os ph er e to ana l t i t u d e o f 80 km.The thermosphere i s lo ca te d above th e mesosphere and exte nds t o ana l t i t u d e of 800 km.The exosphere i s t h e o u t e r l a y e r of t h e a tm os ph er e, o r t h e d i s s i p a t i v elayer , and i s located above the thermosphere. Gases h e r e a r e so r a r e f i e d a n da t the high temperatures o bs er ve d t h e r e h av e su ch h ig h v e l o c i t i e s t h a t t h e i rpa r t ic le s (hel ium and hydrogen) break away f rom the e ar th 's a t t ra c t i v e forc eand move in t o in t er p l an et ar y space .Thus we have a b r i e f d e s c r i p t i o n of a s t ru c t u r e of t he a tmosphere .Atmospheric condi t ions ar e cha racter i zed by the var ious meteorologicale lements a tmosphere pre ss ure , temperature , humidi ty , c loud cover , pr ec ip i t a t i o n , wind, e tc . The atmosphere may be cha rac ter i zed as a variable medium.

5

As a re s u l t of unequal heat in g of the a i r masses a t the equator and po les ,f lows a re formed which r e s u l t i n the passage of cold a i r toward the equator andwarmer air to ward t he p o l e s . The e f f e c t of t h e e a r t h ' s r o t a t i o n i n th enorthern hemisphere causes the a i r f low to d ev ia te t o the r i g h t and move from

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7th e south t o the southwest , whi le approaching 30 N it moves t o th e west.T h er ef o re , f l i g h t s from w es t t o e a s t o ve r th e t e r r i t o r y of t h e USSR a r eaccompanied by t a i l winds, whi le east - to-west f l i g h t s encounter head winds.The s h i f t from wes te r ly winds t o ea s t e r ly occur s a t a l t i t u d e s a r o u n d 20 km.Whereas p i s t o n a i r c r a f t f l y on ly i n t h e l ow er tr o po s ph e re , j e t a i r c r a f t , i nc o n t r a s t , f l y i n t h e u pp er and t o a c e r t a i n e x t e n t i n t h e low er s t r a t o sphere.

The f u r t h er development of h igh- speed av ia t ion w i l l i n t h e n e ar f u t u r ep er mi t us t o f l y a t su pe rs on ic speeds corres ponding t o Mach = 2.5-3. A t t h i sp o i n t , f l i g h t s w i l l b e i n t h e s t r a t o s p h e r e .B ef or e t h e p e r f e c t i o n i n g of j e t a i r c r a f t , i t w a s assumed that a t highal t i tu de s the f l i g h t s would encounter favora ble weather cond i t ions . However,

it w a s found that a t a l t i t u de s o f 10 ,000 - 12,000 m cloud cover and bumpinesswere sometimes encountered. To th es e well-known phenomena, t h e re were addedt h e j e t s tr e am s c h a r a c t e r i s t i c o f a l t i t u d e s of 9-12 km.The j e t s t reams are the broad expanses of zones of very st ro ng winds

observed i n the upper l a ye r s of t h e t ro p os p he re , u s u a l l y a t a l t i t u d e s of9000 - 12,000 m . Post -war s t ud ie s showed th at the minimum vel oc i t y of the j e tstream (along i t s ax is ) eq ua ll ed ap prox ima tely 100 km/hr, wh il e t he maximumw a s 750 km/hr (over th e P a c i f i c Ocean). Over th e U S S R , the wind speed i n th ej e t stream reaches 100 - 200 and sometimes even 350 km/hr, while over theNorth Atlantic and Northern Europe it reaches 300 - 400, 500 ove r t h e USA,and 650 km/hr over Japan. The j e t s t r eam i s comparable t o a g i g a n t i c h i g h l yobla te channel wi th a height averaging 2 - 4 km and a width of 500 - 1000 km.These f lows run bas i ca l ly west-east, bu t i n ce r t a i n sec t ion s they may varys i g n i f i c a n t l y.

Fl igh t speed may be increase d by the s e le c t i v e use of j e t s t ream t a i lwinds, whi le f l i g h t a ga ins t the head wind should be one o r two km above o rbelow the axis of t h is s t ream. A s a r u l e , t h e j e t s tr e am s a r e t o be f ound i nth e region where the tropopaus e i s s i t u a t e d .I n s t u dy i n g a i r c r a f t f l i g h t and de te rm in in g t h e f o r c e s a c t i n g on a i r c r a f t ,we may consider the a i r a s a continuous medium.A t s e a l e v e l , t h e a i r co ns is t s of a mixture of n i t rogen (78.08% of thevolume o f d ry a i r ) , oxygen (20 .95%) and ins ig n i f i ca n t qu an t i t i e s o f o the rgases (argon, carbon d ioxide , hydrogen, neon, hel ium, e t c . ) . The a i r a l soconta ins water vapors .In the troposphere and s t ra to sp he re th e temperature , press ure and

d e n s i t y of t h e a i r v a ry w i t h i n r a t h e r br oa d 1 i . m i t s as a funct ion of the geog r a p h i c l a t i t u d e of t h e l o c a l e , t h e t im e of ye ar , the t ime of day and th eweather .In o rde r t o ach ieve a common concept of t he c h a ra c te r i s t i c s of theatmosphere (p ressu re, temperature and den si t y) , he standard atmosphere w a s

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a r r i v e d a t t h e a r b i t r a r y d i s t r i b u t i o n , i n t h e a tm os ph ere , o f p r e s s u r e , / 8den s i ty and t empera tu re f o r d ry , clean a i r ( c o n t a in i n g n e i t h e r m o i s tu r e n o rdus t ) o f a cons tan t composi tion app l i cab le f o r eng inee r ing . -- pr imar i lya v i a t i o n c a l c u l a t i o n s w i th r e s p e c t t o t h e i r c o m p ar a b il i ty ( f o r exam ple, i nc a l c u l a t i n g t h e l i f t and d rag and f o r g radua t ing va r ious aer ia l nav iga t ioninst ruments such as al t imeters and o the r s ) .In th e s tanda rd a tmosphere , t he a l t i t u d e i s computed f rom sea level .Normal conditions a t sea l e v e l are: a tmospheri c p r es su r e p 0 = 760 mm Hg, a i rd e n s i t y p = 0.125 kG sec /m 4, emperature t0 -- 15OC (o r To = 288OK) and3s p e c i f i c w ei gh t o f t h e a i r y0 = 1.225 kG/m .V a r ia t i on s i n a i r p r e s s u r e a nd d e n s i t y w i t h a l t i t u d e , which pr oc ee d i naccordance with a s p e c i f i c l a w , are c a l c u l a t e d p e r e ac h a l t i t u d e a cc or di ng t osp e ci a l formulas . The air t empera ture i n t he s t a nda rd a tmosphere up t o ana l t i t u d e o f 1 1 ,0 00 m drops uniformly by 6.5OC per 1000 m. Above 11,000 m ,the temperature i s cons ide red cons tan t and equa l t o -56.5OC. In fac t , however, a t t h i s a l t i t u d e it may reach -8OOC. Resul t s of c a l c u l a t i o n s a r e

given i n the tab le . Below w e p r e s e n t an a b b r e v ia t e d t a b l e o f t h e s t a n d a r datmosphere.TABLE 1. STANDARD ATMOSPHERE (SA)

-A l t i - f Tempera- Mass l e l a t i v Ao.7 Speedtude ,I t u r e dens i ty l ens it y o f,m (tH) > O C7j kG/m3 m 4 km/hrII

1000 21.5 854,6 - 1,3476 1,1374 1,096 12420 15 760 j: 1O332,3 1,225 0,1250 1,oo 12251 000 8,5 674 9164,Z. 1.11 0,1134 0,9074 12112000 2,o 596 8105,4 1,006 0,1027 0,8215 11973000, -4.5 526 7148,O 0,909 0,0927 0,742 11834000 I -1 1 462 6284,2 0,819 0,0636 0,6685 0,754 324.7 11685 000 -17.5 405 ii 5507,O 0,7362 0,0751 0,6007 0,70 . 320,7 11546 000 -24,O 354 4809,5 0,659 0,0673 0,5383 0,648 316,6 11397000 -30,5 308 4185.3 0,589 0,0601 0,4810 0,599 ~ 312,4 11258000 I -37,O 267 3628,4 0,525 0,0536 0,4285 0,553 30S,2 1110go00 i -43,5 230 3133.1 0,466 0,0476 0,3805 109410000 -50,5 188 2694,O 0,412 0,0421 0,337 107811000 i -56,5 i 169,6 2306.1 0,363 0,0371 0,297 106312000 1 -56,5 144,6 1969,5 0,310 0,0317 0,253 106313000 * -56,5 123.7 1682,O 0,265 0,0270 0,216 106314000 -56.5 105;6 1436,5 0,226 0,0231 0,185 106315000 -56,5 90,l 1226,9 0,193 0,0197 0,155 106316000 -56.5 77,l 1047,8 0,165 0,0166 0,135 106317000 -56.5 65,8 894,8 0,141 0,0144 0,115 106318000 -56,5 56,2 764,2 0,120 0,123 OI09S4 106319ooa -56,5 48 ,O 652,7 0,103 0,0105 0,084 106320 000 -56,5 40,9 557,4 0,088 0,009 0,0717 1063

Tr. Note: Commas i n d i c a t e d e c i m a l p o i n t s .

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5 2. Cmpressib i 1 i t y of A i rCompressib i l i ty i s t he p roper ty o f gases (and f lu i ds ) t o change th e i ri n i t i a l volume (and , consequen tly , dens i ty ) under the e f f e c t o f p r essu re o r achange i n temperature.In so lv in g t echn ic a l prob lems , compress ib i l i t y i s t aken in to account i nthose cases when changes i n volume (density) are considerable by comparisont o t h e i n i t i a l volume ( d e n si t y ).If the volume of water w i t h an i n c r e a s e i n p re s s u re of 1 a t . withconstant temperature changes an average of only 1/21,000 o f i t s i n i t i a l v al ue ,i . e . , only 1/210 of a percen t , a i r , which has a h ig h c o m p r e s s i b i l i t y , r e q u i r e sa change i n pressu re of only one one hundredth that of atmosphere (0.01 at .)t o change i t s volume by 1%under normal atmospheric con dit i ons .Therefore, a l l gases are consid erably more compressible tha n droppingli qu id . For example, i f t h e pr e s su r e i n a given m a s s of g a s i n c r e a s e s i nsuch a way that i t s temperature does no t vary during t h i s change, the volume

of the gas decreases . When the i n i t i a l pressur e i s doubled, t h e volumedecreases by 50%. .The change i n volume f o r gas i s equal ly h igh dur ing heat ing .Di f fe r ences i n com press ib i l i t y o f l i q u id s and gases a r e exp la ined byt h e i r m ol ec u la r s t r u c t u r e . I n l i q u i d s , t h e i n t er - m o le c u l ar d i s t a n c e i s small ,i . e . , the molecules a re ra th er dense , which determines the small c a p a b i l i t yliquids have of compressing. By comparison wi th l i q u i d s , ga ses have anextremely low de ns ity . For example, the dens ity of water i s 816 times that ofa i r . The low de n s i t y of a i r and o ther gases i s e x pl a in e d by t h e f a c t t h a t i ngases the in ter -molecula r d is tan ce sub st an t i a l ly exceeds the d imensions ofth e molecules themselves. Therefo re, when th er e i s an in c rea se i n the p ressu re,the volume of t he gas decreases due to the decreasing d is tan ce betweenmolecules . Thus a r i s es the e l a s t i c i t y which gas possesses .In av ia t i on problems, the need to account f o r a i r c o m p r e s s i b i l i t y r e s u l t sfrom t he f a c t t h a t a t h ig h f l i g h t s pe ed s i n a i r , s u b s t a n t i a l d i f fe r e nc e s i np ressu re a r i s e which are t h e cause o f su bs ta n t i a l changes i n i t s dens i ty .To eva lua te the e f f e c t o f compress ib i l i t y , l e t us examine the speed ofsound.

3 . T h e Propagation o f Small Disturbances i n Air. Sound and Sound Waves.The proper ty of compre ssib i l i ty i s in t imate ly re la ted to the phenomenonof th e propagat ion of sound i n gases . The speed of th e propagatio n of soundp lays a v i t a l r o l e i n h igh -speed aerodynamics. The e f f e c t o f compress ib i l it y

on the ae rodynamic charac te r i s t i c s of a i r c r a f t i s a funct ion of t he degreeto wh ich the f l i g h t speed of t h e a i r c r a f t appro ache s the spe ed of sound. Whenair flows a t speeds g re a t e r t han th e speed o f sound , qu a l i t a t iv e changes occur /10i n t h e c h a r a c t e r o f t h e f lo w.The s en sa t i on which w e p e r c e i v e as sound i s t h e r e s u l t o f t h e e f f e c t , on

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our aud i to ry appara tus , o f t h e o s c i l l a t o r y mot io n o f a i r caused, f o r example,by t he motion of some body i n it. The disp lacement of each p a r t i c l e of a i rduring i t s v i b r a t i o n i s i n s i g n i f i c a n t l y small. T h e p a r t i c l e s v i b r a t e a r o u n dth e i r equ i l ib r ium conf igu ra tion , which co rr esponds to th e i r i n i t i a l s t a t e .However, the lab ora tory proc ess i s propagated a very long d i s t ance .The human ear perce ives as sound those d is turb ance s which a re t rans mi t tedwi th a f requency f rom 20 t o 20 ,000 v ib ra t i on s pe r second. Those with afrequency of less than 20 pe r second are ca l l ed inf rasound , and those above20,000 pe r second a r e ca l l e d u lt r a sound .By small d i s tu rbances w e mean s l i g h t changes i n the p res su re and den s i tyof the medium (gas o r l iq ui d) . Disturbances being propagated i n t he medium,such as a i r , a r e ca l l e d waves (due t o the s im i l a r i t y o f t h i s phenomenon t owaves on th e su r f ac e of water ) .The speed of the propagat ion o f t h e d is turban ces i n space ( t he wavev e l o c i t y ) i s qu i t e su bs ta n t i a l . The speed o f p ropaga tion o f a sound wave,i . e . , small changes i n de ns i ty and p res su r e , i s ca l l ed th e speed o f sound.

I t i s a func t ion of th e medium i n which th e sound i s being propagated andof i t s temperature.In high-speed aerodynamics, sound i s considered as waves of p e r t u r b a t i o n sc r ea t ed i n t h e a i r by a f l y i n g a i r c r a f t .The speed of sound i n gases i s a funct ion of temperature . The h igher thegas temperature, the l e s s compressed it i s . Heated gas has a hig h e l a s t i c i t yand the re fo re i s more d i f f i c u l t t o compress . Cold a i r i s easi ly compressed .For example, a t a gas temperature T = 0 (o r t = -273OC), the speed of soundequals zero because under th es e condi t ions the gas pa r t ic le s ar e immobile ande x e r c i s e o n ly s l i g h t d i s t u r b an c e s , w i th t h e r e s u l t t h a t t h e y c an c r e a t e no

sound .The dependence o f th e speed o f sound in a i r on temperature may bedetermined according t o th e fol lowing approximate formula:a = 20 JTm/sec .

With in the l i m i t s of t roposphere, the a i r temperature decreases wi tha l t i t u d e . Consequent ly , i n the t roposphere the speed of sound a l so decreasesw i t h a l t i t u d e . On t h e e a r t h ' s s u r f a c e u nd er s t a n d a r d co n d i t io n s (p = 760 mmHg, t = 1 5 s e c ) , a = 340 m/sec. With an i n c r e a s e i n a l t i t u d e f o r e v er y 250 m , -11the speed of sound decreases by 1 m/sec.

A t al t i tudes above 11 ,000 m , the temperature i s ( according t o thesta nda rd atmosphere) con sidered cons tant and equa l t o -56.5OC. Consequently,the speed of sound a t t he se a l t i t u de s shou ld a l s o be conside red cons tan t ande q u a l t o a = 20 4273 - 56.5 = 296 m/sec (Fig . 1 ) .

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ec.

1 . The Change i nSpeed of Sound w i t h1 t i u d e .

4. T h e S p e e d of Sound as a C r i t e r i o n f o r t h eCompress i b i 1 i t y of GasesIn gas dynamics, f o r the speed of soundt h e r e i s th e well-known formula:

m/sec,AP

where Ap i s the change i n pre ssu re , Ap i s t hechange i n gas de ns i ty which it ca us es . The morecompressed the gas i s , the s lower the speed ofsound, s o th a t one and th e same change i n de ns i tymay be obtained through a s l i g h t change i npressu re . And, i n co nt ra s t , the les s the comp re ss ib i l i ty of t he medium and the gr ea te r i t se l a s t i c i t y , t h e g r e a t e r t h e sp ee d o f sou nd i nth e same medium. In t h i s cas e, a s l i g h t changei n de nsi ty may be achieved only through a g r e a tchange i n pr es su re . The speed of sound i s takent o c ons i der a t i on i n any c a s e i n which t h e r e i s an eva lba t ion of the e f fe c t ofpre ssib il i t y i n any aerodynamic phenomena, because t he val ue of the speed ofnd ch ar ac te ri ze s the c om pre ssib il i ty of the medium. I f th e medium i sl a s t i c (comp ress ible) , compress ions and expansions w i l l v ar y s u b s t a n t i a l l yom la ye r t o l ay er with t he speed of sound. I f t h e medium i s absolu te lympress ib le , i . e . , fo r any inc re ase in pre ssure the volume o r dens i tyins unchanged, then as can be seen from th e formu la give n above, th e speed

w i l l be q u i t e h igh . In such a medium, any di st ur ba nc es a r e propany d i s tan ce ins tan taneous ly .A s was shown above , the va lue of the speed of sound va r i es i n di f f e r en t

s a nd , i n a dd i t i on , it i s a func t ion of temperature . With an in cr eas e i nt i t u d e , tempera ture and the speed of sound decrease . Therefore, t he e f fe c tf c om pr e s s i b il i t y on t he f l i gh t o f a i r c r a f t a t h i gh a l t i t ud e s s houl d a ppea rven gr ea t e r . Let us in t roduce sev e ra l va lues fo r the speed o f sound a t= 0 C : f o r n i t r o g en it i s 3 3 7 . 3 , f o r hydrogen it i s 1300, and f o r water its 1450 m/sec.

For s o l i d bodi es , which are l es s compress ible than gases , th e speed ofi s s t i l l g re at er . Thus, i n wood th e speed of sound i s 2800 m/sec, whilei t i s 5000 and i n g la ss it i s 5600.

An a i r c r a f t i n f l i g h t , r e p e l l i n g a i r on a l l s i de s , pa r t i a l l y c om pr e s s e st as wel l . A t low f l i g h t speeds , the a i r i n f r o n t o f t h e a i r c r a f t su cc ee dse i ng d i s p l a ce d a nd ada pt s i t s e l f t o t he f low around t he a i r c r a f t so t h a ti s i n s i g n i f i c a n t i n t h i s c as e. A t high e r f l i g h t speeds , however ,a i r compression begins t o pla y a more important ro le . In t h is case , therea s c a l e o f f l i g h t sp ee d w e must use a c h ar a c t er is t ic speed which may /12a s a c r i t e r io n f o r the compress ib i l i ty of th e medium. Such a speed i se speed of sound, inasmuch as i t i s a funct ion of the tempera ture and

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pr op e r t i e s o f t he ga s. 5. T h e Mach Number and i t s Value i n F l i g h t Problems

The r a t i o of t h e f l i g h t (or f low) speed t o th e speed of sound i s c a l l e dt h e Mach number:

L et u s assume t h a t t he t r u e f l i g h t s pee d ( s e e 6 of this Chapter) of ana i r c r a f t a t an a l t i t u d e o f 1 0,00 0 m i s 920 km/hr (255 m/ se c) . Then the Machnumber M = 255 - 0.85, where a = 300 m/sec. I n o t h e r w or ds , t he f l i g h t spee d3 0 0i s 85% of th e speed o f sound a t t h i s g i v e n a l t i t u d e .

Thus, i n comparing th e speed of t he motion of the body i n t he a i r withthe speed of sound under th e same cond i t io ns , w e may determine the effect ofa i r compress ib i l i ty on the cha rac te r of th e f low a round the body. The Machnumber i s t he i nde x o f t he air compress ib i l i ty . The g r e a t e r t h e Mach number,t h e g r e a t e r t h e a i r c om pr e s s i b il i t y s hou l d be du r ing f l i g h t .

To monitor th e Mach number i n f l i g h t , an ins t rumen t the Mach in di ca to r(Machmeter) i s us u a l l y s e t up on t he p i l o t ' s i n s tr um e n t pa ne l. I n h igh-s p e e d f l i g h t , es pe c i a l ly when maneuvers a r e be i ng performed which r e s u l t i na los s of a l t i t u de , t he read ing on th i s ins t rument must be fo l lowed, and th ep i l o t must not exceed the Mach number which th e in s t ru c t io ns permit f o r theg i v e n a i r c r a f t . I f f l i g h t speed remains cons tan t as a l t i t u d e i n c r e as e s , t h eMach number w i l l inc reas e due t o the dec rease i n the speed of sound.

Fa i lur e t o monitor t he Mach number i n j e t a i r c r a f t would r e s u l t i n g r avetro ub le because knowing the i nd ica te d speed ( see 6 of this Chapter) and event h e t r u e s pe ed do es n o t g i v e t h e p i l o t a f u l l unde r s t a nd i ng of th e f l i g h t Machnumber a t any sp ec i f i c a l t i tu de . For example , i f t h e a i r c r a f t i s f l y i n g a t anin di ca te d speed of 500 km/hr a t an a l t i t u d e of 12,000 m, t he t r ue s pe e d w i l lbe around 930 km/hr whi le t h e speed of sound i s 1063 km/hr, so t h a t underth es e given f l i g h t co ndi t ion s the Mach number = 0.875. I f , however, thea i r c r a f t i s f l yi ng with an in di ca te d speed of 500 km/hr a t an a l t i t u d e o f1000 m , t he t r u e s pee d i s on ly 525 km/hr, wh il e t h e Mach number = 0.43.

I n t u r b o j e t a i r c r a f t , a change i n t h e Mach number may be re pr es en te d i nthe following way. A f t e r t a ke o f f a nd r e t r a c t i o n of t h e l a nd ing ge a r andwing f l a ps , t he a i r c r a f t p i c ks up spee d u n t i l i t achieves an i nd ica ted speedof 500 - 600 km/hr and s t a r t s c l i m b i ng . S t a r t i ng a t an a l t i tu d e of around1000 m , the Machmeter shows a Mach number of M = 0.5 - 0.55. As t h e a i r c r a f tc l imbs , the t r ue speed w i l l inc reas e , the speed of sound w i l l decre ase, and /13he Mach number inc rea se. When th e ai r c r a f t reach es an a l t i t u d e of 8-9 km,the Mach number reaches a val ue of 0.63 - 0.66 (depending on th e ac tu altempera ture a t t h a t a l t i t u d e ) . A t a l t i t u de s o f 10-12 km dur i ng a c c e l e r a t i onth e Mach number in cr ea se s t o 0.80 - 0.85. A t high a l t i t u d e s t he Mach number8

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be gr ea te r when t he same t rue speeds are maintained. T u r b o j e t a i r c r a f t ,i k e many ot he r high-speed a i r c r a f t , have a l i m i t t o t h e i r Mach number becausecondi t ions of s t a b i l i t y and handiness (more w i l l be sa id conce rn ing t hele c t io n of th e Mach number i n Chapters 7 and 11). There fore (e spec ia l ly a tit i s i n s u f f i c i e n t t o m on it or f l i g h t s im pl y w it h r e s p e c t t od; th e Mach in di ca to r m u s t a l so be observed.

5 6 . F1 i g h t S p ee d . Corre ction s t o Instrument Readings Necessi tated byCompressibil i tyAircraft speed in d i ca t or s measure d i r ec t l y no t on ly the speeds, bu t t he2= p V / 2 . The a c t u a l f l i g h t s pe ed i s no t t he same a s t h i si s ind ica t ed by th e ins trument , because the a i r - pre s sur e sensort h e e f f e c t of p e r t u r b a t io n s crea ted by th e a i rc ra f t and the a i rcompress ib i l i ty . In addi t i on , th e va lue of the ac tua l f l i g h t speed dependson ins t rumenta l cor rec t ions .The re fore , t o e l imina te th e above -mentioned e r r or s i n th e ins trumentgs , the fol lowing co rre c t i on s a r e int roduced: aerodynamic , which

a cc ount s f o r t he d i f f e r e nc e i n t he l oc a l p r e s s u re s ( a t t he po i n t w here t hea i r - p r e s s u r e s e n s o r i s loca ted) from pressures i n the undis turbed in c id entf low, cor rec t ion s f o r com pres s ib i l i ty , and inst rument cor rec t ions* .The speed which would be shown on an id ea l ( i . e . , er ro r- fr ee ) speedi n d i c a t o r i s c a l l e d t he i nd i c a t e d s pe ed Vi ' The speed which i s read from theinstru ment (read from the wide ne ed le ) , does not as a r u l e e q ua l t h e i n d i c a t e dspeed. Therefore, a sp ec ia l name has been crea ted f o r it instrument speed

' inst 'The t rue a i r speed i s t h e s pe ed of t h e a i r c r a f t ' s m otio n r e l a t i v e t o t h ea i r (and i s read f rom the thin a r row on the ins t rument) .The KUS11200 combined speed in di ca to r, which j e t a i r c r a f t f ly in g a tMach speed s up t o 0 .9 are equipped with, shows the instrument speed and thet r u e a i r speed . During low-a l t i tu de f l i g h t (where the a i r dens i ty i s c l o s et o t h a t of t h e e a r t h ' s s u r f a c e , e qu al t o 0.125 kG - sec2/m4), th e ins t rumentand tr u e a i r speeds agree and both arrows on the instru ment move toge the r,be ing super imposed. With an inc r ease i n a l t i tu de , the t r ue a i r speedsurpasses the instrument speed and the arrows diverge, forming a "fork."Knowing t h e t rue a i r speed and wind speed, it i s poss ib le to de te rmine the /14ground s pe ed , i . e . , t he spe ed o f t he a i r c r a f t ' s d is pl ac em en t r e l a t i v e t o t heea r t h. In f l yi ng and aerodynamic computa tions , both the ind ica ted andins t rument speeds are use d. And what i s th e di ff er en ce between them? Toswitc h from instrument speed t o in di ca te d speed, we must intro duce an aero

dynamic correction and a c o r r ec t i o n f o r a i r compress ib i l i ty :

* M . G . K ot ik , e t a l . , Fl ig ht Tes t ing of Air cr af t , Mashinos troyeniye , 1965(Avai lable i n NASA tr an s l a t io n) .

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'ins t = vi + 6Va + 6Vcomp = vi + 6Va,gwhere Vi = i nd i c a t e d s pee d ,

6 V = aerodynamic correc t ion,a"comp = c o r r e c t i on f o r c om pr e s si b i l i ty , and

Vi = indica ted ground speed.g

For h igh- spee d a i r c r a f t , an e s s e n t i a l c o r r e c t i on i s t h e c o r r e ct i o n f o ra i r co mp re ss ib il it y, whose val ue may range from 10 t o 100 lan/hr. The e f f e c tof a i r compress ib i l i ty inc reases the speed in d i ca to r r ead ing , s o t h a t 6Vcompi s always negative (Fig. 2 ) .

400 600 800 lo00 1200 1.~70 Vi , km/hriFigure 2. Nomogram f o r Determining t h e Correc t ion forAir Compressibil i tyThe aerodynamic co rr ec ti o n may reach val ues from 5 t o 25 km/hr and may b e /1 5i t he r po s i t i v e o r ne gat i ve. Whereas t he c o r r e c t i on f o r com pr e s s ib i l i t y i si d e n t i c a l f o r a l l a i r c r a f t , t h e aero dyn amic c o r r e c t i o n i s ba s i c a l l y a f unc t i onof t h e t ype of a i r c r a f t o r , more s pe c i f i c a l l y , t he pos i t i o n and f e a t u r e s of

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t he engine. There fo re , each a i r c r a f t has i t s own graph of aerodynamicc o r r e c t i o n s .The ind ica ted speed wi th th e co r r ec t ion fo r compress ib i l i t y i s c a l l e d t h eindicated ground speed: V.1 = Vi + 6 Vcomp * A t sea l e v e l , i r r e s p e c t i v e o f a i rgtemperature , vi = vi. Accordin g t o t h e nomogram i n Figure 3 , w e may fi n d th e

.Ef l i g h t Mach number b ein g given th e val ue of Vi , and then determine the t ruegf l i g h t s pe ed : Vt = aM For example, we m u s t determine the t rue speed andf l i g h t Mach number f o r th e a i r c r a f t i f a t an a l t i t u d e o f 10,000 m y Vinst -

= 500 km/hr. Taking the aerodynamic correct ion 6 V = -10 km/hr, we find:aVi = 490 km/hr. For t h i s s peed , according t o the nomogram (Figure 2) , w ego b t a i n GVcomp = -2 3 km/hr. Then l e t us de te rmine the ind i ca t ed speed Vi =

'ins t - 10 - 2 3 = 500 -33 = 467 km/hr. The tr u e f l i g h t speed may be found fromthe fo l lowing express ion :V.1 - 467V = - = 810 km/hr,t & 0.58

where f o r H = 10,000 m , A = 0.337, a dT = 0.58 ( se e t h e t a b l e f o r t h e / 16standard atmosphere) . Or, f o r s p e ed Vi = 490 km/hr, acco rding t o th e nomoggram (Fig,. 3), w e o b ta in a Mach number of 0.75. Knowing th e speed of sound a tH = 10,000 m and t he f l i g h t Mach number, i t i s easy to . de te rmine the t ru espeed: Vt = aM = 300 0 . 7 5 3.6 = 810 km/hr.

The accepted value 6V a = -10 km/hr i s character i s t ic of modern h igh-s pe ed a i r c r a f t w i t h i n t h e r a ng e o f t h e i r i n d i c a t e d s p e ed s o f 220 - 600 km/hr.Later we w i l l d et er mi ne t h e c . o rr e ct i on f o r a i r c o m p r e s s i b i l i t y i n eachconcre te case according t o t h e nomogram i n Figure 2 , while we w i l l assumet h a t the aerodynamic co r re ct i on i s 6 V = -10 km/hr.a5 7. T h e Character o f t h e Propagation o f Minor Per turbat ions i n F l i g h ta t Var ious A 1 ti t u d e s

In an example of a i r c r a f t f l i g h t , l e t us examine the manner i n whichs l i g h t f l u c t u a t i o n s i n dens i ty and p ressu re , i . e . , minor pe r tu rba t ions , w i l lbe propagated i n t h e a i r f low. 'The a i r c r a f t , be ing the source o f the pe r tu rba t ions , has an e f f e c t on the a i r p a r t i c l e s l o ca t ed i n f r o n t of i t andp e r t u r b a t i o n s a r e s e n t f o rw ar d from one p a r t i c l e t o t h e n e x t a t t h e speed o fsound.

L e t us f i r s t t a k e an a i r c r a f t f l y i n g a t below the speed o f sound (Fig. 4a).

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P

Figure 3 . Nomogram f o r De te rm in ing the Mach Number

/ \I _ .I '.-- '\ \Figure 4. Propagat ion Charac te r i s t i cs f o r Sound Waves

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When the a i rc ra f t passes through poin t A t h e p e r t u r b a t i o n s c r e a t e d by i tt t h a t given moment, propa gating along a sphere a t the speed of sound, overthe a i r c r a f t . Af te r a s h o r t t i m e , th e Mach wave rea che s p oi n t B y whilet i m e t h e a i r c r a f t h as s uc ce ed ed o n l y i n p r og r es s in g t o p o i n t C ;i t s f l i g h t s pe ed i s below the speed of sound. Passing through po in t D ,t aga in c r ea t es p e r tu rba t ions which w i l l be propagated with the speed ofand i n a s h o r t wh i le r ea c h p o i n t E . The a i r c r a f t , however, du r ing th i s

w i l l no t have reached poi n t E bu t w i l l be located between poin ts C and. Thus, t he a i r c r a f t r emains c ons tan t ly wi th in the sphere c r ea t e d by i t sI f , however, the a i r c r a f t f l i e s a t t h e speed of sound (Fig. 4b) ,

B i s reached s imul taneously by both the a i r c r a f t and the soundi . e . , t he pe r tu rb a t io ns c r ea t ed by it a t p o i n t s A, C and D .Thus, i n f r o n t of t h e a i r c r a f t t h e r e a r e always Mach waves which,coming superimposed upon each o th er , f o n a dense sec t io n o f a i r c a l l e d t h ecompression shock o r shock wave.If t h e a i r c r a f t f l i e s above the speed of sound, it moves ahead of thesp he r i ca l waves it has c r ea t ed (Fig. 4c ). The a i r c r a f t w i l l r each po in t Ca t t he moment when t he pe rt ur ba ti on it c r ea t ed a t p o i n t A has reached onlyB y w h il e t h e p e r t u r b a t i o n c r e a te d a t p o i n t D has reached poi n t E . Thus,h in d an a i r c r a f t f l y i n g a t supersonic speed a Mach cone i s formed whichco ns is ts of an i n f i n i t e number of Mach waves propagated along the sphere a tth e spee d of sound. However, th e air mass within the Mach cone i s d i sp laced -17r e l a t i v e t o th e e a r t h a t t h e a i r c r a f t ' s s pe ed . The g r e a t e r t h e a i r c r a f t ' sspeed , t he sha rpe r the ang le a t th e t i p of the Mach cone. This angle i sdetermined according t o th e formula (Fig. 4c) :

s i n 4 = -M 'If t h e Mach number i s 1, then $ = g o " , w h i le t h e f u l l a n gl e i s 180" (normalshock) ; f o r M = 2 , s i n 9 = 0 . 5 and the angle $ = 30" ( f u l l a n g l e 6 0 ) .

Compression shocks a r e bot h normal and obl ique. A normal compressionshock i s one whose surface i s perpend icu la r t o t h e d i r e c t i o n o f t h e i n c i d e n tf low, i . e . , which forms an angle B = 90" w i t h it (Fig. Sa). Oblique shocksar e those whose su rf ac e forms an acu te angle of f3 < 90" w i t h t h e d i r e c t i o nof the inc id ent f low (Fig . 5b) .The gr ea te s t speed losse s and increas es i n pressure ar e observed whenthe f low passes through a normal compression shock. The br ak in g of the flowon t h i s shock i s s o su bs ta n t i a l t ha t beh ind the shock the f low ve lo c i t y must /18be below the spee d of sound (by as much as it was above the speed of soundi n f r o n t o f t h e s h o ck ).In an obl ique shock the losses are l e ss than wi th a normal shock,s p e c i f i c a l l y , p r o p o r t i o n a t e l y l i t t l e the more the shock w a s i n c l i n e d i n t h ed i r e c t i o n o f t h e fl ow , i . e . , t h e l e s s t h e an g le B . The in te ns i t y of anoblique shock i s a l s o s u b s t a n t i a l l y l e s s t ha n a normal shock. I f t he ang le B

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i s c lo s e t o 9Qo, hen behind th e obl ique shock th e speed of the f low i ssubs onic, while somewhat g re a te r than t h a t which would be obtain ed i f t h eshock were normal.

oblique compressign

ip e r t u r b a t i o n fy-boundary

Figure 5. Form ation of Normal ( a ) and O b l i q u e( b ) Compress i on Shocks.

S t reams passingthrough an oblique shockchange the d i r ec t ion o ft h e i r mo ti on , d e v i a t i n g .from t h e i r i n i t i a ld i r ec t io n . Dur ing f lowaround a wing o r fus e l a gewith a speed exceeding thespeed of sound, an obliqueshock developes i n f ro n to f t he wing o r fuse l age .

A i r c r a f t i n t e n d e df o r tr a n s - and s up er so ni c speeds must haveaerodynamic shapes whichdo not generate normalcompres sion shoc ks. Theforward edge of th e wingon super son ic a i r c r a f tmust be kn i f e - l ik e , andthe wing i t s e l f must beq u i t e t h i n .

5 8. Trans- or Supersonic Flow o f Air Around BodiesIn the case of low -veloci ty f low around bod ies , the f low i s deformed a t

a su bs ta n t i a l d i s t anc e f rom the body and a i r pa r t i c l e s , i n b r eak ing away, f low /19smoothly around it (Fig. 6a) . When t h i s occur s , t he p ress u re c lose t o theMC 1

Mach

Figure 6 . Subsonic (a ) andSupersonic ( b ) Flow Arounda Wing P r o f i l e .

body va r i e s in s ign i f i can t ly , wh ich pe rmi t s ust o c o n si d e r a i r d e n s i t y as constant . As ar e s u l t o f t h e d i f f e r e n c e i n p r e ss u r es u nd erand over the wing, l e f t i s c rea ted .In the case o f son ic o r super son ic f lowaround a b ody , l o c a l a i r p r e s s u r e and d e n s i t yv a r i a t i o n s a r i s e w hi ch , p r o p a ga t in g a t t h espee d of sound, form a s o n i c o r super son icshock wave i n f ro n t of th e body.This occurs because t he speed of the a i rp a r t i c l e s c l o s e t o t h e body s u dd en ly v a r i e si n both amount and di re ct io n. When t h i soccurs , th e f low i n a sense "encounters" anobstac le which , depend ing on the s i tua t ion ,

may be the body i t s e l f o r an "a i r cushion" i nf r o n t o f i t and form a compression shock

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(shock wave). A t th is compression shock th er e i s an uneven change i n t heb a s i c p a ra m e te rs c h a r a c t e r i z i n g t h e c o n d i ti o n s o f t h e a i r , i . e . , speed V ,pressu re p , dens i ty p and temperature T. Shock waves may b e formed e i t h e ri n f r o n t of t h e p r o f i l e o r c l o se t o i t s t r a i l i n g p o r ti o n. P r e c i s e c a l c u l a ti o n s and measurements have shown t h a t t he thi ckn ess of th e shock waves -o rcompression shocks i s n e g l i g i b l y small and has an order of length of th e f reepa th of the molecu les, i . e . , 10-4 - 10-5 mm (0.0001 - 0.00001 m m ) . 9 . Sonic I'booml'

S up er so ni c f l i g h t i s accompanied by the c ha ra c t e r i s t i c son ic %boom.This phenomenon i s t h e r e s u l t o f t h e f o r ma t io n o f a system of compressionshocks and expansion waves i n f ro n t of t he nose o f a f u s e l a g e , t h e c a b i n , o rwhere t h e wing and t a i l assembly j o i n the fuse lag e.* The most powerful shockwaves are formed by th e a i r c r a f t ' s nose and wing, which dur ing f l i g h t are t h ef i r s t t o e n c ou n te r t h e a i r p a r t i c l e s , and t h e t a i l assembly. These shockwaves are la b el e d bow and t a i l shock waves , r e s p e c t i v e l y ( Fi g. 7 a ). I n t e r imediate shock waves e i t h e r ca tc h up with th e bow shock and merge wit h i t o rf a l l beh ind and merge wi th t he t a i l shock.Behind th e bow shoc k, th e a i r pressure increases unevenly , becoming great e r than atmospheric pr es su re , and then decrease s smoothly and becomes even le ssthan a tmospher ic, a f t e r which i t aga in inc reases uneven ly un t i l it i sprac t i ca l ly a tmospher i c aga in a t t h e t a i l wave.The sudden pr es su re drop i s t r a n s m i t t e d t o t h e a i r a round it i n ad i r ec t io n pe rpend icu la r t o the wave su r f ace . Persons on the ground f e e l th i sdrop as a s t r o n g Ifboom." Sometimes a second Yboom" i s h e a r d t h i s i s there su l t o f t he success ive e f f e c t s o f bo th the bow and t a i l shock waves .

Figure 7. A i r Pressure Changes during a "boom" i nt h e Ver t ica l P lane b e l o w t h e A i r c r a f t ( a ) , and t h eIn t e r cep t ion o f t h e Conic Shock Wave w i t h t h e Ear th ' sSur face ( b ) .

. .* A. D . Mironov, Supers onic "Floc" i n Aircraft . Voyenizdat, 1964.

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Repea ted obse rva t ions have es t ab l i s he d th a t t h e two s u c c e s s i v e s o n i cbooms are d i s t in c t ly hea rd on ly when th e re i s more than 1/8th of a secondbetween them.The l on g e r t h e a i r c r a f t , t he longer the time i n t e r v a l b etw ee n t h eocc urr enc e of th e bow wave and th e t a i l wave. Therefore, two "booms" ared i s t i n c t l y h e ar d i n t h e c a s e of a n a i r c r a f t w it h a long fus ela ge. And, i nco n t r as t , an on ly vague ly sep ara t ed "boom" ind ica te s th a t t h e a i r c r a f t hassmall dimensions or i s f l y i n g a t a r e l a t i v e l y low a l t i t u d e .If t h e a i r c r a f t f l i e s a t a cons tan t super son ic speed , t he " b 0 0 m" i sh e a rd s im u lt a ne o us ly a t d i f f e r e n t p o i n t s on t h e e a r t h ' s s u r f a c e . I f t h e s epo in t s were t o be jo ined by a l i n e , we would ob ta in a hyperbola forming asa r es u l t o f t he i n t e r ce p t i on o f the con ic shock wave wi th t he p l ane o f thee a r t h ' s s u r f a c e ( F i g . 7b) . One hyp erb ola corre spon ds t o t h e bow wave, andt h e o t h e r t o t he t a i l wave. The l i ne s of s imul taneous au di b i l i ty of the"boom" ar e d isp la ced a long the e ar th 's su r fa ce , fo l lowing behind the a i r c r a f t and forming unusual t r a i l s . A t the same t ime, d i r e c t ly below the a i r craf t . t h e r e i s a su bs ta nt ia l l y louder Itboom," which a t te nu ate s as a func t ion -2 1

of d i s t a nce and under c e r t a in c i rcumstances it i s completely ina udi ble . Theground ob se rv er who h ea rs th e 'tboom" from an a i r c r a f t f l y i n g , l e t u s s a y , a tan a l t i t u d e o f 15 km with a speed twice tha t o f sound w i l l not observe thea i r c r a f t above h im; a t an a l t i t ud e of 15 km, it takes sound approximately50 s e c to reach t he ground a t an average speed of 320 m/sec, while duringt h i s time t h e a i r c r a f t w i l l have covered approximately 30 kmTo g e t an i d e a of t h e e f f e c t o f a p r e s s u r e d r o on b u i l d i n g s t r u c t u r e s ,l e t us po in t ou t t h a t t he overp ressu re Ap = 10 kG/m3 c r e a t e s a s h o r t - l i f tload of 20 kG on a door with an area of 2 m 2 , f o r example. A f i g h t e r w it h afuse l age l eng th of 15 m a t Mach 1.5 and H = 6000 m cr ea te s Ap = 11 kG/m2. Aheavy, del ta-winged super sonic a i r c r a f t weighing 70 tons w i l l , f l y i n g a t an

a l t i t u de o f 20 km and a t Mach 2 cr ea te Ap = 5 kG/m2, and a t l ow a l t i t udes(5-8 km) a drop may re ac h 12-18 kG/m2. I t i s a known f a c t t h a t i n t h e i rdes ign , bu i ld ings are planned fo r the s o-c al l ed wind load , which cor respondst o t h e f o r c e of t h e p r e s s u r e o f a i r moving a t a speed of 40 m/sec, i . e . ,g re a te r than 140 km/hr. This type wind w i l l c rea te an overp ressu re of 100 kgon 1 m 2 of wall s u r f a c e . The pres su re i n th e "boomT' a t p e r m i s s i b l e f l i g h ta l t i t u d e s i s 1 /5 th o r 1 /6 th th a t o f t he des ign a l lowance fo r wind load .The ch ar ac te r i s t i c s o f t h e e f f ec t o f p r ess u re d rops i n shock waves du r ing"booms" are g iven i n Tab le 2. For example, on a w a l l with an ar ea of 1 2 m 2during an overpressure o f 50-150 kG/m2, t h e r e i s a s h o r t - l i v e d l oa d o f 6001800 kG. Under t h e ef f e c t of such a load, wooden s tr u c tu re s may co llap se.T h e r e f o r e , a i r c r a f t are f o rb i d de n t o a c c e l e r a t e t o s u p e r s o n i c v e l o c i t i e s below

9-10 km over popula ted areas. I n t h e o pi ni on o f f o r e i g n s p e c i a l i s t s , a s o n i c"boom" wi th an in t e n s i t y of 5 kG/m2 i s the most which can be to le ra te dharmlessly . T h e re f o re , f u t u r e s u p e r s on i c j e t a i r c r a f t w i t h h ea vy f l i g h tweights (140 - 170 tons) w i l l have t o f l y a t a l t i t u d e s of 18-24 km i n o r d e rt o min imize the e f f ec t o f p r essu re d rops . I n th i s case , t hey w i l l have toc limb t o a l t i t ud es o f 9-10 kma t sub son ic l i g h t regimes (Mach number = 0.9 - -2 20.92) , whi le beyond t h a t a t up t o s ch ed ul ed f l i g h t a l t i t u d e a t Mach M = 1.0

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1 . 2 , and only a t t h i s a l t i t u d e w i l l t h ey b e a b l e t o a c c e l e r a t e t o s u p er so n icc ru i s ing speed .T A B L E 2

P re s su re Drop, kG/m2 Relative Loudness and Resultant Destruction0.5 - 1.5 D i s t a n t b l a s t1.5 - 5 Close b la s t o r thunde r5 - 15 Very close, loud t h u n d e r (window glass ra t t lesa n d s h a t t e r s )15 - 50 Large window panes s h a t t e r50 - 150 L i g h t s t r u c t u r e s c o l l a p s eThe sound of th e s o n i c boom i s a f unc t i on o f t he f l i g h t a l t i t ud e , Machnumber, a i r c r a f t ' s a ngl e o f a t t a c k , f l i g h t t r a j e c t o r y , a t mosphe ri c p r e s s u rea t s ea leve l and a t t h e f l i g h t a l t i t u d e , and wind d i r e c t i o n wi th r e s p ec t t oa l t i t u d e . For example, th e t tboom't from an a i r c r a f t f ly in g a t an a l t i t ud e of15 km and a t Mach 2 (V = 2120 km/hr) i s he a rd t o a d i s t a n c e of 40 km from thea i r c r a f t ' s p a t h , w h i l e a t an a l t i t u d e of 11 km i t i s hea rd only to a d i s tanceof 33 km. D ur i ng f l i gh t a t an a l t i t u de of 1 . 5 km a t Mach 1 .25, t h e "boom"

i s he ar d on ly w i t h i n a be l t 8 km wide.A t a i l wind may d i sp lace th e shock wave, r e su l t in g i n d i sp l ace of th ea ud i b i l i t y z one . The climbing and descent speeds and the angle of inclination0 o f t h e t r a j e c t o r y h ave s i g n i f i c a n t effec ts on the s i z e of th e a u d i b i l i t yzone and the loudness of the "boom." F o r exam pl e, i n ga in i ng a l t i t ud e a t anangle of 0 = 15' a t H = 5 km, t h e t'boom't i s heard on the ground a t M > 1 . 2 .

In descending from an a l t i t u d e of 10-11 km a t an angle 0 = - l o " , t h e "boom"reach es th e .ground only a t M = 1.03.In conclus ion, l e t us dwel l on the e f f ec t of t he shock wave crea t ed bya s upe r s on i c a i r c r a f t on a p as se ng er a i r c r a f t i n f l i g h t . A s has already beensa i d , the pre ssure drop during a compression shock i s 5-18 kG/m2. I f f o r t h emean valu e we s e l e c t 10 kG/mZ, i t amounts t o l e s s than 0 .1% of t he a i rpre ssure a t ground lev e l (p = 10,332 kG/m2 = 1 a t . ) . The ve l oc i t y head f o ra j e t pa s s e nge r a i r c r a f t f l y i n g st a speed of 850 km/hr and a t an a l t i t u d eof 10 km i s approximately 1200kG/m2, i . e . , more than 100 t imes the pressur edrop i n t h e "boom." Consequently, such a drop has e s se n t ia l l y no e f f ec t onan a i r c r a f t i n f l ig h t . However, th e r e may be a c e r t a i n e f f e c t on t h e a i r c r a f t ' s b e h a v i o r as cr ea te d by the accompanying j e t from th e a i r c r a f t f l y i n gby; t h i s e f f e c t i s comparable t o t h a t o f a s l i g h t g u st ( a s i n g l e g u st o f"bumpy a ir " ) , d ir ec te d along the propagating l i n e of th e shock wave. As ar e s u l t , h e a i r c r a f t w i l l exper ience s l ight bumpiness .

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10. Features of t h e Formation o f Compression Shock during F l m AroundVarious Shapes o f BodiesLet us now look a t the fea tures of the format ion of compress ion shocksf i r s t with the example of flow around the a i r i n l e t o f a j e t e ng i ne dur i ngs upe r s on ic f l i g h t , and t he n l e t us cons ide r flow a round th e pr o f i l e .The ex is t en ce of a normal shock a t t h e i n ta k e t o t h e d i f f u s e r l ea ds t os u b s t a n t i a l l o s se s of t o t a l p r e s s u r e ( k i n e t i c e ne rg y) o f t h e air e n t e r i n gthe compressor and the combustion chamber.During de c e l e r a t i o n i n t he d i f f u s e r , t h e s upe r s on i c f low i s transformedas it pas ses through th e normal compression shock. When t h i s occu rs, onep a r t o f t he k i ne t i c e ne rgy o f t he a i r i s u s e d f o r i t s compression, while theo t h e r i s t ransformed in t o hea t ( l os t ene rgy) . However, dur ing f l i g h t o ft h e Mach number M < 1 . 5 , l o s s e s a t the shock a re small. A s a r u l e , t h e r e f o r e ,f o r s uc h f l i g h t s pe eds i n t a ke dev i ce s a r e u se d on s ubs on ic a i r c r a f t .A t f l i g h t gr ea te r than 1.5 Mach, however, lo ss es a t the normal shockbecome greater . To e l i m i n a t e t h i s , th e process of a i r d e c e le r a ti o n i n t h ei n t a ke de v i c e i s achieved through the c r ea t i on of sys tems of obl ique shockswhich te rminate i n a weak normal shock. Because o v e ra ll energy losses i na system of oblique shocks are l e s s than i n one normal shock, th e pre ssu re a tt he e nd o f t he de c e l e r a t i on w i l l r e t a i n a high val ue. Thus, th e normal shocki s d i v i d e d i n t o a s e r i e s o f o b li q ue s ho ck s. S t r u c t u r a l l y , t h i s i s achievedth ro ug h s e t t i n g up i n t h e d i f f u s e r a s p e c i a l s p i k e i n t h e s h ap e of s e v e r a lcones whose t i p s a r e d i r e c t e d a cc o rd i ng t o f l i g h t ( F ig . 8a ) .When f l i g h t speed i s de cr ea se d , t he a ng l es o f i nc l i n a t i on o f t he ob l i queshocks inc r ease ( th e angle B tends toward 90 ; see Figure 5 ) . A s speed i sinc reased , the rev e rse occurs , and the se angle s dec rease. This h inde rs the

opera t ion of the input device inasmuch as t h e f r o n t f o r a l l the shocks w i l lno t pass through the inpu t edge of th e cone (Fig. 8b). There fore, sometimest h e s p i ke i s a d j u s t a b l e , s o th a t i n the event of changes i n speed , i t spo s i t io n can be va r ied ax ia l l y , the reby he lp ing the shock t o pass th rough t heleading edge of the a i r i n t a k e a t a l l f l i g h t s p e e d s .On th e wing p r o f i l e , th e formation of compression shocks OCCUTS even

su bs ta n t ia l l y below th e speed of sound. As soon as the f low speed of theconvergent st ream exceeds t h e speed of sound somewhere on th e pr o f i l e , Machwaves appea r which, i n accumulating, form a shock. I t must be noted th a tthis shock wave i s formed f i r s t on the uppe r p ro f i l e su r fac e c lo se t o somepo in t correspond ing t o t he maximum of th e lo ca l speed and th e minimum

23

pr e s s u re on t h e p r o f i l e . As soon as the speed of the f low surpasses the speed-2 4of sound, a shock wave forms on the lower p r o f i l e su rf ac e as w e l l (Fig. 9 ) .1. A t p o i n t C t h e p o i n t of l e a s t p r e s s u r e on t h e p r o f i l e , t h e s pe ed o fthe motion of t h e a i r has a t t a i ne d th e loc a l speed of sound (Fig. 9a) . TheMach waves move from the source of the perturbation toward point C and,

running in t o each othe r , form a weak normal compression shock.

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----F i g u r e 8. Formation of Compression Shocks a t t h e In take tot h e Diffu ser of a Turbojet E n g i n e a t Supe rsonic F l i g h t Speeds:a - l ine drawing o f i n p u t device w i t h cone: O A , BA obl iquecompression shocks, AK normal compression shock; b -opera t ional conf igura t ion of supe rsonic d i f fu se r d u r i n g f l i g h tspeed below i t s design speed.

Figure 9. The Formation of Compression Shocks a t VariousStreamline Flows.2. As th e speed of sound inc re as es somewhat (a t V2 > Vl), th e speed

of the f low a round the pr of i l e in c rea ses (F ig . 9b) . Behind point C y thespeed of th e f low becomes g re a t er than the speed of sound. A s e c t i o nappears where the flow moves a t s u p er s o ni c v e l o c i t y , r e s u l t i n g i n t h eformation of an oblique shock.

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- -

3. A t a speed of V3 (V3 < a ) , reg ions o f son ic and supe rsonic f low a l soform on th e bot tom of th e pr of i l e , r e s u l t in g i n th e forma tion of compressionshocks (Fig. 9c).

4. A t a speed of V4 c lo se t o th e speed of sound, th e compression shocksare d i s p l a ce d towa rd t he t r a i l i n g edge , t her e by i nc r e a s i ng t he s e c t i on o f t hep ro f i l e which encounter s supe rsonic flow p a s t it (Fig. 9d) .

5. When velocity V5 becomes somewhat g r e a t e r than th e speed of sound, abow wave forms i n f r o n t o f t he p r o f i l e and a t a i l wave forms behind it (Fig.9e) .

During flow around a bl un ted body, th e compression shock forms a t a -2 5s l i g h t d i s t a n c e f r o m i t s forward section and assumes a curv i l inea r form( F i g . l oa ) . A t i t s forward edge, the shock i s normal here i t i s perpend i cu la r t o the inc i den t f low. Depending on the d i s tanc e from th e body, th eangle s of in c l in a t io n of th e shock dec rease . During supersonic flow arounda knife-edged body such as a wedge with a la rg e open angle (Fig. lob ) , t heshock i s formed also a t a s l i g h t d i s tan ce from t he bow po in t and a l so has acurv i l inea r form. I f th e open angl e of th e wedge i s small enough, thecompression shock "seats i t s e l f " on t h e sh ar p edges (Fig. 1Oc).

Figure 10. T h e Formation of Compression Shocks a t Id en ti ca lFlow Velocit ies: a - i n f r on t of a b l u n t e d body, b and c -i n fro nt of knife-edged bod ies. 1 1 . C r i t i c a l Mach Number. T h e Ef fec t of Compress ibi l i ty on t h eMotion o f Air F l y i n g Around a Wing

The compress ibi l i ty of the a i r be gi ns t o m a n i f e s t i t s e l f gradua l ly asspeed i s increased. Up t o a Mach number o f 0.4, t h e e f f e c t of compress ib i l i tyon the aerodynamic ch ar ac te r i s t ic s of t he wing i s only s l i g h t and may i nprac tPce be ignored. With a f u r t h e r i nc r e a s e i n s pe ed , t h i s e f f e c t becomesmore and more noticeable and can no longer be ignored. S t a r t i n g a t f l i g h tspeeds of 600 - 700 km/hr and above, drag in cr ea se s sha rpl y because ofcompress ib i l i ty . This occurs due t o the fa c t th a t lo ca l speeds of the mot ionof the a i r over the wing and a t poi n t s where t he wing a t t ach es t o the fu se lages u b s t a n t i a l l y su r pa ss t h e f l i g h t s pe ed . In f lowing around th e convex su rf ac eof the wing, f o r example, t he air streams are compressed and t h e i r

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c ross - sec t ion dec reases . However, because the span across t h e stream m u s tremain cons tan t , the speed i n it i s increased. A t any s u f f i c i e n t l y hi gh f l i g h tspeed, the l o c a l air speed a t any po in t on t h e wing o r o t he r po i n t on t hes t ru c t ur e comes t o equal th e lo ca l speed of sound (Fig. 11) .Lava1 nozzle

/ P r o f i l eloca l=aFigure 1 1 . T h e Formation o f t h e Local Speed of Sound i nFlow around a P r o f i l e .The f l i g h t speed a t which the lo ca l speed of sound w i l l appear anywhereon the wing i s c a l le d t h e c r i t i c a l f l i g h t s peed Vcr, while i t s corresponding

Mach number i s ca l l ed th e c r i t i c a l Mach number Mcr . Higher va lues f o r the -26l oc a l spee ds a r e obs er ve d on t he uppe r a i r f o i l p r o f i l e . A s the speed of thei nc i de n t f low o r t he f l i g h t s pe ed i nc r e a s e s , t he l oc a l s pee d re ac hes t he s pe edof sound fa s te s t a t t h i s p o i nt .

Let us examine the a i r s t ream sur rounding th e p r o f i l e (F ig . 11) . Letus s e l e c t two c ha r a c t e r i s t i c c r os s- s e c ti ons o f t h i s s t r e am : t he l a r ge one Iand the small one 11. The loca l a i r s pe eds i n s e c t i on I1 w i l l b e g r e a t e r t h a nt h e l o c a l s pe ed s i n s e c t i o n I as a r e s u l t o f d i f f e r e nc e s be tw een t he areas oft h e s e s e c t i o n s . If we incr eas e the speed of t he i nc id en t unper turbed f low,t h e l o c a l s peed s i n c r e a s e i n b o th s e c t i o n s , b u t i n s e c t i o n I1 it i s g r e a t e rt ha n i n s e c t i o n I . This i s e xp la i ned by t he f a c t t ha t as a r e s u l t of t hei n c r e a s e i n s pe ed t h e r e i s a drop i n den sit y which i s more i n t e n s e t he f a s t e rthe speed of the s t ream. To r e t a i n t h e s t e a d i n e s s o f t h e mass flow weightr a t e o f a i r al on g t h e s tr ea m , t h e s pe ed i n s e c t i o n I1 must increase addi t iona l l y i n o r d e r t o c ompensate f o r t h e g r e a t d e n si t y d ro p i n t h i s s e c t io n . A tthe thresh old, th e lo ca l speed of the f low of a i r i n s e c ti o n I1 may come t oequal the lo ca l speed of sound.

From t h i s it f o ll ows t h a t du r ing f l i g h t wi th s pee d Vcr, t he l oc a l s pe e dof sound i s achieved a t the na r rowest po in t o f the stream. I t has beene s t a b l i s h e d t h e o r e t i c a l l y t h a t a t t h i s i n s t a n t th e c r i t i c a l p r e s s u r e dropforms between section I and I1 which i s equal t o pI I : pI = 0.528.

I t i s w el l known t h a t i f the speed of sound i s achieved a t the narrowestp a r t of t h e stream, the speed increases and becomes supersonic i f the s t r eamcontinues broadening. Therefore, a fu l l y supe rsonic zone of f low i s formeddown w i t h po r t i on o f t he p r o f i l e s u r f a c e du ri ng f l i g h t w it h M > Mcr .

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The g r e a t e r t h e f l i g h t s pe ed , t h e g r e a t e r t h e z on e o f s u p e r s o n i c s p ee d w i l lbe. However, f a r beh ind t he p r o f i l e t he speed mus t be the same a s t h e f l i g h tspeed. Therefore, a t some poHnt on the pr o f i l e th er e must develop dece le ra t io nof the a i r f rom super sonic t o subsonic speed. Such de ce ler a t ion , asexperience has shown, occ urs only with th e forma tion of a compression shock. 12. T h e Dependence o f t h e S p e e d o f t h e Gas Flow on t h e Shape o f t h e -27Channel. T h e Laval Nozzle

A means fo r obta in in g su person ic speeds i n th e mot ion of t he gas w a s .developed by the engin eer Laval (Switzer land) during h i s work i n the 1880'son improving a steam tur b in e he had invented . Laval obt a in ed a super son icflow of vapor as it flowed from a s p e c i a l n o z zl e .This nozzle , subsequently ca l l ed the Laval Nozzle (F ig . ll), i s a t ubewhich i s f i r s t compressed and then expanded. The narro wes t se c ti o n of th etube i s c a l le d t h e c r i t i c a l s e c t io n . If a vapor o r gas i s run through sucha nozzle a t a s l i g h t p r e s s u r e d r op i n w hich t h e s p e e d o f t h e flo w i n t h ec r i t i c a l s e c t i o n becomes s u bs o ni c, i n t h e ex pa nd ed p o r t i o n o f t h e n o z z l e t h espeed w i l l drop; i n th i s case the Lava l Nozzle opera t es as a t y p i c a l V e n t u r itube. However, i f t h e d i f fe r e n ce i n p r e s s ur e s a t t h e i n p u t t o t h e n o z z l e anda t i t s o ut pu t a r e s u f f i c i e n t l y g r e a t , i n t h e c r i t i c a l s e c t i o n t h e s p e e d o fthe flow becomes equal t o the lo ca l speed of sound. In th i s case , beyond th ec r i t i c a l s e c t i o n , i . e . , i n t h e bro ade ne d p o r t i o n o f t h e no z z le , t h e sp ee d o fthe flow does not decrease bu t , on the cont rary , in cre as es . Thus, it waso bs er ve d t h a t i n s ub - and s u pe r s o n i c f l ow s , the dependence of the speed ofthe f low of gase s on th e shape of t h e channel i s d i r e c t l y o p p o s i t e .Subsonic flow accelerates i n the compress ion channe l and dece le r a t es i nthe expansion po r t i on . In co nt ra s t , however , superson ic f low los es i t sspeed i n t he compression sec t io n , wh i l e i t i n c r e a s e s it i n t h e e xp an si on

s e c t i o nThere fo re , i n F igure 11 we se e the appearance o f super son ic speed a f t e rthe s t ream has passed through th e nar row se ct io n (po in t K ) .However, super sonic speed does not i nc rea se a long the e n t i re length ofthe nozz le ; a t some point it must de ce le sa t e t o subson ic speed . And hereinl i e s the cause f o r the format ion of the compression shock.

13. Laminar and Turbulent Flow o f AirUnder th e e f f e c t o f i n t e r n a l f r i c t i o n d ue t o t h e v i s c o s i t y o f a i r andthe roughness of th e su rf ac e of t h e body around which th e f low moves, th e

speed of air a t t h i s su r f ace becomes equal t o ze ro. Depending on th e di st an cefrom the su r f ace , t h e speed o f th e f low inc rea ses and r eaches the speed o ffr e e f low. The layer o f a i r i n which t h e r e i s a change i n speed from zeroto the speed of f r ee f low i s ca l l e d the boundary l aye r .I t i s w e l l known th a t t he flow of a i r i n the boundary la y e r may be

laminar ( s t r a t i f i e d ) when t he gas f lows wi thou t be ing mixed i n the ne ighboring

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and turb ul en t when th er e i s random mixing of gas p a r t i c l e s throughoute volume o f the f low. The boundary l ay er al s o en t a i l s phenomena such as /28( flow sep ara t ion ) , he format ion of s u r f a c e f r i c t i o n d ra g , a er onamic hea t in g, et c.The in t e ra ct io n of the boundary la ye r and th e compression shocks re s u l t sn t h e f o ll ow i ng . I f the f low in the boundary l ay er i s laminar (F ig. 12) ,

an oblique compression shock developesd i r e c t l y on t h e a i r f o i l p r o f i l e . Behind theshock there i s sep ara t ion and tu rbu lence o ft h e bo un da ry l a y e r ; i n t h e t u r b u l e n t r e g io na normal shock developes. In gen era l , theoblique and normal shocks are combined. Whent h e r e i s an ob l ique shock , t h e in t en s i ty o ft h e normal shock w i l l b e s u b s t a n t i a l l y l e s sbecause the f low approaches i t , having alreadya t t e n u a t e d i t s speed somewhat i n the obliq ues ho ck , w it h t h e r e s u l t t h a t t h e d ra gdecreases , There fo re, 1 ,aminari zed a i r f o i l s ,i . e . , a i r f o i l s w i th v er y smooth s u r f a c e s , a r e12 . Compression s u i t a b l e i n t h a t t h ey o f f e r t h e l e a s t s u r f a c eon the Prof i l e : 1 - f r i c t i o n d r a g and wave dr ag a t s u p e r c r i t i c a lc Zones ; 2 - Com- f l i g h t Mach numbers.Shocks; 3 - S u b -i c Zones. A ft er th e normal compression shock th er ebeg ins t he so - ca l l ed wave f low sep ara t ion ,i s accompa.nied by a d e cr ea s e i n t h e l o c a l a i r s pe ed . T hi s i n t u r nn a s ha rp drop i n th e a i r f o i l l i f t .

During turbulent f low around an a i r f o i l t h e r e i s no oblique shock andThe appearance of loc a l shocks on the a i r f o i ls t i t u t e s t h e s o - c a l l e d s hock s t a l l . Par t o f t h e k i ne t i c energy i n the shocks t ransformed i n t o hea t which i s t h en i r r e v e r s i b l y p ro p ag a te d .

A t h igh f l i g h t speeds , t he c ha ra c t e r i s t i c s o f t he compress ion shock a r efunct i on of th e natur e of the boundary layer . Experience has shown thatow i n a boundary l ay er i s usua l ly l aminar over a c e r t a i n p o r t i o n and t h e nt ch e s t o t u r b u l e n t .The po s i t ion o f th e t r an s f e r po i n t s o f l aminar boundary flow t o tu rbue n t depend on th e shape of t h e p r o f i l e , j.ts th ic kn es s , roughness, e t c . Thea body i n l aminar flow exper i ences l e ss f r i c t i o n and less aeroa t high speeds than does one i n a t u r b u l e n t l a y e r .The s t a t e o f t h e boundary l ay e r i s r e f l e c t e d n o t o n ly i n t h e w ing d r a g ,

i n i t s l i f t i n g c a p a c i t y as wel l . I n th e boundary l a ye r a f low separa t ionw hich d et er mi ne s t h e c r i t i c a l angle of a t t ac k and i t s correspondingl i f t r a t i o .

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14. Pressure Dis t r i-bu t ion a t S u b - and S u p e r c r i t i c a l Mach Numbers /29Pr e s s u r e d i s t r i bu t i on a l ong a wing p ro f i l e unde r f low condi t ions i s showni n Figure 13. The arrows rep rese nt t he value s of th e d i f fe r ence s be tween thelo ca l and a tmospheric pressu resa t each pa in t on t h e p r o f i l e .

b ) y c The p o si t iv e overp ressur e(a tmosphe r ic pre ssure l e s s

-1 I- t ha n l oc a l ) i s i nd i c a t e d byarrows poin t ing toward thecontour, whereas negativep r e s s u r e o r ra re fac t ion (a tmosp h e r i c p r e s s u r e g r e a t e r t ha nl o c a l ) i s shown by arrows pointt iO P \ ed away from the contour.

Figure 13. Diagram of t h e Pressure To determine and computeDis t r ibut ions a long the A i r fo i 1 Pro- th e for ce of th e evacuation onf i l e : a - v e c t o r a l ; b - expressed b y t ho s e p o i n t s o f t h e p r o f i l e a tt he p r e ss u re c o e f f i c i e n t ( 1 - upper which p re ss ur e measurementsw i n g s u r f a c e , 2 - lower su r fa ce ) . were t aken , the pr o f i l e chordf o r a l i n e p a r a l l e l t o t h e c ho rdi s pro jec ted , t he n t h e m easu red va l ue s f o r t he p r e s s u r e a r e p l o t t e d a t as e l e c t e d s c a l e from po i n t s s pe c i f i e d a l ong t he pe r pe nd i c u l a r t o t he c ho rd :pos i t i ve ove r p r e s s u r e i s usu a l ly p l o t te d below and evacua t ion i s pl o t te d above.The poi nt s thus obtai ned th en merge i n a smooth curve.

In diagrams used i n aerodynamics, normally th e pres su re co ef f i c i en ts(F ig . 13b) , which repres ent the r a t i o of the ove rpressure a t any given pointon t he p r o f i l e t o t he ve l oc i t y he ad o f t he t u r bu l e n t f low are p l o t t e d a tp o i n t s on t h e p r o f i l e r a t h e r t ha n t h e o v e rp r es s ur e , as fo l lows :

Pover - Ploca l - P a t .p = - 9 v2

where pl0 c a l - i s t he a bs o l u t e p r e s s u re a t a given poin t ;- i s t h e s t a t i c p r es s ur e i n t h e un p er tu r be d f lo w, i . e . , th ePa t . atmospheric pressure a t f l i g h t a l t i tu d e s ;9 - i s the v e lo c i t y head i n the unpe r turbed f low, de te rminedby t h e f l i g h t s pe ed and a l t i t u d e .From the above it f ol lows t h a t t he p r e s su r e c o e f f i c i e n t c ha r a ct e r i ze s / 3 0t h e de gr ee o f d i f f e r e n t i a t i o n ( i n u n i t s o f t h e v e l o c i t y head ) o f t h e l o c a lpre ssure a t any poin t on the uppe r and lower pr o f i l e sur fa ces from the s t a t i cpressu re i n the unper turbed f low. The co ef f i c i en t w i l l b e ne g at iv e i f t h el o c a l p re ss ur e on t h e g r o f i l e i s below atmospheric pressure. Consequently,a nega t ive va lue f o r p corresponds t o the pre sence on the pr o f i l e of r a re fac t ion, where a po s i t i v e va l ue i n d i c a t e s an i ncr e a s ed p r e s s u r e .

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A t small Mach numbers, the d iagram f o r th e pressure d i s t r i bu t i on f o r eache o f a t t ac k has i t s own co ns ta nt form bec aus e th e a i r compress ib i l i t y haso e f f e c t on t h e n a tu r e o f t h e d i s t r i b u t i o n o f t h e p r e s s ur e c