~miCS N. F. Krasnav
1
Fundamentals of Theory. Aerodynamics of an Airfoil and a Wing Translated from the Russian by
G. Leib
Mir Publishers Moscow
First puhlished 1985 Revised {rom the 1980 Russian edition
© JlaA8TeJII.CTBO .Bhlcmall ml(OJlat, t980
© English translation, !'IIir Publishers. t985
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mooeru craft. The funclnmentllb of nl'rodynamics are u~ed in stu{lyinf,;' the external Dow over \"ariou~ bodies or the motion of air (a ga~l inside various objects. Engineering success in the tields of aviation, artillery, rocketry. ~pace night, motor vehicle tran~port. nnd so on, i.I~. [ields that perlnin to the flow of air or a ~as in some form or other, depends 011 a lirm knowledge of aerodynamicti. The present textbook, in ,uldition to the general law~ of ]]o\\' of a fluid, treats the application of aerodynamics, chiefly in rocketry anll modern hi~h ,peed aviation. The book conHist.:; of tll'O parts, each forming a separlltp mlume. fhe tirst of Lllem concerns Ihl' fUlulaml'ntal concepts and definitions of nerodynamirs and tht- th(>ory of flo\\' over all airfoil alld a wing, il\cllldin~ an unstead.I' Ilo\\' (Chapter~ 1-9), while the ~C('l)nd desfTibl'~ the aerodynamic design of craft and their individual parts (Chapters 10-15). The tll'O p;lrts are (le~igned for u~e in a two-s!'mester ('our~e of aerodynamics. although the first part can be u~ec! indepcn<.lently by those intl'rested in individual problems of thporl'tical Ill·ro<.lynamics . .-\ sound theort-tiral bad'l!:round is important to thl' s~udy of an~' ~ubject bCl'ause (:featin' solutions of pructicul problems, ~cienti1ic re~earch, and <.liscovel'ies are irnpC)~sible without it. Studenl.5 should therefore del'ote special attention to the lir~t live chapters. which deal with the run<.lamental concepts and definitions of lwrodynamics. thp kinematics of a fluid. the fundamentals of !luL<.l dynamks, Ill(' IIl('ory of ~hocks, and the method of ch,aracteristics used widely in iuv!'stigatiu!!, !-uper,'onLc no\\'~. Chilpters 6 and 7, which relale to the rlow ov{'r ail'foils, ar{' al~1) important lu II fumlamcntal ullder~talllliug of tbe subject. Thl's!' chllptel's contain a fairly ('mnplete discussion of thr g('neral theory of flow of a ,!l'a~ in two·<.limensional space (the theory of two·tlilllen5ional now). The inrormalion on the S\lper~onic st!'ady Dow over a win~ in Chapter 8 relates directly 10 th{'se mat{'riais. The Ilerodynamic design of most modern cralt is baser! on .<:t\l(tie,~ ot' such flow. One of the mo~t topiral area~ of mOtlel'n aero<.lynarnic research i~ the ~t\ldy of optimal aerodynlllllic {'ontiguralions of craft and tlu!ir sep:lratl' (isolated) parts (the fuselagt~. wing, empcnnagl'). Therefore, ~I ~mall ~crtion ,u.5) that
~ie~\~efl~;ilt:!:·};~~~ \\;~~~u~i~~ll 11~~~~HI:~1~:~t't~~or~rse~~~~:or;;:p~~t~nn:n~~~r[~:I
and methodological inform.. tion on tlw ('onversion of the llerodynamic coefficients of II wing from on!' a~pl'cl rlltio to ililother. Th!! stully of non·stationary 'l'as floll's is a rllther well developed field of modern thcorl,tical and practical aerodynamics. The results of this ."tudy are widcl~' \Ised in ('alculilling the eOrcl of aerodynamic forces and moments on
Prefollce
craft whose motion is generally charaetl'riled by non-uniformity. and tbe nOllstationar~' aerodynamic characteristics tbus calcul!\ll'd afe used in the dynamics
~I~:t;~s \~lflet~eS~~~~~~~at~i~r c~~f~;i:~~ \ll~ t ~ ~sr;:f\tc~o~ .c~{~~~r)~n! ~icgd:~r~~ atives (stability derivatives) are anulysed, as is the concept of dynamic stability
y
~h~:~~~ ~~o~~ ~:~~ ~dWt~~1~~~~Oa{~:: ~~d~J ~ f~~Ic:~~~; ni\~~~r~~~ ~ i l~~~' t~oc~i ~ atives of a lifting surfact' of arbitrary planform, genernlly with a curved leadin!? edge (i.e. with variable Sweep along the spaul. Both exact and approximate methods of determining the non-stationary aerodynamic cllaracteri~tics of n wing are given. A special plact' in the book is devoted to the most irnl)Ortau~ theort!tical and applied problems of high-spe(!d aerodynamic..~. including the thermodynamic and kiuetic parameters of dissociating gases, the C!quations of motion and en('rg~', and the theory of shocks and its relation to the IJhysicochell\ieal pro]l£!rties of gases at 1iJ.i~h temperaturE's. Considerable attention is giV>'ll Lo shock waves (shocks), which arc a manifestation of th£! specilic properties .)l supersonic flows. The concept of the thickness of Ii shock is discu8.'!cd, and the b.)ok includes graphs of the functioll~ characterizing changes in the parameters of it g;u.o as it pa~s('!< through a shock. Naturally, a textbook cannot reflect the entire diversity of problell1s facing the science of aerodynamics. I have tried to provide the scicmtilie information for specialists in the field of aeronautic and rocket enginCt:ring. This information, if mastered in its entirety, should be suflicient for ~·oung specialists to cope independently with other practical aerodynamic problems that may nplK.'ar. Among these problems., not reflected in the bOOk. are magnetogasdynamic in\'est· igations, the IIpplicatlOn of thE' method of characteristics to throe-dimensional gas flows, ant! experimental aerorlynamics. I will be hal)p~' H study of this text' book leads student!< to a more (omprehensi\'e, independent investigation 01 modern aerodynamics. The book is the result of my experience te
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r:r ~~ell~E~n~aju~f~~_~oil'~;! ~~5~~~~~iif 3Nlne~~ ib~ .~o\;;~~\:;i ~;J~~. s~~c~~~~et~ in rt'SC!areh institutions, design department.". llnd HHlu~trlal entcrrrises All physical quantities are given according to the luterMtiona System of Units (51).
edi:i~th:.~a~~!n til;a~~~~~~ ~~~i:.~ id~~tn aOc~~~~~~fkr~~\~~~~~ ~~'~~k~~e~ ~(r ~f!ih~ valuable suggestions made by the re\'iewer, I exprt'ss m~· profound Iilratitude.
profe~sor
A.\l.
~Ikhiteryan,
to whom
Nikolai F. Krasnov
Preface Introduction Chapter 1 Basic Information from Aerodynamics
1.1. }"orces Acting 00. a Moving Body Surface Force Property of Prl'Mures in aaldeal Fluid Innucmcc of Visco9ity 00. the Flow o[aF~id
1.2. Resultant Force Action Components of Aerodynamic Forces lind ;\1ou1('nt.'" Conversion of Aerodynamic Forees and .\Iaments from One Coordinate S~·Hern to Another 1.3. Determination of Aerodynamic Forces and Moments According to the Known Distribu\ion of the Pressure and Shear Stress. Aerodynamic CoefJicieuts Aerodynamic Forces and ~{oments aDd Their Coefll.clcot.9 1.4. Static Equilibdurn and Static Stability Concept of Equilibrium and Stability Stlttic Longitudinal Stability SULlie Lateral Stability 1.3. Features of Gas Flow U High Speeds Compressibility of a Gas Heating of a Gas Stllte of Air at High Temperatures
Chapter 2 Kinematics of
II
Fluid
2.1. Approaches to the Kinematic Investigation of a Fluid
k~Fe:i~~a:p:r~i~baCh Streamlines and Pathlincs .,2. Analysi!! of Fluid Pa.rticle ~1otioQ 2.3. Vortex-Free MOtiOD of a Fluid
2:) 25 2S 26
28 3(i 36
40
41
.!i1 52
52 53 57 58 58 59 65
71 71
n
73 74
79
g~~;~~li?'o!q~f~h~ Equation cartesian Coordinate System Curvilinear Coordinate System ~~t!naU~Krv~du;~~:ce of Gas Flow Flow Rate Equation 2.5. Stream Function 2.6. Vortex Lines 2.7. Velocity Circulation Concept Stokes Theorem Vortn-Induced Velocities: 2.4.
~~ ~~:t~f ~~jdn~~W3
Chapter 3 Fundamentals of Fluid Dynamics
97 98 98 too
3. L Equations of Motion of a VlscoUJ Fluid Cartesian Coordinates Vector Form of the Equations of
t06 t06
~Iotion
Spherical Coordinates
Equations of 'fwo·Dimenslonal Flow of a Gas Near a Curved Surface 3.2. Equations of Energy and Dif(wion of a GIIS Diffusion t:quation Energy Equation 3.3. Sys~m of Equations of Gas Dynamics. Initial and Boundary Conditions 3.4. Integrah of :'.lotion for an Ideal Fluid 3.5. Aerodynamic Similarity Con~cpt of Similarity Sinlliarity Cri~ria Taking Account of the Viscosity and Heat Con· duction 3.6. Isentropic Gas Flows Configuration of Gas let Flow Velocity Pressure, Dellllity, and Temperature FI\lW Gas from a Reservoir
or •
Shock Wave Theory
82
86
88 89 90 9f 9f 92 :
Parallel Flow Twu-Dimemlional Point Source and Sink Tllrec·Dime[1Sional Source and Sink Doublet Circulation }<'low (Vortex)
Curvilinear Coordinates Cylindrical COOJ'dinate!l
Chapter 4
80 80 81
100 103
US flo!! fi8 US 120 121
12t 124 129
134 138 138 141 149 149 150
152 154
4. t. Physical Nature of Shock Wave For-mation 159 4.2. General Equations for a Shock 162 Oblique Sh:ock 163 Normal Shock 168 4.3. Shock in &he Flow of a Gu with
Contents
Constant Specilic lleats System of Equatioll" Formulas for Ciliculating the Parameters of n. Ga" Behind a shock Oblique Shock Angle 4.4. Hodograph 4.5. A Normal Shock in !.he Flow of a Gas with Constant Specilic Heat~ /j.G•.\ Shock at Hypersonic Velocitie:o and Constant Spccilic Heats or a Gas /j.7. A SI10ck in a I-low of a Gaswitb Var)·. ingSpccificHcat~al1d with Dissoci
Chapter 5 Method of Characteristics
;:;.1. I::quatioTl!S for tht' \"elocity I'otential and Stream "Ul1rtion 5.2. The Caudl)" Problem 5.3. Charaeterj~tk~ COlllp.llihility Conditiuns OelPrllliliation or Characteristics
\!;!~:~f~/.II~I~it~~nO~,rc~a:j~!~~~:~~s
Finite-Span Wing
Chapter 7 An Airfoil in a Compressible Flow
i6()
iSS 1~3
19a
195 196 2(111
2"5 209 209 2(19
213 21 'I
5.5. A pplication of the 'lethod of Cllaract('fisties to the ::lulution 0\ tiw Problem on :-;hapinr; \lIC :\n7Jle~ 01' Super:iOnic Wind Tunnel,
AirfoilBnd
184
in a lIodograph Equations for CIJ1ll'acteristics in a lIol1o/l:raph for l'artiC\llllr Cll~~S of Ga~ Flow 5.-i. Outline of ~ollilion of G~s-Dj"na!Uic i-'r'Jblems Accurdlng to the :'Ilethod of Characteristic~
in an Incompressible Flow
170 176 179
ior
Characte-ri~lic~
Chapter 6
90
1fJ!'l 169
219 ???
230
G.1. TIlil! .\irfoil in an Incolnpre!sible Flow 234 6.2. Transverse Flow uH'r a Thill l'iate 24') 6.3. Thill Plate at lIli .\ngle of Attack 2~3 SA. Finite~Span Willg in an Incompres5ihle Flol\ 2'19 6.;•. \ring with Optimal PllUlionn 258 <:;()nV(' .. ~ioll uf Coe!Jicil'nt~ (v a"~1 (',\".I Irom One Willg .bpe-ct [(nIIO tf) .\nother :!as 7_t. SUbSOllic Flow UI"('f a Thill .\II·j,)il 264 l.ine(lrization of the EqllaliulJ 101 IhC' Velocity Potellli;1I :!G~ IlelatioD Between the l'.I1·,l1l1l'tef~ oj
t'h~~rlff~~i ~~er a:(\ 1:;~:·':~:~!t~iTj LI~ 7.2. Khristianovkh lJethud Content of the llethod
2tiG 269 1G!1
7.3. 7.4. 7.5.
7.6.
-Chapter 8
A Wing in a Supersonic Flow
~.I.
Conversion of the Pressure Coefficient for an Incompressible Fluid to the Number M.., > 0 Conversion of the Pressure Coefficient from M"'l > 0 to M..,~ > 0 Determination of the Critical Number M Aerodynamic Coef6.eients flow at Supercritical Velocity over an Airfoil (M "" > M ..... cr) Supersonic Flow of a Gas with Constant Specific Heats over a Thln Plate Parameters of a Supersonic Flow over an Airfoil with an Arbitrary Configuration lise of the Method of Characteristics Hypersonic .flow over a Thin Airfoil Nearly l:niform Flow over a Thin Airfoil Aerudynamic Forces alld Their Coeflicieuts Sideslipping Wing Airfoil Dclinition of a SideslippUl~ Wing Aerodpwmic Charact('ril!.tlcs of Ii Sideslipping Wing Aidoil Suction force Linearized Theory of Supersonic Flou' OIW a Finite-Span Wing Linearil.;ltioll of the Equlltion for tile ]'oteulial Fllnction Boundar,I' Connitions
~1~;nVO~I~~f~rorp~J~u?;~:sal a~dluX~r~~
dynamic Coerticients l"eatures or S\lprr~(llli(' Flow over Wings 8.2. :'.ll!thod of Sourc,,~ fI..a. Wing with It S~mmelri~ Airfoil and Triangulu l'lanforlll. (2 -u, eYa '" U)
!~~r~ o~~~~i~ih:¥d~:nd witb It Sub· Triangular Wing Symmetrie about lhe 'x·Axis with Subsonie Leadinl~ Edges Scmi-Inlinill' Wing with a Supel'S(Jnic Edge Triangular WingSym.metrie about the x-Axis with SupersoDlc Leading Edges 8.1i. Flow over a Tetragonal Symmetrk Airfoil Wing with Subsonic Edges at II Zero Angle of 'Attack 8.5. Flow over a Tetragonal Syruruetri" Airfoil Wing with Edges~of Different Kinds (Subsonie and Supersonic)
271 272 273
274.
274 278 285 285 291 293 293 ~
299 3(l1 3(5 308 308 310 313 3t5 317 321 321 326 328 330 331
343
Contents
11
Leading aud ~Ii,]d!e Edges are Subsonic Trailing l~lIgo: 15 .supersonic :y'J3 Le~ding Ed~e is Subsonic, .\liddle and Trailing Edgo:s are Supersonic 345 Wing with ,\ll Supl'r~Ollic Edges 346 General I{el,llioll for Calculating the Drag 350 8.6. l'ic!d of Applicatiun of tht Source ~Iethod 3,ll 8.i. Doublet Vi~tributio!\ ~lptLiOJ 3,l3 So!!. fo'lo\\- OH'r .:J. Triallg"lllar Wing with SubsoniC' Lrading Eligt'~ ~ fUI. Flow o\,l'r u Ill'xagOlml Wing \\'ith :'Ilbl'uni(' LcadilJg Dild Supersoni<' hailing I':dgc~ 366 !:l.lo. Flow o\'rr II Ih'Xil!>!"onal Wing with ~npl'nllni<' Ll'ildillg and Trailing Edgl's 3;2 Kit. l)rHIt of Wing!! with S\lb~onic Leadin~ fo:dlft!s ' 381 fI,12. :\l'rodynamir Cilaruclerislil'S or a Hcctanl.,rular Wing :-IE!5 fI.I:J. Ih'\'('r'"'~-t'low :\Il'lhllll 391
Chapter 9 Aerodynamic Characteristics 01 Craft in Unsteady Motion
!'I.I. G('ner.d Ih·la\itJlls ror ll)(' ,\crtlttynalllic Ct1('rtil"il'n\s
!l.2.
!I.a.
'\nal~'si~ of :-'tabilil,\' l)erh'ulivc$ .In,] ,\l'rot! ,'u3mic C~)('lIki(lnl~ Con\"('n-ion' of ~labilily Ikri\'atin~
1I1)I,n ;1 Chall!!l' in IiiI' ro~ilion of til(' [,'m'n' Ih'dncti!))} Centrl' !l.Ii. l'artirulHI' C3~('$ of \lotion LOIH>:iludinal and Lateral \Iotion~ '\lotlO]] of the C(,lltre of 'lass anll Hot;i\ioll ahout It l'itrhill>! \lolioll !'I.~'.lU~·lIa!ni;' ~tJ.lJilit~
395 398 1,04 1,116 4,,6 411, ,i08 !itO
Uelllll\wII 4111 Stabilil,- Cllara'-tcri"lir~ 413 ItG. Ba~k l\('latiun,; for Ln>:
!I.a
~~~~~~i;~~~r:fhod of Caleulating tile
Stability DerivatiVe,", for a Wing in an IncompreSsible flow \"elucil\' Field of an Oblique Horseshoe Vorlex \'ortt'X \lod('1 of a Wing Calclilation of Cireliialory Flo",' ,\crod)'namic Characteristics Ih'[ormation of a Wing Surface
428
43t 431 436 439 440 451
innuence of Compressibility (the Number .11",,) on ~oll-StatiODary flow 9.9. l"lIsteady Supersonic Flow over II. Wing 9.10. Properties of .l.erodynaruic Derivatives 9.11. Approximate .\iethods for Determining the ~on-Station8ry Aerodynamic Characteristics Hypotheses of I1armonicity and Stationarity Tangent-Wedge :\Iethod
Referenc:es Supplementary Reading Name Index Subjed Index
452 456 478 488 488 489
'93
'" 495
'"
Aerodynami.cs is a complex word originating from the Greek words all!' (air) and 6e"a~lLa (power). This name has been given to a scien<:'e that, being a part of mechanics-the science of the motion of bodies in general-studies the la\\'~ of motion of air depending on the acting forces and on their basil" establishes special laws of the interaction between nil' and a solid bod)' mO\'ing in it. The practical problems couia'onting mankilld in cOlilleclioll with flightf' in heavier-than-air craft provirl(>d an impetll$ to the de\'elopment of aerodynamics as a f'cieJlce. The$e prol>letllf' were associated with tht' determination of the forces and motllellt:-; (what we call the aerod)'namic forces and momentf') acting Oll Itlo\'ing bodies. The main task in investigating tile action of forr(>f' "'fl.." calculation of tile bnoyancy, or lift. force. At the beginning of its de\'elopment, f1erod~'llamic~ dealt with tlte ill\'estigation of the lllot.ion of air at quit(> low speeds becalL:
14
Introduction
of motion of air or in general of a gas at high speeds. It W,HI found that if the forces acting on a body moving at a high speed are calculated on the basis of the laws of motion of air at low ~peeds. they may differ greatly from the acLuBI forces. It became lle('e~~ary to seck the explanation of this phenomenon in the nature itself of the motion of air at high speeds. It consist!:! in a change in its Eiensity depending on the pressure. which may be quite ('ollsiderable at such speeds. It is exactly this change that underlies the property of COnIpressibility of a gas. Compressibility causes ft change in the internal energy of a gas, which must be consid('red when Cuiclllating the parameters rlNermining the motion of the fluid. The change in the internal energy associated with the parameters of state and t.he work that a ('Olllpressed gas ('an do upon expansion is determined by the lirst law of thermodynamics. Hence. thel'modynamic relations were tlsed in the aerodynamirs of a compressible gas. A liquid IHld Hir (a gas) differ from each other in their physiral properties owing to t.heir mole(,ular structure being different. Digressing from tlH'se features. we ('1111 take into account only the hasic difference lletween a liquid and a gas associated with the deb11'ee of their compressibility. A('c_ordingly, ill aerohydromechanics, which deals with tJH! motion of liquids and gases, it is customary to lise the term fluid to (Iesignatc both a liquid and a gas, distinguishing between an incompressible and a compressible fluid when necessary. Aerohydromeclumics trents laws of motion common to both liquids and gases, which made it expedient and possible to combine the studying of these laws within the bounds of a single science of aerodynamics (or aeromechanics). In addition to the general laws ('ltal"aCterizing the motion of fluids. there are laws obeyed only by a gas or only by a liquid. Fluid mecha.nics :
Introduction
160
aerodYllamic~
studying the laws of motion of air (a ga!ol) at high subsonic and supersonic speed~. and also the laws of intcractioll betweell a gas and a body trayelling in it at such speeds. One of the founders of gas dynamics was academi<"ian S. Chaplygin (1869-1942), who in 1902 published an olltstanding sdentific work "On Gas Jet-OJ" (in Russian). Equations are derh'ed in this work that form the theoretical foundation of modern gas dynamics and entered the world's science lmder the uame of the Chaplygin equat.ions. The development of theoretical aerodynamics was attended by the creation of experimental aerodynamics de"oted to the experimental investigation of the intera,1-ion between a body and a gas flow past it with the aid of nriolls tcdlllical meaJllo; slI,h as a wind t.unnel that imil-lIl.c tlw now of ail"\I'afi. Under the guidllnce of professor S. Zhllko\"sky (1817-Ht!1). the first aerodynamk laboratories in Hus."'ia wcre el'e,tpd (at t.hc ]\IOSt'OW Statc University, the Mosrow Higher Tcehnical College. and at Ku,l!ino, lIear Moscow). In 1\118, the Central "\l'roh:-'drodynftlllic Instit.ute (1's.'\(;'I) was organized hy Zhukovsl.:y's iuit.intive with tlte (Iirect aid of V. L('uin. At prcselll it is olle of the major world ,enlres fo the sciplJ('c of al'l"odYlIfllIlirs h(>ariug Ihe )Iallie of :\. Zh\lko\"~kr. The developmcnt of adatioll, artillery, amI rocket.I'Y. and t.he mntllring of the theoretical fllndamcntnls of aerodynamirs changed the nature of acrodynamk instal latiolls, from the first, comparath'ely small and low-speed wind tunnels up to lhe giant high-speed tunnels of TsAGI (HMO) and mocil'rn hypcrsonit" installations, and also special facilities ill whirh a snpersonir. now of a beated gas is artificially created (what wC call high-temperature t.unnels. I-tho,k tUllnels, pJasma instaJIatioJls, ek,). The nature of tbe il1terartioll between a gas and a hody moving in it may vary, At low speeds, t.he interactioll is mailll~' of a fm'('e nature. With a growth ill the speed, the force interartion is attended b~' heat.ing of the surface owing to heat transfer from the gas to the body: this gives rise to thermal interaction. At YCry high spce
The inve~tiguliou of all .kind::: of intel,{\(·tion hNwl'l'lI il gas and a ('raft allows one to perform aerodynamir ,alcnlatiolls fI~so{'jatl'd with Die evaluation of the quantitative ("riteria of thil-l interartion.
16
Introdudion
Ilamely, with determination of tile aerodynamic forces and moments, heat transfer. and ablation. As posed at present, this problem consists not only in determining the overall aerodynamic quantities {the total lift force or drag. the total heat flux from the gas to a surface, etc.), but also in evaluating tile tlistribution of the aerodynamic properties-dynamic and thermal-over a surface of an aircraft moving through a gas (the pressure and shearing stress of friction, local heat fluxes, local ablation). The solution of such a problem requires a deeper investigation of the flow of a gas than is needed to determine the overall aerodynamic action. It consists in determining the properties of the gas characterizing its flow at each point of the space it occupies and at each instant. The modern methods of studying the Dow of a gas are based on a number of principles and hypotheses established in aerodynamics. ()ne of them is the continuum hypothesis-the assumption of the continuity of a gas fiow according to which we may disregard the intermolccular distances and molecular movemeuts and consider the continuous changes of the basic properties of a gas in space and in time. This hypothesis follows from the condition consisting in that the free path of molecules and the amplitude of their vibrational motion are suffici~ntly small in comparison with the linear dimensions characterizing flow around a body. for example the wing span and the diameter or length of the fuselage (or body). The introduced continuum hypothesis should not contradict the -concept of the compressibility of a gas, although the latter should seem to be incompressible in the absence of intermolecular distances. The reality of a compressible continuum follows from the circumstance that the existence of intermolecular distances may be disregarded in manyinvostigations, but at the sarne time one may assume the possibility of the concentration (density) varying as a result of a change in the magnitude of these distances. In aerodynamic investigations, the interaction between a gas and a body moving in it is based on the principle of inverted Row according to which a system consisting of a gas (air) at rest and a moving body is replacod with a system consisting of a moving gas and a body at rest. When one system is replaced with the otlier, the condition must be satisfied that the free-stream speed of the gas relativo to the body at rest equals the speed of this body in the gas at rest. The principle of inverted motion follows from the general principle of relati'\'ity of classical mechanics according to which forces do not depend on which of two interacting bodies (in our case the gas or craft) is at rest and which is performing uniform rectilinear motion. The systt'm of differential equations underlying the solution of problems of flow over objects is cllstomarily treated separately in
Introduction
17
modern aerodynamics for two basic kinds o[ motion: frt'(' (in\'isrid) flow and flow in a thin layer of the gas adjacent to a wnll or boundary-in tho boundary layer. where motion is {'onsidered with (1('('01111 t taken o[ viscosity. This dh'ision of a nO\\' is based on the hypothesis of the ab$encc of the I'f'v('rse innupnce of the boundary layer on the free flow. According to this hypothesis, the parameters of IJl\'i!'icid flow. i.e. on the outer !'iurface of the bOllndaryla~'er. are the same as on a wall in the absence of this layrr. . Th.e linding of the aerodynamic parameters of craft in unsteady motiol! dlarar\.r>ri1.cd by a ehango in the kinematic parameters with time is usually a \'ery intricate t.a.'lk. Simplified ways of solving this problem are used lor practical pllrposcs. Such simplification is possihle \vllen the change occllrs .~lJfJi('ienlly slowly. Thif: is chararterif:tic of IlHlIlY craft. "'hell determining thcir aerodrnamic chn.racteristits. w(' ran pro('('ed rl'om the hypolhe!iis o[ steadiness in accordanee wilh which the:-;(' dl,II'(H'tcristic~ in unsleady lIlotioll nrc assumed to be thl' ~alllC;JS ill ~teallr Illolion, ilud are determined by the kinematic pnraml'ler.<; of Ihis ll1olioll {It 11 gh'cn inst.anl. \rhell performing lIero(lynaml(' experiment.s ali(I cnklliations. account must be Ilikell of \"ari01ls dl'CIIIlHllanc9S n~~ociatcd with. the ph~'sical ~imilitllde of the llow phenomena being studi('!!. Aerodynamic C;llcuiatlOlls of fll\l-~rfllc (T11.rt (rod,;et!'. airplanc:-;) arc based on prelinlinary wiue;;pr();ul iJ1H'sligatiolls (theoretical and experimeutal) of flO\.... over models. The ('ouditions tll(lt ml}.~t Ill' ohsl'n'ell in such investigations Oil models III'C fOllud in tile th('or~' of dYl1allli('. simili~ tude. allll t.ypical aud convenient paramel('I".~ dl'll'rLllillillg the basic conditions of the proce:-;scs being studied arc f'_~[
i;.:ed basic scientific trends have taken shn.pe in aerodYlLamics. They are (lSSOCiflled with the aerodynamic investigations of craft as a whole and their illdi\·i~ dual structural elements, and also of the most characteristi\. kiuds of gas flows and processes attending the flow over a body. It is qnite natural that any classification of aerodynamics is conditional to a certain extent becauso all these trends or part of them are interrelated. :"-Jevcrtheless, such a "branch" specializa(.ion of the aero~ dynamic science is of a practical interest. The two main paths along which modern aerodynamics is developing can be determined. The first of them is what is called force aerodynamics occupied in solving problems connected with the 2-017\$
18
Introduction
force action of a fluid, i.e. in iillding the distribution of the pressure and shearing stress over the snrface of a craft. and also with the distribution of the resultant aerodynamic forces and moments. The data obtained are used for strength analysis of a craft as a whole and of individual elements thereof. and also for determining its flight characteristics. The second path includes problems of aerothermodynamics and aerodynamic heating-a science combining aerodynamics, thermodynamics, and heat transfer alld studying flow over bodies in connection with thermal interaction. As a result of these investigations, we find the heat nuxes from a gas to fl wal1 and determine its temperature. These data are needed in analysing the strength and designing the cooling of erflfl. At the Sflille time. the taking into account of tlte changes in the properties of fl gAS flowing over a hody under the influence of high temperatures allows us to determine moro precise!)' the quantitative criteria of force interaction of both the external flow and of the houndary layer. All these problems are of a paramount importance for very high air speeds at which the thermal processes are very illtensiv(!. Even greater complicaliolU; are introduced into the solution of sudl problems, however, because iL is associated with t.he need to take into consideration the chemical processes occurring in the gas, alld also the influence of c.hemiral interaction between the gas and the material of t.he wall. If we have in view the range of air speeds frolll low subsonic to very high supersonic ones. then, as already indieated, we can separate the followillg basic regions in the science of inve~tigatilJg flow: aerodynamics of an incompressible fluid. or fluid mechanics (the Mach number of the flow is M - 0). and high-~peed aerodynamics. The laLLer, in tllfn, is divided into subsonic (M < 1), transonic (M ~ i), supersonic (M> 1) and bypersonic (1)1 ~ 1) acrodynamics. It must be noted that each of these branches studies flow processes that are characterized by certain spedflc features of flows with the indicated :\Iac.h nnmbers. This is why the investigation of such flows can be hased on a different mathematical foulldation. 'Ve have already indicated that Aerodynamic investigatiolls fire hased on a division of the [Jow ncar bodies into two kinds: free (extE'rnal) inviscid flow and the boundary layer. An independent section of aerodynamics is devoted to each of them. AerodynamiCS of an ideal fluid st.udies a free flow and investigates the distribution of the parameters in inviscid now over a body t.hat are treated as paramet.ers 011 the boundary layer edge and, cOlisequE'lI tly, arc the bOllndary conditions for solving the differential equations of this layer. The inviscid parameters inclnde the pressure. If ,ve know it.s distribnlion. we can find t.he relevant resultant forces and moments. Aerodynamics of an ideal fluid is based on Euler's fundamental equations.
Introduction
19
Aerodynamics of a boundary layer is one of the broadest and illost dc\'e-Iopcd sect,iolls of the sden('o of a fluid in mot.iOIl, 1t. studies \'is('ous gllS now in 1\ bOlilldal'Y laym', The 1'oluliOIl of [he problem of flow in a boundary III:,-'cr makes it possihlc 10 lind Ih£' di1'tributioll of the shearing stresses and. ("onscIllIcully, of the resultant aerodynamic fof("l's IIlid momcnts (',\USNI hy fl'klion. It also make::: it. possihl(> to (",II("lIia(.r. the tra!lsfer of helll from Ihe gas nowing o\"el' a hody to a houndary, The ('on('lusi(Jn~ of t.he bOlludarr la,..'cr theory ('illl also ill.' used Cor corrL'('ting lhc ~ollllion 011 in\"iscicl now, pm'lklllarly for lililling the corrert ion to tile pre!'.'1llre distrihution rlllC 10 Ille iunllen('e of the boundm'y layer, The modC!rll Ul(!oryof tht' hOlludary layer is hased Oil fundalllental iln'csligat.ions of A, XIl"ier, G, Stokc!ot, 0, HeYl101ds, L. PI'andtl, and T, "Oil Karmali, ,\ substantifll ('outribut.ioll to the de\"elopment of the boundary Illycr theory was lUade h)' the So\'iet ~cientists A, DoI'OdnitsYII. L, Loihi,yansky, A. :'Ilelnikov, 'X, Kochin. (i, Petro,', V, SII'llminsk~'. atlll otllrol'!-', TIIC'Y !"l'eat.ed a harmouium: theol'Y of the boundary laycl' in a compl'e~sib)e gas, worked alit. method!! of ('alrlllilting til(') no\\' of a \"i!-'cou~ nuid o\"er "arious hodi('s (t\\"o- and thl't'c-dimensionlll), in\"estigilted problem!> of tJle transition of a laminill' I>oundlu'y layel' into a tlll'bulcnL one, and studied the complkalc:i pl'Ohll'llI~ of tllfhulent lIlotion. In ilel'odYlUunil' iH\'cstigatiotls in\"olving low airspeeds, the thermal proe(lSS(l!-t in the hOUtHlil!')' layer do not havC! to bro taken illio ,H'('Olltll ul'('aw<e of tlwit' low iut.enllity, "'hen high !oipl'Nis arc iuvoh-ed, howen'l', lIt'rount Illlt:-;t be taken of heat transfer and of tite influence of the high hOlllldllry layer temperatur('!'! 011 rrktioll, It i!'l quile natltral tbat abumlallt attention is being gi\"en to Ihe :'!ollltion of slldl problem!;. especially recently, In t.he SO"iet L'nion, professors L, Kalikhman, I, Kibei. V, lye\"Jev and others al'e developiug the gas-dynamic theory of Ileal. transfer, stlldying tllc ,'is('olls Row 0\'("1' ,'arious bodic!'! ot high I("mpel'ature!'! of the boundal')'layer, Similar prohlems arc also heing :o:ol\"ed by U Ilumber of foreign scienti!'!t~.
AI hypersouic flow speptls, the prohlt'ms of 1I("l'oclyuflmic heilting liot the only on(l~, That ionization Q('t'III'S fit· sneh sp('eds because of the J1igh temperat.ures and the gas begins to ("(lIuluct ele<"lridty ('illlSell lIew probll'lIIs a~!'!ocialt·d with control of till' plllllma flow with thc ~id (If 11 magnetic. lieJd, When df'~rihillg the prOl'CI"!:.I!!' o[ inlel'a('tion of 11 moving body witlt pla!!-ma, the rcle"alli. flCl'otirunmic 111'('
('ulrulation!'l musl1.ak(, into flCCOllitt (.ll'("tromagu('1 i(' [()rc'('~ in addition 10 gas-rlynamk. one!'. The!
20
Introduction
of aerodynamics (tre treated ill a fundamental work of L. Sedo\': "Continuum :\lechanirs" (ill HII~:-iall-a textbook for universities) [21It must he noted that the continuum hypothesis holds only for conditiolls of flight at low altiludes, i.e. in sufflcient.ly dense layers of the atmo:-phcre where the mean free path of the air molecules is small. At high altitlldes in (',ond itions of a greatly rarelied atmosphere, the free path of molecules becomes : of flow O\'er craft 8ud. cOII~(>lluently, their aerodynamic c.hara(~teriRlics \'
Introduction
21
edge of the aerodynamic charaderislics of the s;eparatc constituent parts of an aircraft. Aerodynamic calculalions of tile> lifting plnnes of wings is the subject of a spec'in! branch of the nerodynamic science-wing aero· dynamics. The ollt!!tlmding Russian scienlists ami mechanics)i. Zhuko\':oky (Joukowski) and S. Chap1rgin arc by right. con.!liderec1 to be the founders of the aerodynamic 1I1eory of a wing. The beginning of the 20th centul'r WfiS noted by the remarkable disco\'ery by Zhuko\'sky of the natllfe of the lift force of a wing; he derived a formula for calculating this force that bears his name. His work on the bound \'ortices that. arc a hydrodynamic mode] of a wing was far ahead of his time. The series of wing profiles (Zhukoysky wing pronles) he den·loped w{'re widely used in d{'signing airplanes. Academician S. Chfiplygin is the author of mfiny prominent works on wing aerodYllnmks. In 1910 ill his work "On the Presslll'f> of a Paranel Flow on Obstacles" (in Hussian). he laid the foundations of the theory of an inflllite-span wing. In 1922, he published the scientific work "The Theory 01 n Monoplane Wing" (in Ru!!'sian) that sels out tJle Ihcol'Y of a Ilumber of \.... ing proli1es (Chnplygin wing profiles) nnd al:oo de\"elop~ the theory of 1>tability of il monoplane wing. Chaplygin is the founder of the theory of a finite-span wing. The fundamental ideas of Zhuko\':oky and Chaplygin were de\-cloped in the works of Soviet scientist!!' specializing in aerodynamics. A."sociate member of Ihe USSR Al'aclemy of Sciences V. Golubev (188-1.-1954) i1l\'estigaLed the flow past 1>hort-span wings and various kind:s of high-lilt devices. ImportaHt rcsnlts in Lhe potential wing theory werc obtained by aeademiciau .i\1. Keldysh (1911-1978), and also by aeaciemic.ians 1\1. Lln'rentyev aud L. Sedo\'. Academician A. DorodlIi15yn summarized the theory of the lifting (loaded) line for a sideslipping wing. Considerable ult::; they obtained can be l:onsidered as particular C Sl'S of the general theory of flow developed hy Khristiauo\'ich.
22
Infrodudion
A great contribution to the a('rod~{nami('s of a wing wa!> marie by academician A. \'ekraso\' (t883-UJ:i4), WIIO de\'eloped a harmonioll!> theory of 11 lifting plane in an ullsteady now. Keldysh and La\,rentre" !'o\"ed the important prohlem 011 the now over a \'ibrating airfoil by generalizing Chaplygill's method for a wing with varying circulation. Academician Sedoy estAh1i!>hec\ general formulas for ullsteady aerodynamic forr,C!'I and moments acting on :m arbitrarily moving wing. Profe!'1sol'S F. Frankl. E. Kra!>ilsllchiko\"a. and S. Falko\'ich developed the theory of steady and unstead~' supersonic now O\'er tllin wings of various conligurations. Important results in studying unsteady aerodynamic."I of a wing were obtained by professor S. Uclolserko\'sky, who widely used numerical methods and computers. The results of aerodynamic in\'estigations of wings can be applied to the calculation of the aerodynamic characteristics of the tail unit. and also of elevators and rudders shaped like a wing. The specifiC features of flow over separate kinds of aerodynamic elevators and rudders and the presence of other kinds of conlrols resulted ill the appearance of a special branch of modern aerodynamics-the aero-
dynamics or controls. Modern roc.ket-type craft often have the conliguration of bodies of revolution or are close to them. Comhined rocket systems of the type "hull-wing-tail unit" ha\'e a hull (body of revolution) as the main componellt of Lhe aerodynamic system. This explains why the aerodynamics of hulls (bodies of revolution), which has become one of the important branches of today's aerodynamic science. has seen intensive development in recent years. A major contribution to the development of aerodynamics of hodies of revolution was made by professors f'. Frankl and E. Karpovic.h. who published all interesting scientific work "The Gas I)ynamir~~ of Slender Bodies" (in Russian). Tile Soviet scientists I. Kihei alld F. Frankl. who specializefl in aerodynamic!';. developed the method of characteristic.Oj that macie it possihle to perform effective calculations of axisymmetric supersonic flow past pointed bodies of re"olutioll of an arbitrary thickness. A gl'oop of scientific workers of the Institute of Mathematics of the USSR Academy of Sciences (K. Babellko, G. Voskresensky, and others) de"eloped a method for the numerical calculation of threedimensional supersonic now o,'er slender hodies in the general case when c.hemical reactions in the now are taken into account. The important problem on the supcrsonic flow over a slender COlle was solved by the foreign specialists in aerodynamics G. Taylor (Great Britain) and Z. Copal (uSA). The intensivc dewllopment of mo(lern mathematics and computers and the illlprovemeni on Ihi~ ba~is of the methods of aerodynamic
Inb-odudion
23
in\"esligations lead to greater and greater f:ucces~ in soh-ing m.my (:omplicaled problems of aerodynamics including the determination of till' o"eralJ aerodynamic charac.teristics of a craft. Among them are the aerodynamic derh'atiYes at subsonic speeds, the finding of which a work of S. Belotserko"sky and B. Skripach 14.1 is devoted to. In addition, approximate methods came into u~e for appraising the effecl of aerodynamic interference and calculating the releyant corrections to aerodynamic characteristics when the latter were oblaim'd ill the form of an addiLi"e sum or the rde"ant characteristic's of th(' illdi\"idual (i!lolated) elements of a craft. The solution of such prohlems is the subject. of a special brand. of til(' aerodynamic
scieurc-interferenee aerodynamics. At low slLper!'onic speeds, aerod:'<'lIamic heating is comparatively small and rannot. lead to destrurtion of a <'raft member. The main prohlem ~ol\"ed in the given <,_ase is associated with the choice of the cooling tm' nlaintllining the required boundary temperature. More in\"oh·ed pro hI ems appear for "ery high airspeeds when a mO"ing body has a trementious store of kinetic energy. For example. if a cralt has an orhital or e~rape speed, it is suflic-ient to ron\"ert. only 25-a!.l"" of t.hi~ energy into hell! for t.he entire lllll('rial of a structural memher (0 evaporate, The main prohlem that appears, partie-1l1arly, in Ol'gallizing the safe> re-entry of n cl'a[l into (he delise-layers of the atmllsphN'e rOllsist~ in di~~ipatillg this energy so that a minimum part of iL wilt be absoi"l)cd in I-he form of heat. b~' t.he hOlly, h was fOlllld that blunt-Ilos('d bodies have Stich a property, Tllis is exactly what resulted in the d('\'cloprncmL or aerOd;\'IHHnie- ~t\ldics of sllch bodiC'!';. An important contribution to investigating the prohlems of aerodynamics of blunt-nosed bodies was mndt' by SOl'iet scientistsarad(>nlicitlll~ A. Dorodnilsyu, G. Cherny. 0, Ue]ol'ierkovskr. and otil('rs, Similar investigations WI!re perrorml."tl by :\1 Lighthill (CorN\\. 13l'it,ain). P. GlU'ftbeclian (USA), and other foreign scit'ntist.s, Ullluting of the frout surlare must be considered in n ('ertsin seUi'e as a way of thermal protection of a craft. The blnnted nose experiences the most iuteush'e thermal action. therefore it requires thNlllai Pl'ot.e<'tion to even a greater extent thaH the peripheral part of the craft. The most effecth'c protection is i\5sociatcd with the use of v3rioull rontings whose matarial at. the relcnwt temperatures is gradultllr del'troyed nntl ablated. Here a con:-liderable part of the en('rgy ~\\pplied by the heated air to the craft i~ absorbed. The development of the theory and pract.ical methods of calculating ablation rplal('s to a modern branch of the aerodynamic f:cience-aerodynamics of ablating surfacE'S. A broad range or aerodynamic problem!': is QRsociated with the determination of the interartioll of a Ruid with a craft ha"ing an arbitrary preset shape- ill the general ca!':6. The ~hape~ o[ cralt sur-
It
Introduction
faces can also be chosen for special purposes ensuring a definite aerodynamic effect. The shape of blunt bodies ensures a minimum transfer of heat to the entire body. Consequently, a blunt surface can be considered optimal from the viewpoint of heat transfer. In designing craft, the problem appears of ehoosing a shape with the minimum force action. One of these problems is associated, particularly, with determination of the shape of a craft head ensuring the smallest drag at a given airspeed. Problems of this kind are treated in a branch of aerodynamics called aerodynamics of optimal shapes.
1.1. Forces Acting on a Movtng Body SlII'face Force
Let us consider the (orces oxerted by a gaseolls \"i"cou$ continuum on a mo\'ing body. This flclion coo;<;j;<;ts in the llniform dbtribution over the body's surface of the forcp.s P n produced by the normal and the forces P, produced by the shear stresses (Fig. 1.1.1). The surface' element dS being considered is acted UpOJl by a resllltant force called a surface one. This force P is rleterillined according to the rule of addition of two vectors: P" Hnd P-r' The force P n in addition to the force produced by the pressure. which does not depend all theviscosity, includes a component due to friction (Maxwell's hypothesis). In an ideal fluid ill which visl'Osity is assumed to be abl'lcul., theaction of a force on an area consi."l:; only in that of the forces produc.ed by the normal stress (prI'SSHre). This is ob\'iolls. because if Ihe forcedeviated from a normal to the arCH. its projccLioll onto this area would appear, i.e. a shear !itress wOIl!d exist. Thc latter, however. is absent in an ideal fluid. In accordance witll tlte prillciplc of inverted' flow, the errcc.l of the forces will be the ~allle if we consider a hod\' at rest alld a uniform flow oyer it having H \'c-locity at infinity eqllal to the speed or the body before in\'ersiou. We shall c.aH this \'elocity the \'cloclly at infinity or the frcc·stream "clocH), (Lhc \'clodty of the lIudistUl'l:!CJ flow) and shall dcsignate it by -V,.,,, ill contrast to V (the ,"eloeity of the body relativc to the undisturbed now), i.e. I V I -= I V <» I. A free stream is characterized by the undist.lIrbed pilramctcrsthe pressure por,. density P<x, and temperature T x differing from their counterparts p, p, and T of the flow disLurbed by the body (Fig. 1.1.2). The physical propel'ties of tJ gas (air) are also rharacLerizl'(i by the following kinetic parameters: the dynamic \'iscosity )l- alld the coeIlicient of heat COlldtu'lh'ity f.. (the tlJldis\lll'Led p;lrlHlwter." al't' fI "" and j""", l'espE'cti\'cly). as WE'll as uy thermodYIll'llllic para-
26
Pt. I. Theory. Aerodynllmtcs 01 lin Airfoil lind a Wing
Fig. t.t.t Forces acting on a surface element or ~I moving body
__________ 1!.:!.:!_________ _
V~_::_,_T,,:
Fig. t,t,l
£~~~~rlb~~li~l~d (::nJi~~~~b~~rno;.! meter!): the specific heats at conslant pres.sure c p (c p .,,) and constant volume c,. (c/.,.,) and their ratio (the adiabatic exponent) k = cplc(} (k 00 ~ cl'
To determine the property of pres!'lll'cS in an ideal fluid, Jet liS take an elementary particle of the flnid having the shape of a tetrahedron .1foMl.lf2M3 with edge dimensions of 6o,T, tJ.y. and tJ.z {Fig. 1.1.3) and compile eqnations of illation for the particle by equating the product of the mas.<; of this elelllcnL and its acceleration to the sum of the fOITes aCling on it. We .shall write these equations in projections onto the coordinate axes. We shall limit ourselves to the equations of motion of the tetrahedron in the projection onto the x-axis, taking into account that the other two haye a similar form. The product of the mass of an element and its acceleration in the direetion of the x-axis is PIl'· tJ.W dV:.Jdt, where Pay is the average density of the fluid contained in the elementary volume tJ.W, and dV:.;ldt is the projection of the particle's acceloration onto the x-axis, The forces acting 011 our particle are determined as follows. As we ha"e already pstablished. these forces include ,,,,hat we called the surface fOI'('e. IIere it is delermined hy the action of the pressure on the faces of our particle, and its projection onto the x-axis is Px tJ.S x -
/'
p" 60S" cos (n,x).
Ch, I, Basic Information Irom Aerodynamics
27
Fig. t.1.3
:"i'(Jrmall>tr('~('s acting on 11 fa('(' 01 an ~'Iementary fluid Jlartid~'
hllvin~
Ihe ShllPI' of a tctl'ahed-
I'on
Allotlu'[' fOl'ce nctillgon the il>olatt'd tluid nllulI[e is the mass (body) force proportional [0 the mR~li of th€' particle ill Illis \'olume, \[ass forccs indude gravitatinnoi ones. if it is lIll electric condurtor lionized) and is in an elel'tromaguetir rteld, Hcrc we shall not consider tlte motion of a gali !lnder the action of SUell rOn'eli (see II :"pccifll ('OIlI':"C ill magnetogasdyuHmics). ill thc Ullie being cOll!-;idefcd. we shall writc the pl'ojection of Ihe mass rorrr! onlo the .1·-Hxis in the forUl or Xp:o" 6. IV, denoting hy X the pl'ojPetion of tlte' ma.<.:s fOl're reillted 10 II unit or l11aliS. "'illl lIl'('onnL takl'lI of these \'alocs for the Pl'OjC'(,tions or tht, surfal'e and OIalili [orres, we obtain an eqllalion of motioll fl;w
~Hl~""'" X!,~,. ~W
,- fl.,' 6.S, -
P"
6.S" ('OS
(;;.i)
where 6.S,. and I~S" arc thp <1rC'as o[ [accs .lI(I.11~.11;1 and ,llIJlI~.ll:l'
,
resp(lrti\"clr, cos (11,,1'), is\lit'co:-;iIlC or tll(' allg-If' hetwecn a normal n to f~cc :\JI.'ll2,ll:1
!1S, -, 6.S" ("os (1I~.r). 1f't w< paliS 0\"(>1' to the lilllit \\"itll 6.,r. !!.!I. lind j,z telldiug to zero. COllsequcutl~'. thc [('1'111:< conillilliug ~l-V/6.S,~ will 111so tend to ZPI'O b('~'1111:;(! !!.W is l\ small qllautitr or the third ()rt\C'r. while 6.S,. is a small <jlllllltity of till' :,<('("ond o!'dC't' in ('Otllpal'i:"ol1 with the lillCR!' dim(,lIsiol1:< of the sUI'[ac(' l'h-'m('lli. A.'> a rt'sult, we ha\'(' I',,' - fl" _.- 0, alld. therdor(', p". - 1'", \\'!I('11 cOJlliiderillg til(' I'qllfltious of mo\ioll in proj('('tiolls onto I.he y- aun ':;;·I\\('li, we lind Ihnl Py ,. PI' lind p, p". Sillt"l, ollr ,<.:ul'[ace ei{'OIf'nl with 1111' normal II is oriented arbitrarily. \\'(> ('all al'l'i\'p a\ the follo\\'illg r()1\t'llision frOIll the r('sult:'l obtained. The prel>li\lI'e at,
28
Pt. I. Theory. Aerodynamics 01 an Airfoil end eWing
all snrface elements passing through this point, i.e. it does not depend on the orientation of these elements. Consequently. the pre.'lSure can be treated 8S a scalar quantity depenliing only on the coordinates of a point and the time. InDu.aca 01 VIKOSHy on th. Flow of • fluid
Laminar and Turbulent "'low. Two modes of Dow are characteristic of a "iseous fluid. The firsl of them is laminar Dow distinguished by the orderly arrangement of the individual liIament~ that do not mix with one another. Momentum, heat. and matter aro transferred in a laminar Dow at the expense of molecular pro(',es..ws of friction, heat conduction, and diffusion. Such a flow usnally appears and remains stable at moderate speeds of a fluid. If in given conditions of flow over a surfar.e the speed of the Dow exceeds a certain limiting (critical) value of it, a laminar flow stops being stable and transforms into a new Idnd of motion characterized by lateral mixing of the fluid and. as a rc!;ult. by the vanishing of the ordered. laminar flow. Su('h a flow is called turbulent. In a turbulent flow, thc mixing of macroscopic. particles having velodty components perpendicular to the direction of longitudinal motion is imposed on the molecular chaotic motion characteri!ltk. of a laminar now. This i!l the basic distinction of a turbulent flow from a laminar One. Another rlistin('tion is that if a laminar 110w may be either !lteady or unsteady, a turbulent flow in its essen('c has an unsteady nat.ure when the ,'elority and otlll~r parameters at a given point depend on the Ume. Acrording to the notions of the kinetic theory of gases, random (disordered, chaotic) motion i!l c.hara('teristic of the partieles of a fluid. as of molecules. 'WIlen studying a turbulent flow. it is convenient to deal not with the instantaneous (actual) velocity. but with its average (mean statistical) value during a certain time interval tz_ For example. the component of the average velocity along the x-axis is Vx = = [1/(t 2
-
tt)]
.,J Vx dt, where Vx is the ('omponent of the actual "
velocity at the given point that is a function of the time t. The components Vy and ~ along the y- and z-axes are expre~ed similarly. using the con('ept of the average velocity. we can represent the aetua) velocity as the sum V:t = V:t + V~ in whlcll F~ is a variable ad(litional component known as the lIuetuation velocity component (or the velodty Ouetuation). The fluctuation components of the velocity along the y- and z-axes are denoted by V; and V;, respectinly. By placing a measuriDg instrument with a Jaw inertia (for example~ a hot-wire anemometer) at the reqnirtld point of a flow. we can record
Ch, 1. Basic Ill/ormation from Aerodynamics
29
or measure tne fluctuation speed, III Cl turbnlent flo\\". tile instrllment registers the deviation of the speed from the mean \"alllC'-the fluctuation speed. The !.:ineti<'. energr of a tllrl.l1l\enl nO\\' i:-; determined by the -"HID of the !.:inetk. cncrgie!< \~alclllated "ccorning to the nll>an and fiuctuation speeds, For a point in ljUC'.qiOll. the I,inetk energ~' of a fluctuatiou flow c·an be determined as a quantity proporlionallo the mean "alue of the mean square tll1ctuation \'elocitie~. If we resohe 1.he nne.tuation flow along the axes of 11 coordinate .~y;,lenl, the kinetic energy of each of the components of snch a flow will be proportional to the rel(>,'ant mean square componpnts of the Ilu('tllution velocities, designated lly fJ, find 1';2 [lod determinl?d from the (Ixpr('ssion
T'"7.
V;."!",:)""
I~~II
I,
JV~·~y,t)dt I,
The concepts of <\\"C'rage and fim'!lIltling qoantities cau he ('x tended to the pre~~\1!"C unu other phy:.it"ai IlaL'amete)'s. The existl.'))I'e of fluctuation velocitie!i lead$ to additional normal and !'hear :;trcsses and to the more intensh'c lransfer of heat. anti ma~s. All this has to be taken iuto account whell running experimenl:-!' in aerodynamic tUIlHeis. The turhulence ill the ntmosphl?rc wa.:; fouliel to 1m relatively small and. conseq!H;~ntly, it. should be just. as :mHdl ill the working purt of a tunnel. An inaeascd turllulencc ,,(rech Ill .., results of an experiment adversely. The natllff' of this innllence uepentl ..:; on the turbulencf' 1(;\'('1 (or the initial turbui('nc('). detrrlll\(n'd from the expre!'~ion
(1.1.1)
where V is the overall ll\'erage speed of the turbulent fiow at the point being considered. In modern low-turbulence aerodynamic tunnels, it is possible in practice to reach an initial turbulence close to what is observed in the atmosphere (e ~ 0.01-0.02). The important characteristics of turbulence include the root mean square (rills) fluctuations V"f;?, and V"'V;Z. These quantities, related to the overall avera~e speed, are known as the turbulence intcn-;ities in the corresponding directions and aro denoted as
1""17,
'. ~ V~iV. ',~ VV;"iV.
'. ~ VV;FiV
(1.1.2)
Using these characteristics, the initial turbulence (1.1.1) can be expressed as follows:
e=V(e~+e~+S:)/3
(1.1.1')
30
pt. I. Theory.
Aerodynemics of en Airfoil end _
Wing
(0)
FI,.
t.t ...
}o"Jow of 8 vist.oll$ fluid ov('J' a body: a-51"lIpnmlle vit'w
or
ftl>W; ,. laminar houndary lal'l.'.:
~~;:~U~~~r~1-~;~~:~~l~:?i~~~rnii~~br~s£ifJ:i\f::{~~~:~t~~!:
01 the boundary itlYt".
Turbulenc(' is of a \,or(e); nature, Le. mass, momentum, and energy Arc tran1'fel'l'ed by fluid particles of a \'OI'tC:( origin. Hence it follows that f1l1rtnaHons nre rllara('terized by a ~latislical !l!lSOdAtion. The correlation eo('tl'ieient betweell f1ucluations at points of the region of fl disl1ll'bed flo\\' being studied is a qllantitative lll(,llS1Ife of this M:'iociatioll. In the general form. this (',oefftrient helween two I'AmJom fluctuating qU8utities If and 1f is written as (see 151) R-
(1.1.3)
If there is no statist-iral aS50dation between the quantitics If and 11'. then R .. 0; iI. (·on\'ersely. these quantities arc regularly ass/)dated. the ('orrelation ('oeffldent R .,.., 1. This characteris\.i(, of turbulence is ('alIce! a two-point correlation cOf'flicicnt. Its expression clln he Wl'itten Wig. 1.1.4c) for two points 1 find 2 of a fluid \'olnme with the relevllllt f1ur_tllations V~l aud V;1 in the form
R=V~tV;zi(I/'Vj l,r~)
(1.1.3')
Whell studying a tJlree-dimensional turbulent now. one usually ('onsiders fI large llumber of such coefliciellts. The ('oncept of the turbu)('nc(' scale is introduced to characterize this now. It is delt'rmined by the expres.."ion L-
r
, Rdr
(1.1.4)
The turbuleuce sCflle is a linear dimension characterizing the length of the section of a flow on which fluid particles move "in
Ch. I, BlSic Information from Aerodynamics
91
Association", Le, ha\'e ~t8tistkally associated OU('.tulltiOIlS, By loving together the points being eonsidel'ed in a tllroulent flow. in the limit at r - 0 we can obtain a oDE'!-point corr€>lation coefficient. 'Vith this condition, ('1.1.3') aequires the fOL'ln
(1.1.5) This ('oefncip.llt rbat'ilctf'rizC!< Lhe I'ILntisl,ical assorialiou oetween at a point and. as. will he showu belo\\', tlil'C'('lly determille~ the shear slre~~ ill Il tnrhulellt flow. Turbulence will he homogeneous if its a\"('t'aged chal'actf'ristic:s found for II point (till' len" and iutensit,:,>' of turbulence. the OIlCpoint "orrelatioll cocflil'icnl) M(> lhc ~alllc fOl' the enlire now liu\'ilrian('c of tlac rharildcristk~ of tul"lmlt!Il('" in Il'flll~f(lr~), 1I0nlogC'lIcou:: tllrhllielU'c i" isotropic if it~ ('hm'a('((,"l'i~li{'s do 1I0t depend Oil the dil'e(,tion for whi(,h Ihey 111'0 ("llIl'lilalCld (lIn'al'ian("c of the ('hal'art.el'isti<'~ of tul'imlell("c in 1'0Ia(.io!1 and I'efl('('tion), Parti(,ularly. t.he following ('()lldiLion ii" !,;Clli~r.l'd fOl' an i~olropic now: fluctuation~
17-' f7 V;i If thh; condilion i!' ~llti:.::fied foJ' all poillt:.::. Ih(' IIII'bui(,II(,(, i~ homogeneous .md isotropic. For ~uch IlII'bulellce, tile constancy of the two-point. ("orreJalioli ('oeflil'ient is I'l't.llined willi \'tll'iolls directions of t.he )jill' ('olllle('tiug t.he t.wo points in t.he Ililid "olume' being considt"red_ Fol' ClII isotropiC'" now. I,he ('ol'relation ('oemdent. (1.1.5) ('an be
expre:';:';ed in t.erm~ of the tnrbulen('(' le\'el
f
--:
V~,.'t':
R=~/Vf=~(V;!e;!)
(l,l.!i)
The introduction of the ("ou('rpt of t!Ycl'aged pal'ameters 01' \l1'Opel'ties appreciably facilitlllCs the iu\"cstigatioll of tUl'bulent flows, Indeed, for praetical p\lI'pO~t"S, there is no need to kno\\' t.llt~ iustantaneo\l:'; values or the velodtie~, pressures, or ~henr ~tresscs, and we can limit 01lrse}\"cs to Iheir time-avcragcd \"all1c~, Thc lise or a\'('I'aged pal'ametcr~ simplilil's Ihe relevanl ~>IJ\li.ltiOIl!; motiOIl (11)(' Reynolds eqllation~). Sudl e
or
32
Pt. t. Theory. Aerodynlmics of an Airfoit and a Wing
molecules attended by transfer of the momentum from one layer to another. This leads to a change in the flow velociLy, Le. to the appearance of the relative motion of the fluid particles in the layers. In nccordanc:.e ,vith a hypothesis first ad\'anced by 1. Newton, the shear slress for given conditions is proportional to the velocity of this motion per llnit distance between layers with particles moving Telath'e Lo one another. If the dislnnce between the layers is An. and the relatb'c speed of the pnrtides is Au. the ratio Aul An at the· limit when An - O. Le. when the 1a)'C1'g are in contact, equals t.he derivative (kliJn known as the normal "clocity gradient. On the basis >of this hypothe$is. we can write N"cwton':;: friction law: (1.1.7)
where ~ is a proportionality factor depending ou lhe properties of a fluid. its tempeoralurc and pressure; it i1' better known as the dynamic viscosity. Tht' magnitude of !L [or a gas in aeeQl'dllnce with the rormula of the .kinetic theory is ~ = O.499pcl
(1.1.8)
At a given density p, it depends on kinetic characteristics of a gas 5u('.h as the mean free path l and the Illean speed of its molecules. Let IlS consider friction in a tmbulent flow. We shall pro(',eed from the simplified scheme of the appearall('e of additional frictioll forces in turbulent flow proposed llr L. Prandtl for an incompressible fluid. and from the semi-empirical nature of the relations introduced for these forces. Let us take two layers in a ollc-dimensional flow -characterized by a change in the average<1 velocity only in one direction. With this in view, we shall assume that the velocity in one of the layers is stich that Vx =f=. 0, Vy = Vz = O. For the adjacent layer at a distance of Ay = I' from the first one, the averaged velocity is V ~ + (dV;r;/dy) I'. According to Prandtl's hypothesis. a particle moving from the first layer into the second one rctains its \'elocity v:, and, consequently, at the instant when this particle appears in the second layer, the fluctuation \'elocity V~ = (dV)dy) I' is ob-served. The momentum transferred by the fluid mass pV; dS through the area element dS is pV~ (V;r; V~) dS. This momentum determines the additional force produced by the stress originating from the fluctuation velocities. Accordingly, the shear (friction) stress (in magnitude) in the turbulent flow duo to fluctuations is
c
+
I'. I = pV; (V,
+ V;)
Ch. 1. Basic Information from Aerodynamics
33
An'rngiug this e;..pression, we ohtain _
.\
12
l~tI·-", t:~~1
,~
V;dt-i-
~
II
.\
V~V~dt~':flV);~,,;.pV~V;
"
w"~ra V:V~ \~ the a ... er:[\gNi ,-nIne ()[ till) pto(\I\C\ of
t.he [\uc\,uati.on
n~lo('jties. and V~ is Ihe averaged value of the nu('lliation velocity.
Wo shall show that this vallie of the velocity equals zero. Integrating llH' l·quality I"l ~ T V~ termwise with respect to t wit.hin the Iimils from I, to 12 and then dividing it hy 12 - II' we find 0-::
_ , _ .\" V" dl:-. - '1.--1,.' II
'I' V" dl
'~-tl. I,
- - '-
t!-t l
'\'~ V!; tit
•
II
=1'!I -:--''\'~ J"dt '2-'1' " 'I
Out Since, by delinition, V!I'
V~
=
~
"
.,J V~ dt
"
~ .\ 1'1/ dt, it is obvious that
" = O. lIonee, the averaged nduc of tIm sllCar
-
h~' tile relation I "[ I = = pV~V:, Illat i:; the generalize,/ Heynolll:-; rorrnlila. J ts forIn does not d()p~_tlt! Oli any spedf'lc assumptions Oil 111(' slrllr\tlre of the lurbulencc. Tile shear strcss determined by this (orlmdil call I)e expressed directly in terms of t.hc corrDlation coefficient. In ,,,'col'dance with
stress Jue to nllctlilltions can he expl'cs."{'fl
(1.1.5),
We
have
IToI~ pRVv;. VV;' or for au isotropic now for which we have V~~,
Ittl = pRv;! ..--: pRe 2VZ
(1.1.9)
Vl';f. (1.1.9')
According to this expression, an additional silent' stress due to [Juctuations doos not necessarily appeal' iJl allY 110w characterized by a certain tllrbuicnce lovel. Its magnitude depends on the measure of the stati.::lic
34
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
can be transformed as follows: _
l:td=pV:cV;= tl~tl (d:u,..)2
tit
_
j l'1ade=Pl2( d~,..)2
(1.1.10}
" coefficient
Here the proportionality a has been incl uded in the averaged value of l', designated by l. The quantity l is called the mixing length and is, as it were, an analogue of the mean free path of molecules in the kinetic theory of gases. The sign of the shear stress is determined by that of the velocity gradient. Consequently,
'. - pl' I d'V,ldy I d}'x1dy
(1.1.10')
The total value of the shear stress is obtained if to the value "t') due to the expenditure of energy by particles on tbeir collisions and chaotic mixing we add the shear stress occurring direc.tly because of the viscosity and due to mixing of the molecules characteristic of a laminar flow, Le. the vallie 'tl = ~ dV:tldy. Hence,
, ~ " + '. ~ ~ dV,ldy + pl' I av,ldy I dV,Idy (1.1.11) Prandtl's investigations show that the mixing length l = xy, where x is a constant. Accordingly, at a wall of the body in the flow, we have (1.1.12) It follows from experimental data that in a turbulent flow in direct proximity to a wall, where the intensity of mixing is very low. the shellr stress remains the same as in laminar flow, and relation (1.1.12) holds for it. Beyond the limits of this flow. the stres!! "t'l will be very small, and we may consider that the shear stres... is determined by the quantity (1.1.10'). Boundary Layer. It follows from relations (1.1.7) and (1.1.10) that for the same fluid flowing over a body, the shear stress at different sections of the flow is not the same and is detel'mined by the magnitude of the local velocity gradient. Investigations show that the velocity gradient is the largest near a wall because a viscous fluid experiences a retarding action owing to ito; adhering to the surface of the body in the fluid. The velocity of the flow is zero at the wall (sec Fig. 1.1.4) and gradually increases with the distance from the surface. The shear stress changes accordingly-at the wall it is considerably greater than far from it. The thin layer of fluid adjacent to the surface of the body in a flow that is charact.erized by large velocity gradients along a normal to it and, consequently. by considerable shear stresses is tsllt'!d it boundary layer. In t.his layer, t.he viscous fortes ha\'e a magnitude of the
Ch. I. Basic Information from Aerodynamics
3&
same order as all tho other foJ'("os (for example. t.he fOl'COS of inertia alld prossure) governing motion and, therefore, taken int.o 3r.count in t.he eqnations of motion. A physical notion of the boundary layer can lie obtained if we imngine the surface in the flow to he coat.e(1 with a pigment ~olHhl& in the fluid. It is obvioHs that 111(' pigment diffll:-iel< inlo Uw nuid and is simultaneously carried downstream. Con:;:eqnently. t.he colonred zone is a layer gradually t.IIid.:.t'ning downstrctlm. The coloured region of t.he nuid approximately coincides with I.JJC bOHO!lnry Inyer. Thi,.:; region leaves the surface in the form of n coloured wake (see Fig. '1.1 ...~a). A,.:; :;:howll by observations, for n turbulent flo\\" the difference of tile coloured region from the bonlldal'~' lay!:!r is cOlllpfll'ati\'rly:;:mall, whereas in a laminar flow this differenco may be very significant. J\('cordillg to theoretical and rxpe-l'imental investigationf-;. with an increase, in the velocity. the thickuess of the layer diminishes, and the wake becomes narrower. The nature of the velocity dil'tributiol1 over the CI'OSS ~e(':tiol1 of .1 boundary layer depends on whether it is lnlllinor or t.urbulent. O\ving to lateral mixing 01 the particles and also to their collb;ions, tJlis dist.ribution of the vriocity, more exactly of its timf"-avcraged value. will be appreciably more uniform in a turbulent flow than in a laminar one (see Fig. 1.1.4). The distribution of the velocities near the surface of a body in a flow also allows liS to make the conclusioll on the higher shear stre!>." ill a turbulent. houndary layer determined by the increased value of the velocity gradient. Beyond the limits of the boundary layer, there_ is a part of the flllw where the velocity gradients and. consequcml.l},. thr forces of frictiol1 arc small. This part of the flow is known as the external free Dow. In investigation 01 ali extel'nal flow. the influence of the viseous forces is disregarded. Therefore, such a flow is also considered to be inviseid. '1'ho velocity in the boundary layer grows with an increasing distance from the \.... all and asymptotically approaches a theoretical value correspouding to the Oow over the holly of an i"viscid nuid, Ln. to UIC \'alue of the velOCity ill the exterual flowat the boundary of the layer. We have- already not.ed that ill direct proximity to it a wall hinders mixing. and. consequently, we may RSSllme that the part of the boundary layer adjacont to the waH is in conditions close to laminar one~. Thjs thin section of a quasilaminar boundary layer is called a viscous sublaycr (it is also sometimes called a laminar sublayer). Later investigations show thot fluctuations are observed in the viscous subbyer that penetrate into it from a turbulent core, but there is 110 correlation between them (the correlation coefficient R ~" 0). Therefore, a('.cording to formula (1.1.9), no additional shear stresses appear.
F1g.t.t.5 Boundary layer: I_wall or a body In lbe B-outer edgt> or tbe layer..j
flOWI
The main part of the boundary layer outside 0;" the viscolls sublayer is called the turbulent core. The studying of the motion in a boundary layer is associated with the simultaneous investigation of the flow of a fluid in a turbulent core and a viscous sublayer. The change in the velocity over the cross section of tile boundary layer is characterized by its gradually growing with the distance from the wall and asymptotically approaching the valne or the velocity in the external Row. For practical purposes, however, it is convenient to take the part of the boundary layer in which this change occurs snIrlciently rapidly, and the velocity at the bouudary of this layer differs only slightly from its value in the external Row. The distance from the wall to this boundary is what is conventionally called tile thickness of the boundary layer 6 (Fig. 1.1.5). This thickness is usually dermed as the distance from the contour of a body to a point in the boundary layer at which the velocity differs from its value in the external layer by not over one per cent. The introducHon of the concept of a boundary layer made possible effective research of tile friction and heat transfer processes because owing to the smallness of its thickness in comparison with the climensions of a body in a Dow it became possible to simplify the differential equations describing the motion of a gas in this region of a flow, which makes their integration easier. I
t.l. ResuH-ant Force Action COIIIponents of A.rocIyn.mlc Forcu ..... MOInIlnfs
The forces produced by the normal and shear stresses continuously distributed over the surface or a body in a !low can be reduced to a single resultant "ector RII of the aerodynamic forces and a resultant vector M of the moment of these forces (Fig. 1.2.1) relative to a reference point called the centre of moments. Any point of the body
Ch. 1. Basic Infonnlltion from Aerodynamics
37
FI,. U.t
Aerodynamic rorces and moments acting nn a craft in the nighl path (.Tn' Ya, and la) and body- axis (%, y. and z) t:onrdinate systems
be this centre. Particularly, when testing craft in wind tunnels, the moment is found about olle of the points or monnling of the model that muy coincide with the nose o[ the body, the leading clige of Il wing, etc. When stlld~'ing real cases of the motion of such craft ill thc atmospherc, one call determine the aerodynamic moment about their centre of mass or some othe-r poillt that is a ceJllre of rotation. In engineering practice, in!\tead of cOllsidering tlte vectors R. and M, theil' projections onto the axe~ of a coordinate system are usually dNl.lt wilh. Let \IS cOII.~ider the flight path lind fixed or body axis orthogonal coordiuate I'Yl'tcm~ Wig. 1.2.1) encountcl'ed most often ill aerodynamics. In the night path sy~t(>m. thf' aerodynamic forces and moments are usually gh'ctI becanse the investigation of many problems of flight dynamic!; is eonnected with the m:e of coordinate axes of exactly !;lIch a system. Pal'ticlllHrly, it is convenient to write the eqll.atjon~ of motion of a eraft"s centre of mass in projections onto these axes. The flight path axis Oz, of a velocity system is always directed along the nlocily vector of a craft's centre of mass. The axis 0Ye of the flight path system (the lilt axis) is in tIle plane of symmetry and is direeled upward (its positive direetion). The axis Oz. (the lateral axis) is direeted along the span of the right (starboard) wing (0 right-handed coordinate system). Tn inverted flow, the night path axis OZa eoincides with the direr-li!")n of the now velocHy, while the axis OZa is directed along CHII
38
Pt. I. Theory. Aerodynamics of on Airfoil ond 0 Wing
the span or the lell (port) wing so as to retain a right-handed coordinate syslem. The latter is called a wind coordinate system. Aerodynamic calculations can be performed in a lixed or body axis coordinate system. In addition, rotation of a craft is usually investigated in this system because the relevaut equations are written in body axes. In this system, rigidly fixed to a craft, the longitudinal body axis Ox is directed along the principal axis of inertia. The normal axis Oy if: in the plane of symmetry and is oriented toward Lhe upper part of the craft. The lateral body axis {)z is directed along the span of the right wing and forms a right-handed coordinate system. The positive dircction of the Ox axis from the tail to the Ilose corresponds to non-inverted flow (Jo'ig. 1.2.1). The origins of both cOOl'dinate systems-the night pHth (wind) and the body axis systems-are at a craft's centre of mass. The projections of the vector Ra onto the axes of a flight path ooordinate system are called the drag force X., and lift force }'a' and the side force Za. respectively. The corresponding projections -of the same vector onto the axes of a body coordinate system are
+
+
Ch. 1. Basic Inrormation from Aerodynamics
39
Fig. t.U
Determining the position of a craft in space
the axis Ox and the projection of the vector V Olito the plane xOy. and the second is the angle between the vector V and the plane xOy. The angle of attack is considered to be positive if the projection of the air velocity onto the normal axis is negative. The sideslip angle is positive if this projection onto the lateral axis is positive. When studying a Right, a normal earth-fixed coordinate system is used relative to which the position of a body moving in space is determined. The origin of coordinates of this 5ystem (Fig, 1.2.2) coincides with a point on the Earth's surface, for example with the launching point. The axis OoY, is directed upward along u local vertical, while the axes OoXll and OOZr coincide with a horizontal plane. The axis Oox, is usually oriented in the direction or flight, while the direction of the axis Ooz, corresponds to a right-handed coordinate system. If the origin 01 an earth-fixed system of coordinates is made to coincide with the centre of mass of a craft, we obtain a normal earthfixed coordinate system also known as a local geographical coordinate system Ox:y,Zg (Fig. 1.2.2). The position of a cralt relative to this coordinate system is determined by three angles: the yawing (course) angle 11'. the pitching angle tt. and the rolling (banking) angle 'Y. The angle W is formed by the projection 01 the longitudinal body axis Ox onto tbe horizontal plano X;:Oyj (Ox*) and the axis Oxg; this angle is positive if the axis OX;: coincides with the projection of Ox* by clockwise rotation about the axis Oy~. The anglo ~ is that between the axis Ox and the horizontal plane and will be positioie if this plane is below the longitudinal
%toze
40
Pt. I. Theory.
Aerodyn~mks of lin Airfoil lind
II
Wing
body axis. The angle I' is formed upon the rotation (rolling) of a craft about the longitudinal axis Ox and is measured in magnitude as the angle between the lateral body axis and the (lxis OZr; dispillced to a position correspolHJing to a zero yawing allgJ(> (or as the angle between the axis Oz and its projection onto II horilOnlal planet.he axis Oz;). If displacement of the axis Oz~ with respect to the lateral axis occurs clockwise, \.IH~ angle "r is positi\·e. The pitching angle determines the inclinalion of a cfaft to the borizon, and the yawing angle-the rleviation of the direction of its flight from the initial one (for an aircraft this is the deviation from its course, for a projectile or rocket this is the deviation from the plane of launching). ConversIon 01 "erocfYMllllc Forces .. nd Moments from One Coordinate System to Another
Knowing the angles (I. and ~, we can Call vert the components of the force and moment in one coordinate system to components in another system ill accordance with the rules of analytical geometry. Particularly, the components of the aerodynamicforce and moment in a body axis system arc converted to the drag force and the rolling moment, respectively, in a night path system of coordinates by the formulas
X .. = X cos (~a)
+ Y cos (;;:8) ---:- z cos (~8)
1I1:<:a = M;J; cos (x"?a) T .My cos (Y;a)
+
(1.2.3)
M, cos (;;-8) (1.2.3')
where cos (;:;.9), cos (Y;a), cos (i?a) Bfe the cosines of the angles between tlle axis OXa and the axes Ox, Oy, anil Oz, respectively. The expressions for the other components of the force vector, and also for the components of tho moment vector, are written in a similar way. The values of the direction cosines used for converting forces and moments from ono coordinate system to another arc given in Tahle 1.2.1. Table 1.2.1 FUglit path sHtl'U1 Dodyaxis ~)"'t~rn
u·'·a
0,.
0,.
Ox 0,
ces et cos ~ -sinacos~
sin a
-cosasin IS BinaslnjS
0,
sin IS
0
""~
ros.
Ch. 1. Basie Information from Aerodynamies
In ilcrordallce willt the datil of Taule (1.2.3') acquire the following form:
1.~.1,
Xa ....., X cos a. cos ~ - Y Sill a cos 0: cos B - .1I y sin a cos
.1/;1'8 -= ;\l~ cos
~
-X a -' -X cos
0:
cos ~ -
}" sill
0:
-- Z sill ~ .11, sill
~ .c.
For example. for the motion of the aircrnfl Eq. (1.2.4.) yields. with the relevant signs: co.s
41
fq,s. (1.2.:{) ,lilll
.~hOWll
(1.2.4) ~
(1.2.!1')
ill Fig. 1.2.1,
Il .:- Z sin fl
The force and moment componellts* arc cou\'cl'ted ill a similar way from a flight path to a Doll y a"is coordinate sy~tem. For example. by using the data of Tnblp; 1.2.1, we obtain the following cOllversion formulas for thE) longitlldinal force and the rolling moment: X --' Xacosa.cos~ .. Y~sino:-Zaeo!-\o:~illl~ .1/,,,
:-=:
.1!;.:~ cos (.( cos ~
.1/ Y;, sill a.
(1.1.5)
.l! l;, COS':L sin ~ (1.2.5')
We call go O\'er from II local geographical coordinate system (a normal system) to ;J body axis or flight path one, or vicc \'ersa, if we know the cosinE'S of the angles he tween the corresponding axes. Their vailit's can be delermined from Fig. 1.2.2 that shows t-he mutunl arrangemellt of the axes of thel$e coorriinnl(' sysl(!m~.
1.1. Determination of Aerodynamic Forces and Moments According to the Known Distribution of the Pressure and Shear Sh'ess. Aerodynamic Coefficients Aerodynamic forces lind Moment5 and Their CaeHicients
Assume that for a certnin angle of attack lind side;,lip IIllgle, and also for given parameters of the free stream (tile speed V 00, si(ltic pressure p"", density Poc:, and temperature T ...). we know the distribution of the pressure p and shear stress 't ov~r thp ~lIrfBCc of the body in the flow. We want to determine the resultant values of the aerodynamic forces aod moments. The isolated surface elemt'nl dS of the body experiences 1I normal force produced by the exc(>!';s pressure (p - P«» dS and the tangential • Wc sllall omit the won] ·,.omponents" Jx.low for bre\'ity,
it and use fonnulo3 for scalar quantities.
hu~
shall mean
Ag.U.i ActioD of pftMurC and friction
!sr~~ar) forces on an elementary
force 't dS. The sum of the projeetions of these forces onto the x·axis ·of a wind «(]jgnt path) coordinate system is (Fig. 1.3.1) /\
/\
r(p- p",,) cos (n,x a) + "tcos (t,xo») dS (1.3 .1) where n aud t are a normal and a tangent to the olement of area, respectively. The other two projections onto the axes Ya and z. are obtained by a sirnilar formula with the corre.sponding cosines. To find the resultant forces, we have to integrate expression (1.3.1) over the entire !.urface S . Introducing inlo lhe.sc equations the pressure cOPUlcicnt p = (p - p"",)/q"" and the local friction 'actor Cj .:t = 'tlq "", where q"" = p ... V!. 12 is the velocitv head. we obtain formulas for the drag force, the IiCl force, and the side force, respectively:
x .. =
q"",Sr
~ (S)
"
y~ ,.."qocSr)
fp cos (~,~.H-c,. :tcos
(t.i.)JdSIS r /\ r-pcos(n,y.)+c,.xcos(t,Ya)]dSISr
(1.3.3)
1fPCOS (;;,~.)+c, . %cos(r.i.)ldSISr
(1.3.4)
/\
(1.3 .2)
IS)
Za = -q"",Sr
lSi
We can choose a random surface area such as that of It wing in plan view or the area of the largest cross section (the mid·section) of the fuselnge as the charactoristic area S t in these formulas. The intograls in formulas (1.3.2)-(1.3.4) are dimensionless quantities taking into account how the aerodynamic forces are aHected by the nature of the flow over a body of a given goometric configuration and by the distribution of the dimensionless coefficients of pressure and jfriction due to lhis flow . . In formulas (1.3.2) for the force X . , the dimensionless quantity ,lS usually designated by c Xa and is known as the drag coeffaclent.
Ch. 1. gasic Information from Aerodynamics
43
III the othl'r two fOI'lnulos, the corresponding symbols cUa and c'a <1m introduced. The l'",le'-onl quanti lies Me known as the aerodynamic lift coefficient and thE' aerodynamic side-force coefficient. \;Vith ~I view to the obo\'e. we han'
1'8
Xli = c"/lqooSr>
=
cllaq",ST>
Za """ cZ/lqooSr
(1.3.5)
We can oLlain generol relations for tI[(~ moments in the same way 8~ formulas (1.3.2)-(1.3.4) for the forces. Let liS consider as an example slI('h a r('iation for !lu' pitching mom(,nt ,1[ •. It is evident that the e]('mentllry vallie of this moment d;\/zll is d~t.ermined by the sum of thE' morrwnLs about the axis ZII of the forclls acting on an area dS in i\ plane al right allg'ks 10 the nxi~ ZII' If the coordinat.es of t.he Ilrea dS arc Ya
-
/'
/''\
q"",Sr {(p cos In,Ya)-cr.x cos (t,Ya)lxa
/' -!p cos (n,x a) -;- Cr.", cos (t,x -
/'\
iI )
Ya} dSIS r
Integrating this cxpression over the surface S and introducing the dimensionless pOrilt\lptE'l' m!a .."
q~;;L
=
.~
{If cos (n,~,J
-Cr.:.: cos (~,~a)J
XII
<',
-iPcOs(~:~a)---: c,.%cos((,~)JYa} :;~
(l.:i.u)
in which L is a characteristic geometric length, we obtain a formula for the pitching moment: (1.3.7)
Tile parameter 1nza is called the aerodynamic pitching-moment coefrlcient. The formulas for the other components of t.he moment art' written similarly:
,lira =
1n~·aqooSrL
and
MYa = m!laq",SrL
(1.:3.1,)
The dimensionless parameters mX~1 and mila arc call.,d the aerody118rnic rolling-moment and yawing-moment cocffieieuts, respectively. Thp. relevant Mefticients of the aerodynamic forces and moments call nlso he introduced in a body axis coordinate system. The use of these coefficients allows the forces and moments to be written as follows; X c:Jt.g""Sr, Mx = mAooSrL
Y = c"qooSr, Z = cz<jooSr,
Mil = mlllooSrL A-fz = m.q ...SrL
(\.3.9)
44
pt. I. Theory. Aerodynamics of lin Airfoil and a Wing
The quantities cx, cy , and Cz are called the aerodynamic longitu. dinal·force, normal·force, and lateral·force coefficients, and the parameters m x' my. and tr/z-the aerodynamic body axis rolling· moment, yawing·moment, and pitching·moment coefficients, respectively. An analysis of the expressions for the aerodynamic forces (1.3.2)· (1.3.4) allows us to arrive at the conclusion that each of these forces can be resolved into a component due to the pressure and a component associated with the shear stresses appearing upon the motion of a viscous fluid. For example, the drag X II =- X a.p -I- X a.,. where X a . p is the pressurl;' drag and X a.f is the frit'tion drag. Accordingly. the overall coefftcient of drag equals the slim of the coefJicients of
pr~~:~~a~~'~ ~~:;t:oe~o~~~:~;;al:;/'~~'d -~i~~:'t~'rce
coefftcients, and also the moments, can be represented as the slim of two components. The forces, moments, and their coefficients are written in the same way in a body axes. For example, the longitudinal-force coefficient C x "- cx. p •.• C x .', where c x . p and X".r arc the coefftcients of the longitudinal forces due to pressure and friction, respectively. The components of the aerodynamic forces and moments depending on friction arc not always the same as those depending on the pressure as regards their order of magnitude. Investigations show that the inflnence of friction is more appreciable for flow o\'er long and thin bodies. In practice, it is good to take this influence into account mainly when determining the drag or longitudinal force. When a surface in a flow has a plane area at its tail part (a hottom cnt of the fuselage or a blunt trailing edge of a wing), the pressure drag is llsllally divided into two more components, namely, thepressure drag on a side surface (the nose drag), and the drag dlle to the pressure on the base cul or section (the bast' drag). Hence, the overall drag and the relevant aerodynamic coefficient arc
Xa, = Xa,n -+- Xa.b -!. X a"
and
cx~ =
c"a.1I
~
C:tn.b
-+-
c:t n,'
When determining the longitudinal force and its coefficient, wrobtain
X = Xn + X b -I- Xl and Cx = Cx ... In accordance with Fig. 1.3.1, we have Xb= -qot>
JF;,dS
s,
b
and
+ C:t.b ..:... C:t.f
C.~.b ~ q:~r
where Pb = (P.b - poo)/qot> (this quantity is negative because a rarefaction appears after a bottom cut, i.e. Pb < pool. Characteristic Geometric DimE'nsions. The absolute value of an aerodynamic coefficient. which is arbitrary to a certain extent.
Ch. 1. Basic Information from Aerodynamics
:~~e~!':ic I,.-c~ntr"
45
vicw of a wing:
chord, bt-till lind b-Iocal chord
chord.
depends on the choice of the characterigtic geometric dimensions S rand L. To facilitat.e practical calculations. however, a characterb~ tic geometric quantity is dlOsen beforehand. In aerogpace technology, the area of the mid-section (the largest cross section) of the body Sr = SlIllri is usually chosen ag the characteristic area, and the length of the rocket is taken as the characteristic linear dimension L. In aerodynamic calculations of aircraft., t.he wing plan area S r .= = Sw, tlle wing span l (the distance between tlw wing tips) or the wing chord b arc adopted as the clHlracteristic (Iim('nsions. By the chord of a wing is meant a length equa! ") the distance between the farthest points of an airfoil (section). For n wing with a rectangular pianform, the chord equals the width of the wing. In practice, a wing usually has a chord varying along its span. Either the mean gPOmc:tric chord b = b m equal to b m ~ Swfl or the ml'an aerodynamic chord b = b A is taken as the characteristic tlimengioll for such a wing. The mean aerodynamic chorl] is determined ag the chord of the airfoil of an equivalent rectangular wing for which with an identical wing plan area the moment aerodynamic characleristics are approximately the same as of tlw given wing. The length of tile mean aerodynamic chord and the coordinate of its leading edge are determined as followg (Fig. 1.:l.2):
bA=~ Sw
1/2
I b'l.dz, u
When calculating forces and moments according to known aerodynamic coefficients, the geometric dimensions must be used for which these coefficients were evnluated. Should guch calculations have to be performed for other geometric dimensions, the aerodynamic coefficients must be preliminarily converted to the relevant geometric dimension. For this purpose, one mnst use the relations C1Sl = C 2 S 2 (for the force coeflicients). and TnISIL I = Tn 2S 2 {'2 (for the moment coefficients) obtained from tilC conditions of th(' constancy
46
Pt. I. Theory. Aerodynemics 01 en Air/oil end eWing
Fill. t.3.l Constructing 8 "polar of the first kind of a craft: a-e'l'a vs. a: /.I-Cyll. vs. a; c-polsr ot hrst kind
of the forces and moments acting on the same craft. These relations are used to fmd the coefficients C 2 and m 2 , respectively, converted to the new characteristic dimensions 8 2 and L 2: c2
=
c1
(8 1 /8 2 ),
m 2
=
ml
(8 1 L I /8 2L 2)
where the pr~vious dimensions 8 1 , L] and aerodynamic coefficients c1 ' ml , as well as the new dimensions 8 2 , L2 are known. Polar of a Craft A very important aerodynamic characteristic
~h:;af~~i \.f:~;atl~~~~s~r~:I~tf:~ ~~t=Zea~ t~e ~?f~~~da~r~~e f:r~:: o~~ which is the same, between the lift and drag coefficients in a flight path coordinate system. This curve, called a polar or the first kind (Fig. 1.3.3c) is the locus of the tips of the resultant aerodynamic force vectors Ro acting on a craft at various angles of attack lor of the vectors of the coefficient ella of this force determined in accordance with the relation en~ = RII./(S rq (m)1. A polar of the first kind is constructed with the aid of graphs of C!f A versus ct and cVa versus ct so that the values of C!fa and cJlll Me laid off along the axes of abscissas and ordinates, respecth·ely. The relevant angle of attack ct, which is a parameter of the polar in the given case, is written at each point of the curye. A polar of the first kind is convenient for practical use because it allows one to readily flDd for any angle of attack such a very important aerodynamic characteristic of a craft as its lift-to-drag ratio (1.3.10)
(or Y a) and C!fa (or X.) are the same, the quantity K equals the slope of a vector drawn from the origin of Coor(JiIf the scales of
eVA
Ch. I. Basic Information from Aerodynamics
47'
.'
~ "
'.
o
Flg.UA
'x
))rag polar or the second kind
nates (the pole) to the point of the polar diagram corresponding to the· chosen angle of attack. We can usc a polar to determine tbe optimal angle of attack "oPt corresponding to the maximum lift·lo-drag ratio: K maz = tan "oPt
(CUO')·
if we draw a tangent to the polar from the origin of coordinates. The characteristic poinls of a polar include the point cllamaz corresponding to the maximum Jifl force th.d is achieved nl. tIle critical angle of attack Ct cT ' We can mark n point on the- curve determining the minimllm drag coefficient cX.amlll and tht' corresponding values of the angle of all lick and the lift coeflicient. A polar is symroetl'k llbout the axis of ah::cissas if a craft has· horizontal symmetry. For .such n craft, the vattIC of ex min ('orresponds to a zero lift force, c lla = O. ' .• In addition to a polar of the flrst kind. a polar or the second kind is sometimes used. It dWers in that it is plolLecl iii a bo(ly axis coordinate system along \.... bose axis of abscissas the \"alues of the longitudinal-force coefficient COl' arc laid off. and along the axis of ordinates-the normal-force coefficients cy (Fig. Ut4). This curv6 is used, particularly, in the strength analysis of craft. Theoretical anel experimental investigation~ show that in the most general case, the aerodynamic coefficients depclul ror a given body configuration and angle of attack on dimensionless variables such as the Mach number Moo = V oolaoo and the Rernolds number Beoo = V ooLPoo/p,oo. In these expressions, a_ is the speed of sound in the oncoming Dow, poo and ~oo arc the density and dynamic viscosity of the gas, respectively, and L is the length of the body. Hence. a multitude of polar curves exists for each gh'cn craft. For example, for a definite number Be oo • we can construct a family of such curves ench of which corrC!>ponds to a definit.e value of the
48
PI. I. Theory. Aerodynamics of an Airfoil arKt a Wing
Fig. U.S Determination of the centre of pressure (a) and aerodynamic centre (b)
velocity Moo. The curves in Figs. 1.3.3 and 1.3.4 corrospond to a fixed value of Re"", and determine the relation between cYa and cra. for low-speed nights (of the order of 100 mis) when the aerodynamic coeflicients do not depend on Moo. Centre of Pressure and Aerodynamic Centre. The centre of pressure (CP) of a craft is the point through which the resultant of thc aerodynamic forces passes. The centre of pressure is a conditional poinL because actually the action of fluid results not in a concentrated force, but in forces distributed over the surface of the moving body. It is customarily assumed that for symmetric bodies or ones close to them this conditional point is on one of the following axesthe longitudinal axis of the craft passing through the centre of mass, the axis of symmetry of a body of revolution, or on the chord of an airfoil. Accordingly, the longitudinal force X is arranged along this axis, while the centre of pressure when motion occurs in the pitching plane is considered as the point of application of the normal force Y. The position of this centre of pressure is usually determined by the coordinate xp moasured from the front point 011 the contour of the body in a [Jow. If the pitching moment ill: about this point and the normal force Yare known (Fig. 1.3.5a), tho coordinate of the centre of pressure (1.3.11) A moment Mz tending to reduce the angle of attack is considered to be negative (Fig. 1.3.5a); hence the coordinate xp is positive. Taking into account that
M:=mzqooSrb we obLain
and
Y=cyqooSr
Ch, 1. Buic In/ormation from Aerodynamics
49
wheuce
(LUI') The dimensionless quantity Cp dermed as the ratio belwcell the distance to the centre of pressure and the characLerisLic length of a body (in the gi ....en case the wing chord b) is called tlte centre-ofpl'e~ure coefficient. With small angles of attack, wlten the lift and nOl'mal-force coefficients are approximately equal (c Ya ~ c y ), we have (1,3.12)
In the case heing considered of a two-dimensional fio\\' past a body, Lh(' pitcldng-molllent cocfflcieuts in wint.l (Right path) and body axis coordinate sy.<.;Lcms are the same, i.e, ntza moll. Fol' a symmetric airfoil whclI at 0: _ 0 1110 quantHies C,I ant.l m z simliitatll'ollsly takE:> all zero Y[llnes in accordance with the expressions
=
Cy
= (iJc,/oa) a,
lnz
= (ilmz.:aa.)
IX
holding at small angles of attack (llCl'e the (leri\'llti\'es (jcl/.'O(l. and omz'da are COJlstant qU
f.'p
and of the dillll'lI~i()1l1l''''~ cnordinflle I, can 1)(' II~pd to {Iplc'rmilll' the tllis c<,rlll'\'·
:1'(')[
id)O'l1
11,;:1.14.) In\'esligation.~ ~how
Lllat in real c()t1diLiollS of nuw, :\11 appreciable di~pla('eUlent of the centre of preSSllfC (:an he oh.~{'l'yc{1 in craft even upon a .slight change in lhe angle~ of atlnd, Thi~ i~ c::;peciuJly lIoliceable in craft with all asymmetric conflgllration or upon deflection of an elevator, which di::;l.urbs the existing symmetry, III these ronditions, the celltl'e of pressure is not cOllvenient for llSC' as a cha1'acteristic point in e"timat.ing the position of Lhe resultant of the Ilcrod.\'1l8mic foree" and the tlppearin~ pitching Illoment ahoul the ('entre of lIlass. In these condHions, it is more cOll\'enient to a.o:;sess the flight pl'operl.ie,~ of a ('raft a(:cording to the aerodynamic c('JllI'e location. To reveal the meaning of this concept, let U~ t'oll.!'idcr an asymmetric airfoil and e,·[lluate the moment J/zl! about all al'hill'iU'r poillt F" with thl' coordinate :t'Jl 011 \.Iw (",hortl of the airfoil, II follows from Fig', '1,3,50 HUtt
C.
1.1
.... ..., 1.0
~~a!i~ between the moment coefficient mz and normal·foree coefficient ell lor an uymmet· ric aircraft
-or
0.2 mz
or, since -Yxp = ..lf z is the moment ahout the front point 0, we have Jl zlI = YX n + Mz Going over to aerodynamic coefficients, we obtain mzn
= cil (x,/b) + m%
(1.3.15)
For small angles of attack when there is a linear dependence of m; on cil of the form m~ = m;()";' (iJmzIiJc Il ) Cv (t.3.16) we obtain
or (1.3.17)
where m t () is the coeffident of the moment about a point
011
the
leT'~~g s:~~da~~':n=i~ ~~~f:1~j3:~termines
the increment of the moment associated with a change in the normal-force coefficient. If we choose the point Fa, on a chord whose coordinate XII = XF. is determined by the condition (see Fig. 1.3.5b) (1.3.18)
the coefficient of the moment abont this point will not depend on
1ti;i:n;~i'::t ~~l ~:~:~I)t~~J!~~:~~:~~~; ::n7~1 (~C) ~?~~~a;[V~~ab~iJ;: The aerodynamic centre is evidently the point of application of the additional normal foree pl'oduced by the angle of attack (the coef. licient of this for(,e is (Oc,/OCt.) a = cial. The pitching moment about an axis paS$ing through this pOint does not depond on the
e1
Ch. 1. Basic Information from Aerodynamics
augle of attack. Sudl a point is called the angle~of~attackaerodynaDlie ('Clltre or a craft. 'rhe centre of pressure and the aerodynamic centre are reIn ted by the expression Cp
=
m, -c;-=
(1.3.19)
~~~d~l'co~r;eq~e;t'ii.U/~h·e ~:I~tr~ ~r~:~!~irCe ~~~I:~d:t~~~tlt '71~~ ;;r~ dynamic centre. Expression (1.3.19) holds, however, for a spl101etric configuration provided with all elevator deflected through a certain angle Be (see Figs. 1.2.2 and 1.:-:1.5). In this case, the moment coefficient is
(1.:1.20) and the
normal~rol'ce
coefficient is (1.3.11)
where
mf _.
i)m~.:a':./.,
c~;,...: iJc/rJa,
m~~'
umz:dB".
illlll
c!e
=
'-- (lcy,'dB".
If a conJigHl"Mioll
j:; III 1
liot
~rllll\lell"ic,
= m:o -:-
ell = cyo
lhen
1n1a ...!... m~f"Be c:a·: c~e(\
(1.:J.22) (1.3.23)
The point of applicalioll of th(> normal force dm' to the cle ...·atol defit;'('U(}TI angle nud proportional to this angle is known as the t'lcutor·dt'11ection aerodynamic ct'ntre. The moment of the forces about. a lateral axi~ pa~sing through lhis centre is e"ideutly indL"pendent of the anglo be' In the geJleral ense for nil o:-;ymmelric. configuration, its centre of pressure coincides with none of the aero~ d~·lul.mic centres (based on a or B,,). In a particular case, in a sym~ melrk crnft at« --; D. the centre of pressure coincides with the aerod~"Tlamir centre based on Bel'silll! the definition of the aerodynamic centres hased 011 the angle of attack "lid the elevator deflection angle and introducing the corr~pollding coordinates XFa, and X"'II' we find the ("oelflciellt of the moment ahout tile centre of mass. This ('oemcienl is evaluated by formula (L:l.22), iu which (1.3.24)
where XF~ = x,/b and X-;6 = the aerodynamic centres.
xF/b
are the relathe coordinat~s- of
1!i2
PI. I. 1I':eory.
Aeroclyn~mics
of en Airfoil end I Wing
t .... Static Equilibrium and Static Stability CDncept of Equilibrium and Sfilllility
The stllte of static equilibrium is determined by the flight conditiol\;:; and the corresponding force action at which the overall aerodynllluic moment abont the centre of mass in the absence of rotation and wHh the angle of attack and the sideslip angle remaining constant is zel'O (l'1 = 0). Such equilibrium l·orresponds to conditions of steady rectilinear motion of a craft, when the pal·amete!·s of this motion do IIOt depend 011 the time. It is eddelll that for axisymmetric configuration!' over ,,"!lich the fluid 110w:; in Ok longitudinal dil'ection, the equality)1 .= 0 is arhie\·ed ,vith IIlIIlenel'te« ",le,'ators and l'ndder~ and wilh zero angles of aLta.k and :::id~1ip, Hence, ill this rnse, equilibrium, ('allcrllhe trim equilibrium of a craH. sets in tlt the balanc(' angl(' of aHat:k and sidt·slip angle (cz.blll' ~blll) equal to zero. The Med to halallce Ilight al olher angles (0: =F ablll aud ~ =F ~I>III) requil'es the cOl'respouding turniug of Ihe elevators. EquilibriullI of a. craft (pal'licuhu'ly. with the elevators I'xed in place) may be stahle or Ilnstable, Equilibrium of a Cl'I1ft b; considel'ed t.o he stable if arter lhe introduction of a random r;horl-Lil11c (I\."turhance it retnrns to its initial pO!'lition. If these distlll'bnlLcc" cal1se it to delleI'I ~lill more from the initial position, eqllilibrium is said to he l.IIIstabl('. The nature of t.he equilihrium of a. cl'aft is dctcnnirwtl hy its static stability Ol' instability. To rcycal the l'sscnce of slat ic stahility, we call consider the Oow of ail' in a wiud tunnel PMt a c.raft lixed at its ceutre of lDass and capable of turning abottt it (Fig, 1.4.1). For a given elevator angle 6"" a definite value of the aerodynamic nloment :1l z ('orresponds to each value of lhe ungle or del1ectioll of tile craft Ct (the angle o[ attnck). A pos."ible relation between Ct and :Ifz for n c,erLAi .. angle 6(" is showu iu Fig. 1.4.1, where points 1, 2, and 3 det('rmiuillg t.he balance angles Ctlbtll, Ct~bal' and a;3b~1 at which tile aerodynamic moment equal:,; zero correspond to the equilibrium positiolls. The ligure also ~hows two other moment clIr,'cs for the Gh~','alor angle." li,: and a;. Let HS ,'oll."idor equilibrium at floint 1. If UtC craft is de\'iated tbrough nil 1111~k' smaller or lat'ger than a.'bal' the iuduced momenls, positivI) 01' !Lpnnth'c, l'esp('!~th·ol:t', will result ill tm incrensc (reduction) of this mlg-Io to it.;; previolls ,'aille alh,al' i.e. these momonts nrc stahilizin.e; on('.-=. Con~equ(lntlr, lhe po~itioll of eflniliill'ium at point 1 is stahle (thf' ('!':lft is sLatknlly sllIhle). It can he showll !'imilarly that sneh a po~itit)1I of stable c'luilibduln also (,_ol'l'esponds to point 3.
Ch. 1. Basic Information from Aerodynamics
1!13
Fig. tA.t
Dependence of the aerodynamic
~~~~~n~ ~y ~ :'~ftt~~d~~e u~~
rRection of the eh!\'alorl:l lie:
2, 3, ,,_.polnls or inlel'lH't'tion of
In the Ii!':.;t C!l!"C, fl'cc rot.ation or lhc l'l'nh rOlltilllle:o; IIlIlil it o('('lIpie:; the clplilibrimn po~i!ioli ai, point 1. /In(1 ill the sel'ontl ('il~e-at poiut 3, At. point 2 (a 21'nl) the equilihrium i:-; lJIlst;lbll'. I IHlcpd , eXiuuiu· .. Iloll of I"ig, 1..1.1 1'(>\'(,.. 1:-1 Ihnl at nllues of 111(' angle a lilrgN or slllall(,I' than ct 2La l' momellt~ al'(' iJ1(illl't,tI, po~iti\'e or negatiH'. re· sperlively. thai lend 10 illrl'l~fl~e (01' I'et!ul'e) ct. IIt'nre. these mmnents
arC' destabilizing, flnd tlw !'raft. will be
~Iilti<'ally 1l1l~lablc.
Slatil' sl,allilily is I
arc ill\"C~tignle(ilhnl (JC('l1r ill it~ plilnl' of l'yJILJIlf!II'Y in I lit' I\h::;(>II("C' of roll flIHI I'lip. Wltt'll Ilnill~'I<[ng ~Ialk lateral !«lIhilily. the (I il'tIII'IIed Jllo\"('menl~ of 1\ ('I'afl are (,oH~i. Ilirniug of the rmldel'!' .. 1:<0 lead!' 10 rolling" Ther('hu'f' illVel'ti:!atioll fir 1"!I'ral lltllhility i~ Ill"l'orintC'd wilh lIlI alllllpci~ of holh I'olling- allli yawing fIloment:.;:. stallc Longitudinal Stability
"'hell I'llI'll !'tllbilily ('~i~tl", an illtillcc(l longitudinal 1110111('111 abont the cell Ire of ma!':!' will bp stahilizing, In this ('II!'(', the dir('c· tion of thc ('hlluge in tllC' mOlllent .11 z (nnd accordingly of the coef· licienl rtl z) is oppol"it(' to the change ill the angle Ct. Consl'C]ucntl:,!'. tho (':ondition III slnlic 10llgitudillll) stability clm hl' expre~..ed ill llnordan('(' wilh OIl(' of the Cllr\'c!o= !;hoWIi in Fig, 1.4.1 hr ihc in(,lpllIlilies (}JJ z."urx < (J ot' c)m:.'orx :-. m~ < 0 (thc derl\'l\th"(':< arc e\'lllml\(·(1 for thc halnllcc 1I11g1e of nllack r.t. ."" ct:!'al)'
84
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
Flg.l.U
Action of a force and moment in analysing the static stability of a craft: .-stallc stabilil)': b-stalic Instability: c-neutrlliity relath'c to statiC stability
With static longitudinal instability. fI dcstlluilizing (tilting) moment appears that tends to increase the nllgle of attack in comparison with its balance \'nlue. Conscqm'ully. tbe inequalities aM zlfkt. > 0 or mf > 0 arc the rondilion lor 8tati<' longiludillal instability. A rraft will be neutral relati .... e to statir longitudinal stability if upon n small deflection from the balanrc angle of attack neither a stabilizing nor a destahili:dng momen! il'! illdllret!' This angle of atta('k IX =- ahal in Fig. 1.4.1 rorre!<polld~ to point 4 at whirh the moment r.urvt' is tangent to the borilontal axis. It is obviow; that hert' the coefficient of the restoring moment Am~ -,. ma. 61X --: O. ,a.lu.1 Criteria of Static Stability. The derivative mf' on which the magnitude of the !'!tabilizing or destabilizing momrml dependli i~ clllled the coeflleieut (degree) of static longitudinal stability_ This ~labil ity criterion reilltes to configurationl'! both with and wHhout axial symmetry. For axisymmetric craft. we can assume that the criterion of static stability equals the difference betweell tbe distances from the no~e of a crart to its ('entre of mass and centre of preSSlll'e. Le. the quantity y = ,l·(.M - xp. or in the dimensionless form Y = XCli/b - xplb = XC~1 - c p If the coefficient of the centre of pre~re cp is larger than the l'elative coordinate of the centre of mass XCM' Le_ if the centre of pressure is behind the centre of ma~s, the craft is statically stable; when the centre of pressure is ill front (the difference XCli - ep is positive), the craft is statically unstahle; when both centrcl'l coindde, the craft. is neutral. The action of the relevant pitching moments abOllt a lateral In:1S passing through the cenlre of IlHlSS is shown ill Fig. 1.4.2.
e.!!I
Ch. 1. Basic Information from Aerodynllmics
The criterion Y = XcM - cp determines the margin of static stabilit~·. It may be negative (static stability). positive (static instability). and zero (neutrality relative to longitudinal stability). The quantity Y is determined by the formula Y = mzlclI in which the pitching moment coefficient is eyaluated about the centre of mass. For small \'alues of ct, the coefficients m z and cy can be written in the form Tn: = mr;ct, and cy = ~ct. With this taken into account, wc haye Y = mr;/i{ = amzlacy = XCII - Cp (1.4.1) Hence it follow~ that the deriyati\'c amzlac" = m~1I may be consiripred as n criterion determining the qualitative and quantitative characteristics of longitudinal stability. If m~1I < 0, we have static longitudinal stability, if m~" > 0, we baye instability, and if ~~.4 = O. we have neutrality. The parameter m~Y is also called the coefficient (degree) of static longitudinal stabilit~·. To appraise the static stability of asymmetric craft or of .:;ym~ metric craft with denerted elevators, t.he concept of the aerodynamic centre is used. The dimensionless coordinate of this point with respect to the angle of attack is determined by the formula IF", = Taking this into accouut and assuming in (1.3.17) that = the quantity .1.'11 equals thE' coonJinatc :reM of the centre of mass, we obtain
-In>.
mz = Inzo -
CII
(XFa -
XeM)
Differ(,lltiation of this expres.<;ioll with respect to m~Y = -(IF", - XCM)
CII
yields (1.4.2)
Accordingly, the longitudinal stability is determined by the mutual arrangement of the aerodynamic centre and centre of mass of a craft. When the aerodynamic centre is behind the centre of mass (the dif(er~ ence XFa. - XC!! is positive), the relevant craft is statically stable, and when it is in front of the centre of mass (the difference "iF,." - XCM is negative), the craft is unstable. By correspondingly choosing the centre of mass (or by centering), we can ensure the required margin of static stability. Centering (trimming) is central if the centre of mass coincides with the aero~ dynamic centre of the craft. When the centering is changed, the degree of longitudinal stability is m~1I = (m>),
+ Xell - XCl'tI
(1.4.2')
where the primed parameters correspond to the previous centering of the craft.
156
PI. I. Theory. Aerodynamics of an Airfoil and a Wing
Inlluence of Elevator Deflection. Investigations show (:-,ee Fig. 1.4.1) that when the moment curw Mz = t (0:) is not linear. its slope at the points of intersection with the horizontal axis is not the same for different elevator angles. This indicates a difference in the value of the coefficients of static longitudinal stability. A glance at Fig. 1.4.1, for example, shows thnt Ilpon a certain deflection of the elevator (6r.), stability at low angles of attack (0: ~ O:lba)) may change to instability at larger \"alues of them (0: ~ f.tZ ba1 ) and may be restored at still larger angles (0:. ~ 0:3La')' To avoid this phenomenon, it is necessary to limit the range of the angles of attack to low values at which a linear dependence of t.he pitching moment coefficient on the angle of attack and etentor angle is retained. In this case, the degree of stahility docs not change because at all possible (small) ele\"fl.tor lIugl('s the inclination of the moment r.urve to the axis of abscissas is the ~ame. The condition of a linear nature of the mOlllent chal'ar.teristic makes it possible to use concepts stich as the ael'odynamic. centre or neutral centering (XFo: c-:- x, -'1) whell studying the flying properties of Cfflft. These cOl\~epts lo~e their sense when the linear nature is violated. Longitudinal Balancing. Let us considm' a flight at a uniform speed in a longitudinal plane along a curvilinear trajectory with a constant radius of ~urYatllrc. SHrh a flight is rharacterized by a constant angular velocity Q r ahout a lateral (lxis po~sing through the centre of mass. This velocity can exist provided that the inclination of the trajectory chang.:>s insignificantly. The constancy of the angular velocity is dl1e to the eqllilibriulll of the pitching moments about the lateral axis, i.e. to longitudinal baloncing of the craft at which the ('qnoli!y JI z :...:' 0 hold.", i.e. m: o -:" m~O::bal
+ m~1!6~ -j- m?,Qz =
0
This equation allows us to find the elevator angle needed to balanced flight at the given values of o:.onl and Qz; 6c .bal = -(1Im~.) (mto
+ m~o;Lal
..:..
m;:Q%)
eJ\~ure
(1.4.3)
For conditions of a high static stability, we have m~'Qz -%:: mfa: bal ; consequently 6~.bal = (-1/m~e) (mzo ..!- m~abal) (1.4.3') For a craft with an axisymmetric configuration, mzo = 0, therefore 6~.bal = -(m~/m~e) abal (1.4.4) The normal (lift) force coefficient corresponds to the balance nugle of attack and elevator angle, namely, cV• bal = cvo
+ (c: + c:e6e.bal/abal) abal
(1.4.5)
Ch. I. Basic Information from Aerodynamics
5T
The "alues of c yo are usually very small even for as~'mmetri('. COIJfigurations (at low be and a) and equal zero exaet1~' for craft wilh axial symmetry. lIellce, with a sllffIcient degree of nccurary, we can write
Static Lateral Shlbliity
To analyse the lateral stability of a craft. one mus!. con.<:ider joilltly the nature of the change in the rolling and sideslip angles upun thesinHiltaneou~ llction oj the pertnrbing rolling .lI.,. ami yawing M If momellts. If after the stoppillg of sHch action these allgJes diminish, tending to their inilial vallle~. we il[l\'C .<:tatic lateral stahility. Hence, when in\'cstigaling lateral ~tnbilitr. one lllll~l con.o:;;idel' ~illllll taneollsly the change iu the aerudynamic {'ocrficients mx and my. In most praclical casE'S. however, lateral stability cfln be divitled into t\\"o simpler kinds-rollillg !'hlhility alld dil'c('tiollal stabilityand can be !itudif'd .<:eparatl'ly hy cOIl~iderillg tile challge ill the rele\'ant moment ('o('fJirient~ m~ (1') ilnd my (~). Let liS consider static rolling stability. Assllme lhat ill ~1('Ii(]y motion at the allgJc of al Lack ad the cl"lIft is tllrlH·d IIbout the axis (h through a certain rolling [lliglC! y. This tm'll willi [I ('Oll~tflllt Oril'lltat ion oj the axis 0.1." rel[llh'e to Ille velndlY \"('clor V (,flU'<:(>!' the appearance of the allgie nf attllck a ~ ad r().~ l' and the !-;id('siip angle ~ ;::::; ad sill y. The ! 0, II dislurhing mOIll('lIt i~ rOl"rlll'd. ami
mx
~,~~ttl~c r~~~J~~tg t~srt:m!:~YS~~~~ili/:~,.\\·hl'll m~: ,...
1I. 11](' cmrt
i~
11l'IIt.ral
Since a night lIsually OC('llr": at posilin' [lllgJe~ oj" iltlad.:. the ~igll.~ of the derivativ(>s Ill} lind III~ coincide. li0n('c. ill 1\1I,,1~'~ilJg roJlilig motion. we ran lise the derivativc m~ kno\\'1l as the ('o('rfll'h'nt (df'gred of static rolling stabiJHy. Static directional stability is eiJaractNi1.C'd hy 11 cO('i"fIdl'Il1 (dl'gTl'e) dl;ltermincd by the derivative il.lIy/i.i~ {or (/fII!liuf1 .. 11I~:). If Lhe CIuantity //IV < 0, the craft has statir directional ."tabilitr: al /liD> > o. il has static instability. and when =-- O. l1('utrality. The {'oncept of directional stability is af':soriated with the pl"Opertr of a ('fflet to eliminate an appearing side!'Jip unglll~. .At lhe same time. a crail doe,.: not maintain the .<:taiJili{v of it,.: own night. (li"ectioll hecaJIse afte!' ('hanging its direclion of 'motioll lIll(I('r Ihe artilill of v[lrio\I.<: disturbances. it does not retlll"n to its pn'\"iom'
me
ti8
PI. I. Theory. Aerodynamics 01 an Airfoil and a Wing
direction, hilt like a weathercock. turns with its nose part in the direction of the new vleocity vector V. Similar to the aerodynamic centre based on the angle of attack, we can introduce the concept of the aerodynamic centre based on the Sideslip angle \.... hos,e coordinate we shall designate by IFII' 'L'sing Ihis COllccpt, we can represent the degree of static directional -stahility ill the form m~% = -(IFt! -xc",;) (1.4.7)
x
·where F1l '-' IF/l and IeM = ;I."eMIl (l is a characteristic geometric dimension that can determine the wing span, fuselage length, etc.). Hence, the static directional stability or instability depends on the mutual position of the centre of mass and the aerodynamic centre. A rear arrangement of the aerodynamic centre (IFII > .reM) determines statk directional stability (m;, < 0), while its front arrang&ment (IF11 < XClII) determincs static illstabiliLy (m~: > 0). When the two centres coincide (x FfI = XCM), the craft is neutral as regards static directional Rtability (m~. = 0). A particlliar case of motion of a crafl iu thc plauc of the angle of
~~t~~~~~)na:de sC;at~~f::~:~~d \\~)irh\~:)~l;ct~nt~ ~l~fl:~ll~ ~.r~~c~~;~ ~~~ ~ an automatic pilot (Q x ~ 0). The condition for such steady motion is iat('l'al trim of the craft when Ihe yf\wing: moment vanisli('.'i, i.e.
+
my = myO _L me~ ~ m~fbr 'n~y ~2y = 0 All rraft customarily have longitudinal symmetry, tlwrcfure myo :-:: O. When this condition holds, thl' rqllntion obtaincd allows us to determine the balancc rudder angle b r .= <'lr.bal corresponding to the given values of the balance sideslip angle ~bnl and the angular velocity QIoP Most craft have a sufflcicntly higb degree of static directional stability at which the term m;.!lQ y i:; negligible. Thcrefore br.bal = (-rne/mBr ) ~bal (1.4.8) This relation, like (1.4.4), will be accurate for conditions of rectilinear mol iun. t.5. Features of Gas Flow at High Speeds compresslbllHy 01 ill GillS
One or the important properties of a gas is its compressibility"its ability to change its density under the action of a pressure. All thc proce,,~('s associalcd with the now of a gas are characterized by
Ch, 1. Basic Information from Aerodynamics
lS9
a change in the pr(''';~:llrc Hnd, con."cquenlly, hr the innuence of com· preSo.."surc is nol lilrge, Ilnd the {lITerl of ('ornprc."sihiJity mlly 1m ignorcd, To study 10w-sJlt>ell now o\'er ho(lies, one mar U~C tin' l'qllaliolls of hydro(IYllmni<'." that stlldies the laws of motioll of till illl'omprt>s."ihl(' nllid, In IW/lctite. the innllencc of COIllI}l'l'S."ihility lIlay hc igllnred ill lhe range of air ."pecri." fa'om 11 few nWlro!" por !"(>('on(i 10 JOO-t50 Ill!!", III I'Plll ("ondilioll.", Ihis ('orl'c."pond." 10 :'.ill('h IIlIlIIbt'r1! fa'om M ... -x -'(I x () 10 M '" o.a-!U~) (I'Nl' 1I co' is III(' ."peed or sound in Sill ul\fl[sluriH,d now), Idealizatioll of 11,(' Pl"f)('t'l'." c'om,i."t." in m;."lImilig the :'.Iadl III1111hl'r to l'(pu~1 ZN'O within this !'I'gion of !'p(>l'ds b('('tllIse sm;!]1 distul·bllll("C'." (."OIlllci o."l"illillioll:<) Pl'ojJlIgillt' ill 1111 illcompl'(>g· sible nHil1 at illlillih'l~' hi!!h l'p('('(1 IIncl, ('nll!'l'cllIl'nlly, thc I'atio of the night ~p('t'd to thai of !'ol1lul t('lId!' to 7.t'I'O, :\1odel'Jl ('I";\ft lli!\'t' high llirl'p('('(I:< at whi<"h rIow O,'CI' 11U'1II is ollond('II h~' II cOII.~id('rllh\(' dHllIg(' in Ih(' pr(>.":llIfJPl'allll'e, ]n Iho ('011ditioll!' of a !light III high spf'('(I!'. Olll' Illtl~t Ink(' inln ;lC("Olillt the influ('m'(' of rOlllprl'l'."ihilily" whi!'!1 liMY Ill' WIT :lltlll'!'!' of Ilil!i1-!'Jl('('d ;ll'l'flIl~,t1;l111ic:<,
= \'
,Ill
Heaffng of ill GillS
Th(> ilppr(!ciahICl illtl'PH!l'ti('~ uf iil,' Iliriu ;I('ro(lrIll11lli<" in\'(':l'lllo(IYIlClIlli<' par;-Il11l'I('I'S al'll Iduelit' ("oefftdcmts of IIiI', illid also Oil 1111' phr:-ll'odll'mirnl prm'(>:-H':: Ihal Illll)' lJl'oceed [II il wa~ di~I'I'g'm,.ll'd, 1'",11' \'OQ' high (hYP(lI'soliid ."r(>{'(I.", ho\\,c\'t,I'. thl' fpallll'ps Ilsfd with til(' illnu('n('(1 of high ICIllP(,I'iltllrc~ ('Ollie to Ih(> forefront. ([igh tCIllPPI'illlll'(lS il\lPNll' owing 10 dl'('(li(ll'atioli of the gas Sh'l'lUll w!ll'lI Iho kinetic (>1lt'rJrr of ol'dt'reIlIJWl'atlll'c of till! ol'{h.'I· nf I;JlJ(J I" ('xcitalion of th(' \'ibra· tional It\'l'is I)f till' inlf'I'llill (,Il(lrgr of Ih(> O:.:~·gt'll nud llitl'Og('U mole("ul('s ill th(' I~ir hl"collll'~ noticl'ilhle. AI a Ipmpl'I'llhll'l' of I\bnut. 3(100 K illlli a prl'~~\Il'l' of IO~ Pa, llit' yibrational degrl'l'l' of freedom of the oxrgl'll moit,t"I1I(',«. m'(' f'ompll'tl'\r ('will'd, and further e(e\'atioll of til(' it'llIJlt'l"atlll'(' allow." llil' atolll." to SlIl'TllOlUl1 th(' m'lion
eo
Pt. I. Theory. AerodynlmiCs 01 In Airfoil lind II Wing
of the intramolecular forres. As a re.~tJlt, f01" example, a dilllomic molecule breaks up into two individual atoms. This proress is known as dh_socialion. The latter is attended by J"('combination-the formation of a Ilew molecule when two atoms collide (0 2 ~ 20), This heat-producillg reaction is 8('companied b)' the collision of t.wo I'Itoms with a t.hird partide that carries on part of the released energy and t.hus ensures the creation of a stable molecule. In addition, ('.hemical reactions occur ill the air that result in the appearam',eof a ('ertain amonnt of nitrogen Illonoxide ~O. The latter also di5.<;ociates upon fU1"ther heating with the formation of atomic nitrogen and oxygen by thc eqllations
At 1\ temperAtllre of 5000·GOOO K and a pressure of 10~ Pa, the oxygen moll'culcs dh:sociate almost completely. In addiUon. at such a temp(>ratul"e, the majol' part of the lIitrogen llloieclllt's dissociate with simultaneous l'ecombination of the aloms into molecules. This prore:;:s lollows the (''Iuatioll ;\! • .t 2~. The intensity of dis.<:odation is determined by the dt'q:n"{~ of dis,'!Iocialion. It equals t.lle ratio of the Ilumber of (lir pal'tides broken up in : in their tbermal motion. which explains why it i~ also called thermal ionization. Ionizalion is more intcnsh'e with elevat.ion of the tcmperalure aud is attended by a growth in the cOll{'entratioll of the h'ee ele(:trons. The intensity of this process i$ charac.terized by the degrct' or ioni?ation. II equals the rat.io of the Humber or ionized atoms (rnoleclllc8) to their total number. Investigations ~no\V that nitrogen. for example. is fully ionized
Ch, I. Basic Information from Aerodynami,s
tlll'flllally (l11l! d(,~Tt'(' uf iOIJil_alioli i:-; unily) III II h'lIlp('('I\IlU'C
61
Qf
1'j !JIJO 1\ aud tI pl'~'.'<.'-l' lin' ('1It'I't;y of yilJl';ltioll of the iltOlll" in a TllolN'qh', to tlo Iht' w(ld, llwdC'11 10 O\,C'!'('ome 111(' fol'c(>~ oi intt'raclioll helw(>('u al()nl~ lI[lnll lite di~.'ratll!'(J, Till' atul'c on the spf'ritk IWllt nl il ,'011"(,1111 pl'('_-~II1'1' ("Ill i,\'
the
(01'01111.1
where Ille e,'ipf)llelit 'f'
ill IU1'n. ,11'PC'lIli.-<
r>
for \',11111-:-; o( l' lip 10
:2:!O()-~;)(JtI K, ill
\I-hidl Iltp yihralilillill
of f{,N'Ilolil (lI'C rio,"c 10 III(' ~1;llc ()f :o!lIplctC' ('\('i{ntiOIl, When di~'soril1liol1 :,(>1,; ill, Ihe .'-:pc(ilic Ilrat d('pelHb 1I0t oaly 011
thc Icm"('I'III11n~_ IJlII ;11.~0 on tlu! p.'t':;SIII'C, Thc ~I)f'dfll' hCill~ and thc adi nl'{'(llIul. II \\'n~ l'OIl-
62
Pt. I. Theory. Aerodyn.mics of an Airfoil and a Wing
fl,. S.U Change in the specific heat cp of air at high temperntures
~~an~!'~n
the ratio of th@ specific heats k = cp!cp for air at high temp@ratur@1
sidered to be completed I'll T =, "12000 K and p "-.- 102 Pa. Curves plot.ted according to the data ill 16, 71 and characterizing the change in cp and k at high temperatures are given in Figs. 1.5.2. and '1.5.3 (see (81). The data ohtained indicate Lhat at a temperature up to 2000 K and a pressure of 10:' Pa and above, t.he \'alnes of el' and k - c,'/c,. arc determined by the temperature and do not virtually depend on the pressure. The general trend observed upon a eIJange ill the specific heats and their ratio is such that when the pressure drops and the degrees of dissocialion and ionization. consequently, grow, the value 01 cp increases, and lhat of the ralio k ::.= cT,!c p diminishes, although not monotonically.
Ch. 1. Basic Information from Aerodynamics
63
Fig. U.A Change in 'he dynamic visco-
sity of air at high temperatures Kinetic Coefficients. TJIC procelises of friction and heat transfel' occurring in a \"is<'olis heat-conduct.ing gas depend on kinetic parameters of the gas such as the dynml1ic visrosity ~I and the thermal conducthity A. It Wits I!stablished Ulat in the IlbseliCe of dil'sociation. the eocfJicient It depends ollly on the temperature and i8 clptermined by the formula (1 ..;.2)· in which the expollOm" depcnds all Ute tempel'ilture (sec Fig. 1.5.0. In approximate caklilatiolls, tbe mean value "m ::::::: O.i lIlay be u:;Ied for a sufficiently large temperature iutcl'val; hf'l'c the initial \'aille' of !.I. may be takcn equal to ~~"" . .., 1.in X 10-& Pa·s, which corresponds to l' 00 • - 2R8 K. Formula (1.;j.2) i8 1I1'cd fOl' temperatures Up" to 2000-2500 K. WitJI elevatioll of the temperature, this formula gives apprecialJle crrors. Illve.sligatiolls show that for high temperatures up to 9000 K the dYnamic vil'('osity of nir in ('onditiont'l- of equilibrium dissociation {'an he determined with an 9rC'Lll'acr up t.o10% by Suthcrland's~ formula f,/i'~ ~ (TIT ~)" (1
+ 1111T ~)/(t
-'- 1111T)
(1.5.3)
This formliia yields better results for temperatl1res under 1500 K than (1.5.2). It lUll' been established by accurate ralc-ulations IIlat the dynamic viscosity at high temperatures also depend~ on Iht' pres..<;ure. Figure 1.5.4 presents a graph (sec (81) chara('terizillg tllechange in the coefficient ~t at t.emperatllres up to 12000 K withiu the pressure interval from 109. to 107 Pa. Like the viscosity, tbe thermal ('onductivity at temperature!! up to about 2000I( is indepcndent of the preSl>ure Rnd is determined by the power formula All... ~ (TIT ~). (1.5.4)·
064
Pt. I. Theory. Aerodynemics of an Airfoil and a Wing
Fl§!. U.S
Change !n the thermal conductivity ot air at high temperatures
in which the exponeul x depends, as eM be ~een h'om Fig. '1.5.1, on the temperature. For nPPl'Oximate calculation!!. one may take the OW::Il! \'all1e Xfll l'l}ual to 0.85. and the value of A_ corresponding to T", .c' 261 J": equal to 2;5.2 W:'(III' K). Di!-lsoriating air is characterilert by a depelutcJI(:e of tbe thermal conduclivity 011 the temperature
Ch. t. Bnic Information from Aerodynamict
66
state 01 Air at High Temperatures
Equation of State. Im'e:;UgaLion of the flow of air oycr bodies sho\\'s that the relations of con"entional aerodynamics based on the constancy of the thermodynamic characteristics and of the physicochemical structure are sumciently reliable as long as the air remains comparatively "cold", its specific heats change insignificantly, and, consequenUy, we may apply the thermal equation of slate of a perfect gas p = RpT or p -= RopTI()lm)o (1.5.5) where Rand Ro are the individual and universal (molar) gas constants. respectively (Ro = 8.314 X 1()8 J/(kmol·K}l. p is the density, T is the temperature, and (ftm}o is the mean molar mass of air ha"ing a constant composition. A gas satisfying Eq. (1.5.5) is said to be thermally perfect. The caloric equation of state i ~ pc,,!(pR), which determines a relation for the enthalpy, corresponds to Eq. (1.5.5). If we take into account that cplR = kl(k - 1), we h8"e i ~ [k/(k - 1)[ pip
(1.5.6)
A gas whose 1:Itate is determined by Eq. (1.5.6) corresponding to the condition at which Cp and cl> arc constant and independent of the temperature is said to be calorically perfect. It must be taken into account that the need to consider the change in the specific heats with the temperature sets in before the Ileed to use an equation of stale differing from that for a perfect gas. For example. calculations show thnt the rhange in the specific heats with the temperature when passing through a normal shock ,..-ave begins from froe-stream Mach numbers of .41. .... = 3-4. At M"" = 6-7, the equation of state for a perfect gas retains its significance, as does the equation for the speed of sound a' - kRT (1.5.7) hecause the composition of the gas heated behind the shock waYe does not change. Hence, in this case a gas is not calorically perfect, but does have the properties of a thermally perfect gas. For a gas with a varying heat capac.ity, the caloric equation of state (1.5.6) gives a large error. The actual relation between the enthalpy and temperature is determined by the function h(T} that is more complicated than (1.5.6). The following equation holds for dissociating air, for which the specific beats aud molar mass are functions of its state: p = (R,/~m) pT (1.5.8) in which IlIII = 1'2. (p, T).
"r---t-:T": Fig. t.S.6
All i-S diagram for dissociating air
FI,. t.S.7 Change in the mean molar mass of air at higb temperatures
The caloric equation of a dissociating gaseous medium also acquires a more complicated nature, and in the general form it can be written as i = (P. T). Such a fluid no longer has the properties of a perfect gas for which the enthalpy depends only OIl the temperature. The thermal and caloric equations of state of a dissociating (real) gas are still more complicated because they additionally take account of the forces of interaction hetween the molecules, and also the proper volume of the latter. These equations are solved by numerical methods with the aid of computers and are usually presented in the form of tables or phase diagrams. The results of calculating the parameters of state of air in conditions of thermodynamic equilibrium for elevated temperatures at pressure intervals from 10 2 to 107 Pa are given in [6, 7]. These results
'3
Ch. 1. B..sie Inlormalion from Aerodynamics
67
of calculations were usell to compile an atlas of phase diagrams (81. The mogt wide::;pread u::;e for thermal calculations in aerodynamics has been found by the i-S diagram (enthalpy-entropy diagram) of dissociating air shown in Fig. 1.5.6. This diagram, which presents the caloric equation of 1';tate in a graphic form. contains curves of p = const (isobars), T -: const (isotherms), and r = const (i50chors-dashed lines). Sometimes the caloric phase diagram depicting i against p with curves of T = con-"t, p = ('onst, and S '-- const may be more convenient for calculations_ Such a diagram constructed according to the data of an i-S one is given in [8]. It graphically depicts the thermal equation of state in different variants. Graphs allowing one to determine the mean molar mass ~lln of dissociated and ionized air (Fig. 1.5.i) and also the speed of souud (Fig. 1.5.8) a~ a function of p IIml T [7] or of i and S [8l are important for practical calculations. Examination of these graphs reveals that the speed of sound changes noticeably \vith the temperature and to a smaller extent depends on the pressure, which is explained by the insigllilicant influence of til{' structure of air on the nature of propagation of small disturbances. At the same time, the change in the structure of air upon its dissociation appreciably affC('t~ the molar mass, which is expressed in the strong influence of the pressure on the value of ,",m. Three characteristic sections on which ~lm diminishes wi til increasing temperature can be noted in Fig. 1.S.i. The first of them is due to dissociation of oxygen, the second-of nitrogen, and the third-to ionization of the components of the air. The general trend of a decrease in the mean molar mass of a gas that dis..,ociates and becomes ionized is due to the breaking up of the molecules into atoms, and also to the detachment of electrons. An increase in the pret'sure leads to more illtenshe recombinations. This results ill a certain growth in 11m. Diatomic Model of Air. A moupl of thr air i;: sometimes u::;ed in aerodynamie inw!;tigalions that is a diatomi<: gas consisting of a mixture of oxygen and nitrogen in accordance wilh their ma!<s composition. Such a mix lure is treated as a single perfect gas if the mixtnre components are inert and no r('acliong occur ])t'tween them. At elevated temperatures, the mixture of gases reacts chemically becnu-"e the diatomic gas dissociates and the atoms formed participate in recombinatiou. The dissociation is assumed to be in equilibrium. This means that in the chemical reaction determined for the pure dissociating diatomic gas by the simplest equation of a binary process (1.5.9)
Fig. 1.5.1
Speed of sound in nir at high temperatures: o-""Irat the umperltUl'e and pfUSwe; /)_""llIIt the entha py and entropy
Ch. I. Basic Information from Aerodynamics
69
the rates of the direct reaction Tn and of the reverse one TR (the fates of dissociation and recombination respecti\'ely) are identical. Investigations of a dissociating flow are connected with the determination of the degree of eqlliliurium dissociation IX. Its value for the diatomic model of the air being considered is given in chemical thermodYIlamics by the expre~sion a'/(1 - a) ~ (Pd/P) exp (-T,iT) (1.5.10) where (1.5.11) Pd and T dare 111(' characleristic density and tcmpt"raturc ror di.!'sociation, respecthely, n ..\ is the number of atoms of the element A in a certain \'olume, and n.\, is the Illlmber of molecules of the gas A2 in the same \'olume. The characteristic temperature Td = Dlk, where D is the dissociation energy of one molecule of A2 and k is a gas constant related to one molecule (the Boltzmann constant). I nyestigations show that for the temperature interval from 1000 to 7000 K. the values of Td and Pd can be as~umed to be approximately ("onstant and equal to Td = 50000 K. Pd - 150 g/cm 3 for oxygen, and to T~ ....., 11;{ uao K. Pd = 130 g/cm 3 for nitrogen. To obtain an equation of state for the gas mixture appearing as a result of dissociation of diatomic molecules. we must use the expressions for determining the pressure P and molar gas constant R for the mixtllfe of gases ami the partial presi'illfe Pi of a component:
P~~P'. ;
R~2:."R;; j
}
(1.5.12)
Pi'-" PITH j = PIT (klm l) = (XlIV) T (klmi)
where Pi is the dem~ity of n compouent. lIli is the mass of an atom or molecule, Ri is the gas constant for a component, and V is the volume of the gas mixture. Equations (1.5.12) are knO\vn under the general name of Dalton's law. 'Ve shall use the subscript A to designate ali atomic compoHent (t = 1) and the subscript l\l-a 1Il0iecuiar one (i = 2). Since the concentration of 11 component i~ Ci _., l'i,'P. where fl is the density of the mixture, then C1 - c..\ - IX '"' PA/p and c~ - C)I P)I/P = =l-ct. If mM 2mA. and the relation hetweell the 1l1aS-"'e~ .l:; of the components Pi and the (IUantities p and ct is established from the expressions fJA ,...- pa .= :t·..,,'V, all(1 1')1 .. - P (1 - ~) = XM/l". from (1.5.12) we obtain an equation of state in the following form: P = pTk Ip,\/(rmA) --!- p)l/(pmA)l -.= 11.:.·(2m.,dJ p1" (1 -j- a) (1.5.13) C
70
Pt. I. Theory. Aerodynllmics 01 lin Airfoil lind II Wing
where "·'(2m,,) is the gas constant for 1 g of tlle component A2. in the mixture; \yhen multiplied by 1 + a, it yields the gas constant R (the inrliYidual gas constant) for 1 g of a mixture of the components A and A',!. whose masses are in the ratio of 0./(1 - a). IL is knmvn from thermodynamics that the mean molar mass of a mixture of gases is (1.5.14) where C; is the mass concentration of a component, Jl; is its molar mass, Rnd the symbol ~ determines the number of moles in the mixture. For a diatomic gas, we have ~ c, = c1 + C2. CA + eM = 1. 0-
Since
CA
.....: a, c),[
Jlm
of
=:
.=
1 - a, and
(cA/IlA
2~A
+ CW').lM)-l
= 11M = (i1m)o, then .-=:"
(j.l-m)o/(l
+ a)
(1.5.15)
This expre.c;sion may be used for tile transformation of the equation ~Latl;' (1.5.8) for a dissociating gas of an arbitrary ('om position: p~ [R,!(~m),[
pT(1 +a)
(1.5.16)
2 Kinematics ofa Fluid
2. i. Approaches to the Kinematic Investigation of a Fluid
Determination of the for('o interaction and heat transfer between .a fluid and a body aboHt which it flows is the main task of aerodynamic invest.igations. The solntioll of this problem is associated with studying of the motion of a fluid near a body. As a result, properties arc found at each point of a flow that determine this molion. Among them arc the "clocilr. pressure, density, and temp('ralure. 'Vith
delinitc prerequisites, this investigation can be reduced to determination of the velocity Ileld. which is a set of yeloC'itics of the fluid particles. i.e. La solution of a kinematic problem . .:-Jcxt the known vC'iocity distribution is Ilsed to determine the other properties,
-I.
(a, b, c, I);
y -
iz
(a. b, c, I);
z.,.., fa (a, b. c, t) (2.1.1)
where a. b, c, and tare t-lw Lagrangian variables. The quantities a. b, and care Yarinhll's determining Ihe pathline.
72
PI. I. Theory. Aerodynlmics of In Airfoil lind II Wing
The components of the velocity vector at eadl point of the pathline equal the partial derivath'c.<; lr"x axlat, Fy --= aylat, and V z = = az.lat, while the components ot the acceleratioIl vee,tor equal the relevant second partial derivatives Wx a~xliJt2, Wy :- iJ2yfat 2, and W z = a2z/iJt 2 • Eulerian Approach
The Eulerian approach is in great.er favour in aerodynamic investigations. Unlike the Lagrangian approach, it fixe:; not a particle of a fluid, but a point of space wit.h the coordinates x, y, and z and studies the change in tile velocity at this point. with time. Hence, tho Eulerian approach consists in expressing the velocities of particles as a function of the time t and the (,_oordinate.., x, y, and z of points ill space, Le. in setting the fIeld of velocities determined by the vector V -: l'xi + Vlfi -J- V:k. wherei, i, and kareunit'\'ectors along the coordinat.e axes, while V;T drldt, Vg ,--: dyldt, and V z = dzldt are the velocity vector ('omponents given in the form of the equations
V.
~
t, (x, y, z, t),
V, ••
t, (x, y, z, t),
V,"
t, (x, y, Z, t)
(2.1.2)
The quantities x, y, Z, and t arc called the Eulerian variables. By solving the simultaneous differential equation!:!
*=/d:C. y, z, t);
-f=f2{:C' y, z,
t);
*=fs{x, u, z,
t)
(2.1.3) we call obtain Aquatiolls of 8 family of pat.hlines in t.he parametric form coinciding with Eqs. (2.1.1) in which a, b. and c are integration const.ants. Hence, in the Eulerian approach, we can go ovor from a description of the kinemat.ics to representat.ion of a now by t.he Lagrangian approach. The roverse problem associated with the transition from the Lagrangian approach [Eq. {2.1.1)1 to the Eulerian one [Eq. (2.1.3)] consists in differontiation ot Eqs. (2_1.1) with respect to time and the following exclusion of tho constants a, b, and c by means of £qs_ (2.1.1). By calculating the total derivat.ive of tlte velocity vector with rospect. to time, we obtain the acceleratioll vector
w= ~~ = ~~ +V~
:: +Vg :: +V% ::
(2.1.4)
Projection of tbo vector W onto tho axos of coordinates yields the components of the resultant acceleration. In tho oxpanded form.
Ch. 2. Kinematic5 of II Fluid
7a
these components are as follows:
Wx =
lJ!:x
-i
ar/l.
WY=at
Wz=
r'
VI<: d~;'t ...:.Vy iJ~~;'t .:.1'~ a~~:\, } aV/I, "1'" . aI'"
V." ii:i""" :-V""""FiJ-:.1'!---;J;"""
(2.1.5)
a::! ; l' x ~~T .;·l'" ~~: '7- V; i1~~!
RclntiOlls (2.1.:") for the U('('(llm'nLions rlll'respond to 1.\ now characteriz(l(l hy a chunge in thc vc)odty at a gin'll point with time and, cOI1~eqncnLIYT by the inf!(lnaJity iJV/iJt '=fo 0. Snch a flow of a nuid is ('alled unsteady. A now of a flllid in whirh t.he velocity and other properties at a givcn point ,Iro independent of t.he Lime(fJV/ol 0) is ('alice! st('ady, Streamlines lind Palhllnes
A.I any in~ln11\' we ('all inHlgine a (',urvc in a now huviug the property !.llnl, cvery pill'tidc of tire nuid Oil it has A ,·elority tangent to the C1\l·\·~. 811('h a C\ll'\'~ is (~tllled a sll·pamlinc. To obtain fI 1'.trcamline, one mnst pro('ced til- fnllnws. Tnh a point, A 1 (Fig. 2.1.1a) ill the flow at. tilt! instant t "" tn and express the velor.ity of tire partide tit this point by tile vettor VI' :'\exl take a point A2 aoja('ent t.o At on the voct.or VI' As:mnl(' that. at. the in!;t.tlnt t = to. the velodty \'ector at this point. is V 2' -"ext consider a point A3 on the vedor V 2. t.he velocity at whirh is del ermined at the sameiUI>t.ant by the vect.or V3' ami f'0 on. Such a construt'liOll yields a broken line consisting of .segments of velocity vectors. By shrill!dng the size of these segments 10 zero alld simlllt.uneollsly in('rcasing their number to infmity. at the limit we obtain a CUI'V(, cliveloping the entire family of velocity vectors. This is exactly n streamline. It is obvious that a defmit.e streamline corre...poudf' to each inst.ant. To obtain an equation of II ~treamlinc. let. liS tal,a advantage of tI property 8c.cording to which at earh point. of this line the (lirec!iOlls of the velodty vee·tor Vane! of the "ector ds =- i cL: -i- j dy k dz. where dx, dy. and dz are the projection:<; of the st.reamlinearc clement. ds. must eoincidf!. Conseqnenllr. the ('ross product ds X V - O. Le.
+
+
I d~
V%
dy :ZI=i(Vzdy.-VydZ)-jWzdX-VxdZ) VII V:z: +k(Vlldx- V:rdy)
~.·O
'74
Pt. I. Theory. Aerodyn.mlc5 of .n Airfoil .nd a WIng
(4J
"1",loU
COD.3truction of a sirl'amline (a) and a stream tube (b): l-stn!amllnps: 2-cnntnllr
Ilence sinre, for example, V, dy - Vy dz = O. and so on, we obtain fI system of differential equations
dx/V.
~
dy/V,
~
d::/V.
(2.1.6)
The soliltion of the problem on determining streamlines thus (;onsists in integration of the system of equations (2.1.6). Each of the integrals of these equations FI (x, y. Z, G1) -= 0 and F'l. (3', y, Z, G2) = 0 is a family of surfaces depending on one of the parameters, C1 or C2• while the intersection of these surfaces forms a family of streamlines. Uuliko streamlines, which aro constructed at a fIxed instant, the
Unlike a ~olid hody whose motioll is determined by its translational motion together with its centre of mass and by rotation about an instantan('ous axis passing through the centre, the motion of a
Y;.~
Y"
,-------' , \I.
~:
VI:
"
v..
V,
"v..
flg.I.U
!\fotiOll of a Duid particle fluid particle is characterized. in addition, by the presence o( strain -changing the shape of the particle. Let us consider a fluid particle in the (orm of all elementary parallelepiped with edges or dx. dy, and th. and analyse the motion of face ABeD (Fig, 2.2.1). Sillce tile coordinates of the \'ertires of the foc.e arc different. the ,"clorities detcrmined at 1\ certain instant t -- to are also different:
V. A - V. (.<, y), V.B ~ V., (x'" rho y) } VyA ~ Vy (x, y). VyB :"": VII (x .1-. dr. y) V."l:C .= V;r (x -t- dr, y , dy). l'.rD = V;,; (x. y --;. dy) V vc = Vy (x, dJ.·. y,dy). "V YD = Vy(x, y,dy)
(2.2.1)
where x and yare the coordinates of point A. Let us expand the expression!'! for the velocities ililo n Taylor series, leaving only small qnantities of the nrst order in them. i.e. tel'ms contnining d:c. dy, and dz to a power not higher than the first one. Assuming that nt point..1 the velodty V;,;A V,· and V"A ..: V y • we obtain .
V'\'B~~V;,; ... (~~."I:/i).r) d.l'~'"., Vyn-Vy-(iHl/liJx)d.r r ....
~"D:v;r--:--(OV:'J~Y)dY-_"'''1 1I/D,·V,+(iJVI//(ly)dy....
(2.2.2)
Vx<:=Vx " (aVxlOx) dx T (aVxhJy) dy -t- ... Vile' '-VI/ --:. (Dill/lax) d:c ,. (ov/ay) dg ;-- ...
Examination of these expressions re\"eals that. for cxample, at point /J the velocit.y ('.omponenl along tile x·axis diffcrs Crom t.he value o( tile velocity at point ~1 by the quantity (av/ax) d:c. This signifies that point 11, while participating in translational motion at the velocit~ Vx in the direction of the .r·axis together with point A, simultaneously mo\'('s relative 1.0 it in the same direction at the
76
PI. I. Theory. Ae,odynamics 01 an Ai,foil and
FI.. l.2.l
Angular strain particle
of
II.
Wing
Ouid
velocity (cJV,/cJx) rh. The result is a linear strain of segment AB. 'fhe rate of this strain is 9 x = iJV;cliJr. Point B moves in the direction of the y-axis at the velocity V" together with point A and simultaneously moves relative to this point at the linear velocity (aVylax) dx determined by the angular velocity of segment AB. Considering point D, we cau by allalogy with point B determine that the relaLh·e linear velocity of this point in the direction of the y-axis is (iJVy!ay) dy and, consequently, the rate of linear strain of
::~e~~~~p p~:II.lt ~e ~Y -=a~~!£:~tl~~h~i~~r;u~~:nv~!~~~trn~~ ~~~o~~~;
that point D rotates abollt point A ill a direction opposite to that of rotation of point 8). Hotation of segments AD and AH causes distortion of the angle DA 8 (Fig. 2.2.2), i.e. angular strain of the particleis produced. Similltaneoll!:'ly, bisector AM 01 angle DAB may turn, the result being the appearance of an angle d~ between it and bisector AN 01 the distorted angle D'AR'. Hence, the particle rotates additionally. The angle of rotation of the bisector (Fig. 2.2.2) is d~~l-<
where y ---' 0.5 1:t/2 - (da~ + da l )] and "t --:: :t/4 - da 2• The angles da. z and dell shown in Fig. 2.2.2 equal (av,/{)x) dt and (av,Jay) dl, respectively. Consequently d~ ~ 0.5 (cia, - cia,) = 0.5 (OV,IO. - aV,IOy) dt Hence we can find the angular velocity d~/dt of a fluid particle about the z-axis. Denoting it by fil z, we have .. , = d~/dt = 0.5 (av,/a. - avJay) (2.2.3)
Ii we consider the motion of edge AD relative to segment AB. the angular velocity of this edge will evidently be 2e,
= fJv,iax
-7- aV:c1ay
(2.2.4)
Ch. 2. Kineme!ics of e Fluid
77
The quantity (2.2.4') is called the semi-rate of downwash of the right angle DAB. Let liS apply this reasoning to three-dimensional flow and consider point C belonging 10 a particle in the form of an elementary parallelepiped with edge lengths of dx, dg, and dz. The \'elocity at this point at the instant I = to is a function of the coordinates % -i- dx, y ...- dy, and z ..- dz. Writing the velocity components in tnc form of a Taylor series in whifh only small terms of the lin::t order are retained, we have V,c = 1'., .,- (aI',18r) dx -'- (W,18y) dy T (aI',18,) (2.2.5) Vue ""1'. V" -I- (al'"lax) d:x + (aVlllay) dy + (aVylaz) dz V,c = V, .,- (W,18x) dx.L (8V,liiy) dy -;- (8V,I8,) Let lI~ introduce a notation similar to that adopted when analysing the motion of a two-dimensional particle. We shall assume that e~ = aV 2 laz. This quantity determines the rale of linear strain of a three-dimensional particlt, in the direction of the z-axis. Let us also introduce the notation
dz} dz
Wx
= 0.5 (i)ViDy
-
aI-'ulilz),
wu::""::
0.5
(aV~.IrJz -
aV:ld..c)
(2.2.6)
Tht' quantities Wx and Wv are the componenls or the angular \'elocity of a particle along the x- and y-axes, respectively. The components of the angular velocity of a particle (Un WII' and 00: arc considered to be positive upon rotation from the x-axis to the y-axis, from the y-axis to the z-axis, and from t.he z-axis to the x-axis. respectively. Accordingly, the signs of the dcrh·al.ivcs aVllliJx, avz/ay, and arxfi)z coincide with those of the angular velocity, while the sign!'! of the deriva.tives aVxlay, aVulaz, and aVzhJx arc opposite 10 thosE' of the angular velocity. By analogy with (2.2..1,'), we have the value.
= 0.5 (aV:lay + aVlllaz),
Ell
0..:::
0.5 ({)VxlfJz
+ aViiJx)
(2.2.7)
that equal the semi-rate of downwasll of the two right angles of the parallelepiped in planes yOz and xOz, respectively. By performing simple transformations, we can see that al'zlay = lH'vlaz
£:.:
-r-
= Ex -
Cil:.:;
lJv:r/az =
ell -:- 0011 ;
w:.:i aViax = Ell -
00 11 ;
aVlll1h:
= E: + Wz
aV,,/ay = £~
-
Wz
With account taken of these expressions, the velocity components a.t point C can be represented in the following form: ~-~+~dx~~dz+~.T~dz-~.} + e z (];x; £:.: dz -1- 00: dx - Wx dz (2.2.8)
Vile :..., VII -i-- 0Il dy
+
~=~+~dz+~.+~dx+~.-~dx
78
Pt. I. Theory. Aerodynamics of In Airfoil and a Win;:
Hence. the motion of point C can be considered as the result of addition of three kinds of motion: translational along a pathline together with point A at a velocity of V (V.n V". VJ:), rotation about it at the angular velocity 0) = O):ti + O)lIj + O)J:k (2.2.9) and pure strain. This conclusion forms the content of the Helmholtz theorem. Strain. in turn, consists of linear strain characterized by the coefficients e:t. e,l' and au and of the angular strain determined by the quantities elt , til' and 8:. The linear strain of the element's edges causes a change in its volume 't' = dx dy ch, which is determined by the difference d't = dxl dYI chI - dx dy dz where dx 1, dYI' and dz i are the lengths of the edges at the instant t dt. Introducing the values of the lengths, we find
+
*-~+~*~~+~.~~+~.~-*
••
Disregarding in this expression infinitesimal terms higher than the fourth order. we obtain d. - (ex + 0, + 0,) dy • dt Hence we can determine the rate of change of the relative volume. or the rate of the speeific volume strain 0 = (1h)d'tldt, which at each point of a flow equals the sum of the rates of linear strain along any three mutually perpendicular directions: e - 0. + 0, + 0, (2.2.10) The quantity a is also known as the divergence of the velocity vector at a given point: div V 0. + 0, + e, (2.2.11) Hence. the motion of a fluid particle has been shown to have a complicated nature and to be the result of summation of three kinds of motion: translational and rotational motion and strain. A flow in wllich the particles experience rotation is called a vortex one, and the angular velocity components 0)=. (I)", and wI-the vorta components. To characterize rotation, the concept of the \"elocity curl is introduced. It is expressed in the form curl V = 21\). The velocity curl is a vector (2.2.12) curl V = (curl V):r i + (curl V)" l + (curl V)z k
*
°-
whose components equal the corresponding double values of a vortex component: (curl V)x '"'" 2(1):t, (curl V)" = 26>11' and (curl V)z = 2wJ:
Ch. 2. Kinematics of ,. Fluid
79
U. VOIteI·Free
MotIon of • Fluid
When investigating the motion of a fluid, we may often take nl) account of rotation because of the negligibly small angular velocities of the particles. Such molion is called vortex-free (01' irrotational). For a vortex-free flow, (0 = 0 (or curl V = 0), and, consequently, the vortex components 00;.;, OO Ut and 00: equal zero. Accordingly, from formulas (2.2.3) and (2.2.6) we obtain aVxliJy = aVuJax. aVz1iJy = aVulaz, 8V x!{)z = dV:,'8z (2.3.1) These equations arc a necessary and sufficient condition for the
~i~e~~~t}~~~~~~~~lil~~:Cte~z~g ~ft:Kot :r:ad~l;i~ I;~ :h~o::~~i,~:~e~; th~e:~:;~:lngC~~;of~~nc~~o~:Ci~ ~~~ :~r~ ~z'(x, y, z, t) and considering the time t as a parameter, we can write the expression dq> = Vxdx
+-
l!.,dy
+ l':dz
On the other hand, the same differelltial is d~ ~ (8~Jax) dx -i- (8~J8y) dy + (a~,a,) dz Comparing the last two expressions, we fmd Vx o'fiiJz, VII = oqJJag, V: = {)cp/iJz
(2.3.2)
The function qJ is called a velocity potential or a potential function, and the "ortex-free flow characterized by lhis function is called a potential ftow. Examination of relations (2.3.2) reveals that the partial derivath'c of thc potential
80
Pt. I. Theory. Aerodynamics of In Airfoil and a Wing
respect. Lo this direction Off/OS. The quantity OCf/oS can be considered as the projection onto the direction s of a vector known as the gradIent of the function
or V
~
grad
~
(grad
~).
I
-+-
(grad
~), j
+ (grad ~), k
(2.3.4)
where the caetrldents in parentheses on the right-hand side are the projections of the vector of the velocity gradient onto the coordinate axes: (grad
Simplifies the indetermining three V" it is sufficient possible to com-
2,4. Continuity Equation G ...... Form of the Equation
The cquation of continuity of motion in the mathematical form is the law of mass conservation-one of the most general laws of physics. This equation is one of the fundamental equations of aerodynamics used to nnd the parameters determining the molion of agas. '1'0 obtain the continuity equation, let us consider a moving volume of a fluid. This volume, varying with time, consists of the same particles. The mass m of this volume in accordance with the law of mass conservation remains constant, therefore pm"t = const, where Pm is the mean density wHhin the limits of the volume "to Conse.quently, the derivative d (pro"t)/dt = 0, or, wit.h a view to the density and volume being variables, we have (1/Pm) dpmldt + (11,) d,ldt = 0 This volume relates to an arbitrary finite volume. To obtain a relation characterizing the motion of a fluid at each point. let liS go over to the limit with "t _ 0 in the last equation, which signifies contraction of this volume to aD internal point. If the condition is observed that the moving fluid completely lills the space being stud-
Ch. 2. Kinematics of • Fluid
81
ied and, consequenlly, no voids or discontinuities are formed. the density at a given point is a quite determinate qnantity p. and we obtain the equation (2.4.1) (1/p) dpldt + div V '-'" 0 in which the value of
;i!
I(ilt) d./dt! has been replaced according
to (2.2.10) and (2.2.11) wHh the divergellce of the velocity. Relation (2.4.1) is a continuity equation. It bas been obtained in a general form and can therefore be used for any chosen coordinate system. Cartesian Coordlnaht 5yshrll
Let liS consider the continuity equation in the Cartesian coordinate system. For this purpose, we calculate the derinLivc dp (x. y. i. t)/dt in (2.4.1) in accordance with (2.2.11). As a result. we obtain the continuity cquation op/at 0 (pvJ;.)/ax -:-- a (pVy)/OY T () (pVr)/oz '-' 0 (2.4.2)
-r
Introducing the concept of the divergence or the veclor pV div (pY) = iJ (pVx)/iJx .j- fJ (pVy)'uy -1- iJ (pVz)/Oz we obtain up/iJt .!. di\' (rV)
(2.4.a)
instead of (2.4.2). Continuity equation (2.1.2) dcscrihl~s all UlI~I{'atl~' flow. For a steady flow. aplat = 0 and. consequent I.'" , iJ (PV.~ViJ.r a (pVy)iiJy .!- II (PV1)''B,;:;'~ 0 (2.4.4)
+
IIi\' «(IV) ."" 0
(2..1.4')
For a two-dimensional flow. the continuity C'qunlion is o(p V:r)/iJ.J: 'r a «(I v u)/(Iy
For an incompressible flow, f' iJVJiJx
'-~ con~l.
"VyiiJy
~
0
(2..1.5)
h('II(:('
i- iJl'l"h
div V :;.." 0
(2.4.6)
(2.4.7)
For potontial motion, lhe (:oulinuily ('lllIillioJl i~ transIorml.'ll with account taken or (2.:1.2) to IiiI' fldlowing form: (1/p) dp/dt .L rJ'!lr/ax~ "p{r'r7!1~ M!_ {j2(("ZZ . IJ (:lA.8)
For an incomprN!siblc fluid. /' - I:onsl. tilC'l'l·forC' rJ2(r-'().T~ i· iJ~{1 ii1!J~ (I'!/,. 'rJ Z2 (l
(2.-'1.8')
82
Pt. I. Theory. Aerodynamic5 of an Airfoil and a Wing
The expression obtained is called the Laplace equation. The solution of this equation is known to be a harmonic function. Consequently, the velocity potential of an incompressible Row lP is sucb a harmonic function. Cumll.." Coordinate Systein
Conversion Formulas. It is more convenient to solve some problems in aerodynamics by using a curvilinear orthogonal coordinate system. Such systems include, particularly, the cylindrical and spherical coordina te systems. In a cylindrical coordinate system, the position of a point P in space (Fig. 2.4.1) is determined by the angle "i made by the coordinate plane and a plane passing through point P and the coordinate axis Ox, and by the rectangular coordinates :r and r in this plane. The formulas for transition from a Cartesian coordinate system to a cylindrical one have the following form:
x = x. y = r cos "i.
Z
= r sin "i
(2.4.9)
In a spherical coordinate system, the position oC point P (Figure 2.4.2) is determined by the angular coordinates a (the polar angle) and 'IJl (the longitude), as well as by the polar radius r. The relation between rectangular and spherical coordinates is determined 8(1 follows: x = r cos e, y = r sin 0 cos'lJl, z = rsin e sin 'IJl (2.4.10) The rele\'ant transCormations of the equations of aerodynamics obtained in Cartesian coordinates to a curvilinear orthogonal system can be performed in two ways: either by direct substitution of (2.4.9) and (2.4.10) into these equations or by employing a more general approach based on the concept of generalized coordinate curves (see (91). Let us consider tbis approach. We shall represent the elementary lengths of arcs of the coordinate curves in the vicinity of point P in the general form:
cis1 = ht dq1' ds,
= h2 dq2,
d·'t3 = h, dqa
(2.4.11)
where qn (n = 1, 2, 3) are the curvilinear coordinates. and hi are coefficients known as Lame's coefficients. For cylindrical and spherical coordinates, respectively. we have
ql = Z, q2 = r, qa = "i, q, ~ r. q, ~ e. q, ~ 'i>
(2.4.t2) (2.4.13)
It follows directly from Fig. 2.4.1 that Cor cylindrical coordinates
dB,
~
tho dB,
~
dr. dB,
~ r
dy
(2.4.14)
Ch. 2. Kinematics of a Fluid
83
Fig.l.U An elementary nuid particle in a cylindrical coordinate systcm
Fig. 1.4.2 An elementary noil! particle in .1 !'pberic1l1 coordinate system
Figure 2.4.2 allows us to de{f!rmine the length of the arcs of the relcvant coorrlinate lines in a sphcrical system' dS I = dr,
dS 2 = rdO,
(b3
= r sill €I d1jJ
(2.4.15)
Consequently. tor cylindricnl coordinates, Lame's coefficients ha vc the form (2.4.16) hi = 1, h',l= 1, hs=r and for spherical ones hi
= 1,
h',l = r, h3 = r sin €I
(2.4.17)
Let us consilier some expressions for vector and scalar quantities· in curvilinear coordinates that are needed to transform the continuity equation to these coordinatE's. The gradient of a scalsr function <J> is
where fl , f2' and ia arc unit vectors along the relevant coordinate lines.
84
PI. I. Theory. Aerodyn..mics of .. n Airfoil .. nd .. Wing
Having in view formulas (2.4.11), we obtain grad4l=.*.
:~
it
+*". ::; +*-. :~ 12
is
(2.4.18)
For further transformations, we shall use information from the
-course of mathematics dealing with vector analysis. Let us find the divergence of the velocity vector, writing it as the sum of the components along the coordinate lines: 3
V =Vti,
+ VIi: + V,i,= ",.-.1 ~ V",i n
(2.4.19)
Performing the divergence operation for both sides of this equation, we obtain
,
divV=
L
(Vndivin;'-i .. gradV",)
(2.4.20)
n=1
To determine div in. we shall use a known relation of vector analysis (2.4.21) where n takes on consecutively the values of n = 1, .2, 3 which the values of m =-= 2. 3. 1 and j = 3, 1. 2 correspond to. This relation includes the vectors curl im or curl i, to be determined. for which we shall introduce the common symbol curl in. Using (2.4.18) and the general methods of transforming vector quantities, we nnd (2.4.22) Let us introduce (2.4.22) into (2.4.21). For this purpose we shall first replace the subscript n in curl in with m and j •. and also correspondingly arrange the subscripts on the right-hand side of (2.4.22). As a result, we have (2.4.23)
Let us inlroduce these values into (2.4.20). Having in view that in this formula. in ar,cordance with (2.4.18), we have in .grl\d Vn ,--= (1/kn) aVn/Oqn we find the fonowing expression for the div('rgence: divV:.-: h)ihS'[ a(~:~2h3)
-r
a(~~:h,) +.~~~::~hl)]
(2.4.24)
Ha.ving thiS expression. we cnn transform the contiuuity equation (2.4.1) to various forms of cllrvilincar orthogoll
Ch. 2. Kinematics of a Fluid
8~
Cylindrical Coordinates. In cyJindriral coordinates, Lame's coefficients are given by expressions (2.4.16), and the values of qn (n -'" 1, 2, 3) hy relations (2.4.12). The velocity components along the axes of cylindrical coordinates are
Vi
c,,",
~
V:t
dx/dt, V 2
~,
V, -_ drldt, V3
~
V ... ,.".., r dy/dt (2.4.25)
Taking this iuto Recount, the di\'ergence of the velocity (2.4.24) becomes di\'V,., d.!:,:t
_I.
Q;;~
++.
f~,~'" _l.~
(2.1..26)
To transform the deri\'8th'e dp'dt in the continuity equation (2.4.1) to cyJindrical coonlinates. we use the transformation formula
in which
f
'*
=
::•. ~~I
-%- -+
-;- ::2 .
-~
d;t
::3 .
ddqt
(2.1.27)
(x, y, z, t) is a function of Cartesian coordinates aO(l
time. We obtain
-1i-
*+*.*:4tf-.*r-*.* .*.
:.--%t.~-Vx"*-:
Vr* 7-
l~Y
(2.t,.28)
Suhstituting their values for cli ... V lind dp.'dt in E/}. (2..1.1), wc lind
4/i-+ (J(~:'x) +- fJ(~!'r)
~+. a(~~:;) +~'--.O
(2.t..2\))
Let us write this equation in a somcwhat tlHfcrcnt form:
1'"* + 8(r;
zl'x)
++. ++.
+ a(~~1!r)
r)(r;~'~) =0
(2.4.30)
For the perlicular ease of steady motion a(~:rx)
+ a(e;rVrl
iJ('8t~)
.. 0
(2A.31)
For potential motion, substitutions may be made in the continuity equation: V. ~ a~/ax, V,., a~/ar, V, c·, (11r) (a~/iJy) (2.".25') If the motion of a gas is simultaneously axisymmetric as, for example, if it flows about a body of revolution at a zero angle of attack, the Row parameters do not depend on the angular coordinate y. and the continuity equation has a simpler form:
a (prV.)18x
+ a (prV,)/ar ~ 0
(2.4,32)
86
Pt.
r.
Theory. Aerodynamics of an Airfoil and a WIng
Spherical Coordinates. Using relation (2.4.24), having in view formulas (2.4.13) for spherical coordinates, and also relations (2.4.17) for Lame's coefl\cients and the expressions for the velocity components
VI=Vr=4t-. Vz=Vo=+~. (2.1.33)
V,=VIJ:=r 13 inS!!J!.
we obtain the following expression for the divergence of the velocity:
diVV=;-.
D(;:r~) + r8~n9' "(V~inO) + r8:nl:l' ~'"
(2.4.34)
We shall write the total derivativo for tho density dpldt in accol'd· anee with (2.4.27) in the following form:
!!£-~-;~+{-'-¥r-+~'*+ ~~ . :: =*".!.Vr~+ 1:0 r::9' :~
.*"+
(2.4.35)
Iintroducing the values of div " from (2.4.34) and the derivative dpldt from (2.4.35) into (2.4.1) and grouping terms, we have
*'i-*.
-i- rs:no'
8(P;/2)
"(P~saSin8) +
rS11no .
8(PJrr r 2)
=0
(2.4.36)
.For steady motion, the parLinl derivative iJpliJt
+.
.,~;q.)
+~.
8(P Vas/inO)
+ Si!9
= 0, hence
. 8~;ot) =0 (2.4.37)
In the particular case of an incompressible fluid (p "'"" const)
+. i/
Si~(I' 8(V~~inO) +~. ~~'f
=0
(2.4.38)
The following substitutions must be performed to transform the continuity equation for potential motion:
Vr = ~~
•
Vo=+' ~: •
V.=" rs:nlj . ::
(2.4.39)
Contfnulty Equation 01 Gas flow .1.,. • Curved Surt.ce
Let us consider a particular form of the continuity equation in .curvilinear orthogonal coordinates that is used in studying flow over a curved wall. The x-axis in this system of coordiuates coincides with the contour of the wall, and the y-axis with a normal to this wall at
Ch. 2. Kinematics of
/I
Fluid
B7
Fig.l.U
To the derivation of the
COli
tinuily equation in curvilinl'.1r coordinalA's
the point being considered. The coordinates of point P 011 the plane (Fig. 2.4.3) equal the length x measllred ~dong the wall and the distance y measured along a normal to it.. respectively. Assume that the wall is a surfRce of revolut.ion in an Rxisymmetric flow of a gas. The curvilinear coor(linates of point. Pare:
ql = X, q2 = y, q, = )'
(2.4.40)
The elementary lengths of the coordinate line arcs are ds1 = (t
+ ylH) dx,
ds z = dy, ds a = r d)'
where r is the radial coordinate of point P measured along a normal to the axis of the surface of reyoilltion, and R is the radius of curvature of the surface in the section being considered. ConsequenLly, Lame's coefficients are hi = t -I- y/ll, h z = I, h3 .= r (2.4.42) Let us llSC formula (2.4.21t)Jor div V in which the velocity components arc VI = V:r = ds1idt = (t .:" y/ll) dx/dt. tt"2 = Vy = dy/dt. V3 = 0 (2.4.43) Substitution yields divV=
(1+~_R)r {8(;;"r)
+ iJll'-yr~:7U!R)I}
(2.4.44)
Let us calculate tile total derivathe for the densit.y:
*-0'=*+*-.*+-*.*
=-"*"1- 1;=/ll ."*-7- Vy"*
(2.4.45)
88
PI. I. Theory. Aerodynllmic$ of !In Airfoil and a Wing
Introducing expre!;sions (2.4.14) and (2.4,45) into (2.4.1), after !;imple transformations we obtain the continuily C'quat.ion *r(t....!...y/R) + iJ(r;xv -,\,) ~ arpr(I-~-:!R)Y'l/1
(2.4..46)
When studying the motion of a gas ncar a wall with a small curvature or in a thin layer adjacent to a surface (for example, in a houndary layer). the coord in ale y« R. Consequently (Op/ot) r
-+
iJ (prV3C)/OX
+ iJ (prVy)/oy
'""- 0
(2.4.47)
The form of the obtained equation is the same as for a surface with a straight generatrix. For steady now d (prV.,JIOx
+ a (prVy)/OY
= 0
(2..1.48)
For two-dimensional flow near a curved wall (a cylindrical surface), the continuity equation has the form *(1-:-yJR)+
For y
«R.
a(~:.\.)
+
arpV'/~y-i
ViR)!
-.0
(2.4.49)
we approximately have op/ot -t- u (p V;c)/ux -/. u (p V y)/uy --. 0
(2.4.50)
Hence, for flow near a wall, the continuity equation in curvilinear cootdinates has the same form as in Cartesian coordinates. For steady flow, aplat -= 0; in thi!; case the continuity equation is D (pV.)IDx + D (pV,)18y _ 0 (2.4.50') Equations (2..1..18) and (2.4.50') can he combine(1 into a single one: d/(pv,..r8)Jax a (pVyr8)/ay 0 (2.4.48')
+
0....
where E = 0 for a two-dimensional plane flow and e = 1 for a threedimensional axisymmetric flow. Flow bte Equation
Let us consider a particular form of the continuity equation for a steady fluid flow having the shape of a !;tream filament. The mass of the fluid in a fixed volume confllled within the surface of the filament and the end plAnE" ('.ross sections docs not change in time because the condition that up/at 0 is observed at every point. Hence the mass of the fluid entering the volume in unit time via the end cross section with an area of 8 1 and equal to PI VIS 1 is the same as the mass of the fluid P2 V 28 2 leaving through the opposite cross seclion willi an area of S2 (Pt and P2 are the densities, Vl and T'2 are the speeds in the firsl and second cross seclions of the lilamenl, respectivE'!0-
Ch. 2. Kinematics of a Fluid
89
Iy). We thus have PIV1St ., P2V2Sl. Since this equation can be related to any cross secLion, in Lhe general form we have p VS = const (2.4..51) This is the flow rale equation. 1.5. Stream Function
The studying of II ,·ortex-rree gas flow is simplilied because it can be reduced to the fmding of one unknown potential fUllction completely determining the flow. We shall show that for certain kinds of a vortex flow there is also a function determining its killemath: characteristics. Let 115 cOllsider a two-dimensional (plane or spatial axisymmetric) steady vortex now of II fluid. It call be established lrom continuity equation (2.4.32') that there is a lunction 'If; of the coordinates x and y determined by the relations a¢lax c -PII'V,; 8¢lay ~ PY'V. (2.5.1) Indeed, introriucing these relations inlo (2.4.32'), we ohtllin ax) ,- 82¢/(8x ay), i.e. an identity. Substituting (2.5.1) into t.he equation of slr('amliu('s Vyl8y .. , V:Jlldx wriLten in the form
8~1J;/(ay
pytVy dx - flytV:Jl dy '""" 0
(2.;").2)
we obtain the expression (a'i'iax) dx -i. (a¢lay) dy = 0
from which it follows that (2.5.2) is a dWerelltial of the fUllcliol! '" and, consequently, Integrating (2.5.3), we find an equation of stream lilies in the form 1f (or, y) """ const (2.;JA,) The function1p, called the stream function, completely determines the velocity of a vortex flow in accordance with the relations V. (l/py') 8¢lay, V, - - (1/py') O¢iaz (2.;'.0) We remind our reader that for a two-dimensional flow, we hav~ to assume in these expressions that e ,. . ,. , 0, while for a threc-diml'lIsional axisymmetric flow e = 1 and y = r. A lamily of streamlines of a potential flow can also be charact('rized by the function '" "'- const that is aSSOciated with the velocity potential by the relations a'Piax ~ (lil'lI') 8¢lay,
a
(lIpy') O¢:ax
(UdS)
"90
PI. I. Theory. Aerodynllmics
or
lin Airfoil lind II Wing
Assuming in (2.5.5) and (2.5.6) that p = const, we obtain the relevant expressions for an incompressible flow: (2.5.7) V, ~ (1Iy') Inplay, V, ~ - (1Iy') a~/ax a~/ax ~ (1Iy') a'Play, a~/ay ~ - (1Iy') a'Plax (2.5.8) Assuming in Eqs. (2.5.8) that e = 0, we obtain the following equations for an incompressible two~dimensional flow; a~/ax ~
a'Play,
a~/ay ~
-a'Plax
(2.5.9)
Knowing the velocity potential, we can use these equations to .determine the stream function with an accuracy to within an arbitrary constant, and vice versa. In addition to streamlines, we call construct a family of equipoten~ tial curves (on a plane) or of equipotential surfaces (in an axisymmetric flow) in a potential flow that is determined by the equation 'Q" = const. Let us consider the vectors g"d 'P ~ (a~/ax) I + (a~/ay) j and grad 'P ~ (a'Plax) I + (a'Play)j whose directions coincide with those of normals to the curves
A vortex line is defined to be a curve s constructed at a defmite instant ill a fluid flow and having the property that at each point ·of it the angular velocity vector 0) coincides with the direction of a tangent. A vortex line is constructed Similarly to a streamline (see Fig. 2.1.1). The only difference is that the angular velocities of rotation of the particles 0) (0)1' 0)2' etc.) are taken instead of the linear velocities. In accordance with this delinition, the cross product .00 X dS = 0, Le.
I :~
w~ ~z l=i((o)~dZ-WZdY)-j ((o),;dz-wzdx)
dx dy dz
+k (w%dy-w~dx) =0
Cn. 2. Kinemlltics of II Fluid
l-Ience, taking into account that, for example, COy dz = 0, and so on, we ohtain an equation of a vortex line: dxlw;r;
=
dylw y = dzlw z
91
U1 z
dy =
(2.6.1)
13y consl-tucting vortex lines through the points of an elementary -contour wilh a cross-sectional area of 0, we obtain a vortex tube. The product wo is called the i.ntensity or strength of a vortex tube. Ot· simply the vorticit~·. We shall prove that the vorticity is coustant for all the sections of a vortex tube. For this purpose. we shall use the analogy with the flow of an incompressible fluid for which div V = O. A corollarr is Hle flow rate equation for a stream filament VIS I = V 2 S 2 :-:: • . . . -= VS = const. Let us consider vortex motion (lnd the expression for the divergence or the angular yclocity div
(r)
= QOl;r;IIJx -i- IJUlyfIJy
+ Qooz//)z
Substituting for the componenls of w their values from (2.2.3) and (2.2.fi), we oblain. provided that the second derivatives of l'x, V y , and V z arc continuous, the expression tii,· w ,..." O. using the analogy with the flow rale equation VS = const, we obtain an cClnatioll for the flow of the vector w along a vortex tube in the form 00 1 0"1
=
U1 2(J2
= ... = wo = canst
(2.6.2)
This equation expresses the IIelmholt~ theorem on the constancy of the vorticity along a vortex tube. The properly of a vortex tube consisting in that it cannot break oD suddenly or terminate in a sharp point follows from this theorem. The termination in a sharp point is impossible because at a cross-sectional area of the tube of 0" _ 0, the angular velocity w would tend to infmity according to the IIelmholtz theorem, and this is not real physically.
1.7. Velocity Circulation Coneepl
Velocity circulation is of mAjor Significance in aerodynamics. This concept is lIsecl when studying the flow over craft and, particularly, when determiniug the lift force acting on a wing. L(>t us consider in a fluid flow a fixed closed contour K at each point of which the velocity V is known, and ent\lIate the integral -over this contour: (2.7.1)
92
Pt. I. Theory. Aerodynamics of an Airfoil and
Wing
where V·ds is the dot product of the vectors V and cis. The quantity r determined in this way is known as the circulation of the velocity around a closed contour. Since V=V:J'I-Vllj-j-Vzkandds=dxi+dyj dzk we have
r,.
I V.dx".V,dy··'-V,dz
(2.7.2)
("'
Taking also into account that the dot product V·ds """ -- V cos (V........ds) as '-' Vs ds, where Vs is the projection of the velocity onto a tangent, we obtain
r.~
\ V,ds
(2.7.3)
(K)
II the contour coincides with a streamline in the form of a circle of radius r at each point of which the velocity V. is identical in magnitude and direction. \,,,'C have (2.7.4) r .-= 2nrVs in
The circulation of the vclocity in a vortcx-frec flow can bc cxpressed lerms of the vclocity polential. Assnming that V", dx "I.:~ V: dz =-: dqJ. we obtain for an open contour
-i- VII dy
l'-= ) dq.-='f!A-lJ"n
(2.7.5)
(Ll
wherc
n.......
Let us consider elementary contonr ABeD (see Fig. 2.2.1) and evaluate the circulation around this contour. We shall assume that the velocities are constant along each edge and are equal to the following values:
V.(AB). V,+ ';" dx(8C). V.+ ';'" dy(CD). V,(AD) ConSidering the counterclockwise direction of circumventing the contour (from the x~axis toward the y-axis) to be positive, we obtain
Ch. 2. Kinemllti<:;s of II Fluid
93
for the circulation
d;;, dX)dY-(V
dr:=V:rdx~-(Vv+
dr. =
x
;"
(dVyldx --uV.,)uy)
d~r
",;y'" dy)dx--V"dy dy
According to (2,2.3), the quantity in p
dr,
~
200, dx dy
(2.7.6)
We can prove similarly that
dr", ,.--, 2w", dy dz and dry"'''' 2w g d:c dz In these expressions, the products of the dilterentials arc the areas confmed by the relevallt elementary contours. With Ii view to the rcsults obtained, for the area clement do oriented in space arbitrarily and confmed within an elementary contonr, the circulation is (2,7,i) where Wn is the component of the angular velocity 1I10ng the direction of the normal n to the surface clemen!. du. In accordance with (2.7.7), the "elocity circulation aronnd an elemcHtary closed ('o\tlollr eq\lals the double strength of ,\ vortex in~ide the Helation (2.7.7) can be used for II conlOHr I-tnile conlining a surface S lIt each poilll of whkh thc (If (')" is H(!r<' the veloCity circulation around !itl' \'ontultr i:-:
r-211 '.'.,
d"
(Ii)
Formula (2.7.8) expresses the Sloi;:C'!" titc-Mc-m: the lation arollnd the closed contour 11 equals tlte .]ouhle strt1ngtll or the vOrL ices passing lhrOltg-ll !he slIrface contoul'. The quantity determined hy tlli...; rormu!;) (vol'l('.'( sll'ell~lh) and i:> ,iesignalC'll hy x_
2~ ,
(')n
(2.i .R')
{ia
'(8i
Comhining formulas (2.i.:\) ;lilt! (2.i.X) I'O!' 1'. We' ohlailll\ relalion expn'ssing the integral oyet· the contour f( ill 1f'I'ms of lilt' inli'g'I'I\\ lhe surface S COllflll{'II within this coulollr A': \ V, ds - 2 .\ (i.:)
i
IS)
<0"
du
(2.7.8")
94
PI. I. Theory. Aerodyn.mics 01 an Airfoil and a Wing
Flg.2.7.t
Simply and triply connected regions on a plane: .. -~imply co~n(>cled r('gion (rK _ circulation ov~r th(> conto\lr K); b-triply connected region 0'1, = cJrcula Hon of tbe eJ:tem a J contour, rK, and r x , - clrculatlon~ over the In_ t~ma J conloul':!l X, and X., respectively)
The above expressions relate to 0. simply connected contour (the region being considered is confIDed within a single contour, Fig. 2.7.1a), But the Stokes theorem can also be extended to multiply connected contours (a region confIDed by one external and several internal contours). Equation (2.7 .8~) is used provided that the external contour is connected to the internal ones by auxiliary Jines (cuts) so as to obtain a simply connected region. Now t.he double integral in (2. 7 .8~) is used for the hatched region (Fig. 2.7 .1 b) , while the contour integral is taken for the obtained simply connected region, i.e. around the external contour, along all the cuts, and also around all the internal contours. In accordance with this, by formula (2.7.8"), and also with a view to Fig. 2.7,1b showinJ! 8 triply connected region, we obtain
Jwnda
r K-rK1 -rK.=2 J
(8)
whence the circulation over the contour K of the region being considered is
r K=rKI+ r X.+2
JJ
wnda
(8)
Using this formula when we have n internal contours, we lind I-
rK =
hr ;
Kt -/-
2
Jj
Ctl n
do
(5)
Vortex·lncilieM V.loeHl..
Appearing vortices produce additional velocities in the Ouidfilled space surrounding them. This effect is similar to the electromagnetic influence of a conductor carrying an electric current. In
Ch. 2. Kinemetics ot
110
Fluid
95
d[
accordance with this analogy, the velocities produced hr a vortex arc said to be induced. This electromagnetic analogy is expre5seu ill that Lo determine the vortex-induced "elocity, a Biot-Savart relation i.!' used similar to the one expressing the known law of eleclronwgnelic induction. Let us consider a curvilinear vorlex of all arbitrary shape (Fig. 2.7.2). The velocity vector dw induce(] by the vortex clement dL at point A whose location is rletermiue(\ by the position vector r coincides in direction with the cro."s prodnct r X dL, i.('. the ,"N'.tor dw is perpendicular to the plane containing the \'ectol'S rand dL. The value of dw is determined with the aid of the Biol-Savart rormula, which in the vector form is as follows: dw = (r/4n)
I'
X dL r3
(2.7.n)
where f is the velocity circulation. The derivation of formula (2.7.9)' is given in [101. Since the magnitude of the cross product I r X dL I = r sin 0: dL. where 0: is the angle hetween the direction of a ,'ortex element and the position vector r, the magnitude of the velocity induced at point A is (2.7.10) dw = (f/4n) sin a dLr z Let us use the Biot-Savart relation to calculate the velocily induced by a section of a liue vortex (Fig. 2.7.3). Since r = h/sin 0:. dL = r da:/sin 0: = h do:/sin z 0:, then U)=
4~h
''>
)
.,
sinada=
4~h
(cosa l -coso:Z)
(2.7.11)
For a vortex hoth of whose ends extend to infinity (an inlinite vortex), 0:1 = 0, 0:2 = n, aud. therefore,
w
~
1'/(2nh)
(2.7.12)
96
Pt. I. Theory. Aerodynamics of
r~~'u!:~
a nuid
of a line vortex on p
.n Airfoil .nd •
Wing
~
"
"
fig. 1.1."
Interaction of \'orliccs: o_vorlicrs wilh (In idpntlcat dirPl"
1;011 of mlalion; b · · \·"rUc(·~ with opposite dir,.~tiun~ of motion
For a vortex, one end of which extends to inrmity, and the other has its origin at point A (a semi-infinite vortex), we have a l -::: 0 and a. t •. n/2. Consequently w ~ r / (bh) (2.7.1:1) If a nuid accommodates two or more vortices, they interact with one another, and as a result the vortex system is in motion. The velocity of this motion is determined wilh the aid of the Biot-Savart relation. Let liS tnke liS an example two infmite vortices with the same strength nnd direction of rotation (Fig. 2.7 .4a). These vortices impnrt to each other the velocities V ~ :-:: - l'/(2nh) and VI -:--: = r /(2nh) that
Ch. 2, Kinem.tics of • Fluid
97
in the theory of the functions of a complex variable are known as the Cauch:,r-Riemann eqltations. They express the necessary and sufficient conditions needed for a (',ombinatioll of the two functions cp .L- i'\f.l to be an analytical function of the complex variable cr .= x iy. Le. to be differentiable at all point~ of a certain region. Let us introduce a symbol fOl" this fune.tian:
+
W(cr) -=
+ i'\f.l
(2.8.1)
The function W (0) that is determined if the fUllction of two real variables If ::-; If (x, y) and W=: W(x, y) satisfies differential equations (2.5.9) is called a complex potential. If we recall that the values of the functions If (x, y) or~ (x, y) allow one to determine the velocity field in a moving fluid unambiguously, then, (',onsequently, any two-dimensional flow can be set by a complex potential. Hence, the problem on calculating such a flow can be reduced to finding the function W (0). Let us ('alculate the derivative of the function W (0) with respect to the complex variable cr: dWId
+ iil-f18x
(2.8.2)
Since iJlffiJx = 1':n 8¢/Ox .= - Vy• tlten dWldo = Vo; - iV"
(2.8.3)
This expression is called the complex velocity. Its magnitude gives the value of the ,reiodty itself, I dWfdo I -= VV~ -;- V~ = v. It is evident that the l'eai velOCity vector V = V.., -j- iVy is thp mirror image relative to the x-axis of the vector of the complex velocity. Let us denote by 9 the angle between the vector dWldo and the x-axis and determine the velocities V % = V cos 9 and VII = = V sin O. Using the Euler formula cos 9 -
i sin 9 = e- fi
we obtain dWfda
= Ve- f8
(2.8.4)
2.9. Kinds of Fluid Flows Let us consider the characteristic kinds of flows of an incompressible fluid, their geometric configuration (aerodynamic spectrum), expressions for the complex potentials, and also the relevant potential functions and st.ream functions, 1-01715
98
Pt. I. Theory. Aerodynamics 01 an Airfoil lind II Wing
'.r.II_1 Flow
Assume that the flow of a fluid is given by the complex potential W (0) - V (cos 9 - i sin 9) 0
(2.9.1)
where V and e are constants for the given conditions. According to (2.8.1) (J) + i1p = V (cos e - i sin 9) (x + iy) whence we find the velocity potential and the stream function:
(2.9.2) (2.9.3)
1p= V(ucos9-xsine)
Examination of the expressions for
= V ~ = V cos e,
O!.fJ/8y = VII
=
V sin e, O(J)/8z
= Vz
= 0 (2.9.4)
Here V is the resultant velocity of the flow, and e is the angle between its direction and the x~axis. Assuming the stream function "p in (2.9.3) to be constant and including V in it, we obtain the
equa~
tion y cos e - x sin e = const
(2.9.5)
A glance at this equation shows that the streamlines are parallel straight lines inclined to the x~axis at the angle 9 (Fig. 2.9.1). Since the velocity components V~ and VII are positive, tIte flow will be directed as shown in Fig. 2.9.1. This flow is called a forward plane~ parallel one. In the particular case when the flow is parallel to the x~axis (9 = = 0, VOl: = V, VII ...;; 0), the complex potential is W (0) -
Vo
Two~DlmenslollllJ
_Sink
(2.9.6) Point Source
Let us consider the complex potential (2.9.7) W (0) - (qI2") In 0 where q is a constant. We can write this equation in the form ~ + i'i - (qI2,,) In (,,>0) ~ (qI2,,) (111 r + tel
en. 2. Kinem!ltics of II Fluid
99
v,
Flg.l.9.t
General view of II forward plane-parallel now
Flg.l.9.l
A two-dimeDSional point source
where r is the distance to a point with the coordinates x and y (the polar radins), anti 0 is the polar angle. It follows [rom the obtained equation that ~ ~ (qI20)
In r
~ (q/2n)
'" ~ (qI20)
In V x' -\- y'
e
(~.U.8) (~.9.9)
'Ve find from (2.9.8) that the radial velocity ('omponent (in the direction of the radius r) is iJcp/iJr = V~ = q/(21(r)
(~.9.!O)
and the componellt along a normal to this radius is Vs = O. We thus obtain a flow whose streamlines (pathlines) are a family o[ straight lines passing through the origin of coordinates (this also follows from the equation of a streamline 1j> = const). Such a radial flow issuing from the origin of ('oordinates is called a two-dimensional point source (Fig. 2.9.2). The rate of flow of a fluid through a contour of radius r is 2:nrl'r = = Q. Introducing into this eqllat.ion the value of V~ from (2.9.10),
100
pt. I. Theory. Aerodyn~mi(;$ of ~n Airfoil ~nd ~ Wing
we fmd Q = q. The constant. q is thus determined by the rate of flow of t.he fluid from the source. This quantity q is known as the intensity or strengtb of tbe source. In addition to a source, there is a kind of motion of a Ouid called a two-dimensional point sink. The complex potential of a sink is W (0) = - (qI2~) In 0 (2.9.11) The minus sign indicates that unlike a source, motion occurs toward the centre. A sink, like a source, is characterized by its intensity, or strengt.h, q (the rate of flow). !hree·m.enslon.r Source and Sink
In addition to two-dimensional ones, there also exist three-dimen· sional point sources (sinks). The flux from them is set by the following conditions:
V~= 4~
'7;
Vu= ,~ . ~:;
v¥=
4~' ~
(2.9.12)
whore R = V Xl + y2 + z.2 aud q is the strength of the source (plus sign) or sink (minus sign). The strengl.h of a source (sink) equals the quantity q defined as the flow rate (per second) through the surface of a sphere of radius R. The resultant velocity is V=VV~+V:+V:=±q/(4nR')
(2.9.13)
and coincides with the direction of the position vector R. Conseqnentl~·. the velocity potential depends only on R and. therefore.
8
~'ields
'I' = 'F q/(4nR)
(2.9.14)
where the minus sign relates to a source. and the plus sign to a sink.
_lot Let us consider a flow whose complex potential is W (0) = (MI2n) (1/0)
(2.9.15)
where M is a constant. In accordance with this equation, we have
I"
Let us transform the right-hand side of this equation. Taking into account that
r:.8 = r(e08ij~UiD8)
+(cos9-iSin9)
Ch. 2. Kinemlltics of II Fluid
101
FJI.:a.U To the definition of a doublet: B-doubll't streamlines: b-Ionnatlon ut a doublet
we obtain
'P
+ ilP = (l"fl2~r) (cos 9 -
i sin 9)
Hence (2.9.16) (2.9.17)
'P = (MI2:n) cos 9/r lP = - (:rfl2n) sin aIr
+
Assuming that til ~~ const and having in view that r = V x'J. y'l and sin f) "-- ylr ...., Y/V .rt + y'1,. we obtain an equation for a family of streamlines of the flow being ('.onsidered: y (x'
+ y2)
= const
(2.9.18)
The family of streamlines is an infmite set of circles passing through the origin of coordinates and having centres on they-axis (Fig. 2.9.3a). To comprehend the physical essence of this flow, let us consider the Dow that is obtained by !!ummation of the flows produced by a souree and a sink of the same strength located on the x-axis at a small distance e from the origin of (',oordinates (Fig. 2.9.3b). For point M (x. y). Lhe velocity potential due to a sOllrce at the distance of r 1 is 'Ps = (q/2n) In rl and due to a sink at a distance of r2 from this point is
+
:~
+
::~ = al(~~~IfRk)
+
al(~~:'Ip~k) =0
Since the functions fpa and 'PBk satisfy the equations {)'Cf.lfJx"
+ O'Cf81{Jy' = 0
alld O"CfSk/or
+ o'lf98);.I{Jy'l. = 0
PI. I. Theory. Aerodynamics of lin Airfoil lind II Wing
102
then {j2.tf/{jx2. .;- {j2q;/{jy2 identically equals 'zero. Consequently, the resultant function cp satisfies the contihuity equation. The resultant potential from the source and the sink is
= (qI2n})n (r/r2)
Since r l =V(x+e)2+ y2, r2=V(x-l::)2i-Y\ then In*={ln
~;~:~:~:: .0rln~=+ln[1+ (I-~:+Y"J.
The value of e can be chosen so that tile second term in the brackets will he small in comparison with unity. Using the formula for expansion of a logarithm into a series and disregarding the small terms of the second and higher orders, we obtain
(2.9.19)
Assume that the source and sink approac.il each other (e __ 0) and simultaneously their strengths increase so that the product q2e at the limit upon coincidence of the source and sink tends to a finite value J!. The complicated Row formed is called a doublet. the quantity M characterizing this Dow is called the moment of tbe doublet and the x-axis-the axis of tbe doublet. Going o\'er to the limit in (2.9.19) for cp at e __ 0 and 2qe -- M, we obtain the following expression for a doublet M x ({l=2i". x3+yt
M
cos6
=zn'-,-
It coincides with (2.9.16). Henre. the Dow being considered. characterized by the complex potentlal (2.9.15), is a doublet. This can also he shown if we consider the stream function of such a combined flow tllat will coincide with (2.9.17). Differentiation of (2.9.16) yields the doublet velocity components:
~~
= V/"= _
~
•
c~~6
,
+.-;-
= Va = _
~~
.
s~~O
(2.9.20)
Let us consider the three-dimensional ease. For a flow produced by a source and a sink of the identical strength q located along the axis Ox at a small distance s from the origin of coordinates. the potentlal function in accordance with (2.9.14) is cp=
where
4~
(*-
r2 = y2 -:- Z2.
~l)= 4~ [J/(~-:)I+rl
Jf~] (2.9.20')
Ch. 2. Kinemlltlc~ of " Fluid
103
For small values·of e, we have If = (ql'm) 2XE./(x·!
+ r~)3/t
Henct' in a limit pro("·ess with E __ 0, cOlls](iering that the prodnct q2e tends to the fmite limit .'\1, we obtain [or the !low produced by a doublet with tht' moment .11 the v"locity potential
cr : . .:;
(.11/1:1) xl(x2
-;... r'!)~/Z
(2.9.21)
or (2.9.21')
Circulation Flow IVortew..
Let
11S
consider a now
~et
by the complox potential -= -ai In cr
HI (cr)
(2.9.22)
where a is a constant. We can write this equation in the form qJ + i't -ai In (rei8 ) .,.-, a (9 - i In r) Hence ~ =
11'
aB -a l.n r
(2.9.23) (2.9.2/1)
It follows from (2.!1.23) th(lt the rallial ,"elocHy component Fr = = iJ'i,/iJr c"", 0, while the component V, normal to tll" radins equ
(2.9.25)
'Ve obtain lhe equation of streamlines (pathlines) from the condition"¢ ",-. canst, from which in accordance with (2.8.24.) we nnd the equation r . .,. (,OLlSt.. This eqnation represents a family of streamlines in the fOfm of l'.oneenlric. cil'cles. Flow along them is positi\"e if it occurs (,Ollnlerclockwise (from the x-axis to thl' y-axi.s); ill this case the coeflicient a in (2.9.25) has the sign pIllS. A flow in which the particles move (circulate) along concentric circles is said to be a circulation Row (Fig. 2.9.4). The ('ircnlation of the Yeiocity in the flow being considered i~ r = = 2",,. d\f!{)s. Since Bifid.\" air, we have r .... 2:1:a, whence a = = r/(2:t). Hence, the pitysi('.al meaning of the constaut a is that it..."1 value is determinl'd by the circulation of the flow which. as we have alread:\' I'stablishcd, equals the vorticity in turn. The flow produced by a vortex located at the origin of coordinates where V.t = aIr __ co is also called a two~dimensional vortex source or a vortex. '0
FII·1....
A circulation flow (point vortex)
We have treated the simplest cases of flow for which we have exaetly determined the velocity potentials and stream functions. By combining these flows, we can. in definite condition!'!, obtain more complicated potential flows equivalent to tbe ones that appear over bodie!'! or a given configuration. Let tI!'! take as an example a flow formed by the superposition or a plane-parallel now moving in the direction or the x-axis onto the flow due to a doublet. The complex potential or the resultant twodimensional incompressible now is obviously W (a) ~ W, (a)
+ W, (0)
~
Va
+ (M/2n) (1/0)
In accordance with this expression ~ + i1jl ~ V (x + ig) + (MI2n)/(x + iy) whence ~ ~ Vx + (MI2n) x/(:i' + g'), 1jl ~ Vy - (MI2n) g/(x' g')
+
(2.9.26)
(2.9.27)
(2.9.28)
To find the ramily of streamlines, we equate the stream function to a eonstant: (2.9.29) ~ Vg - (MI2n) g/(x' + g')
c
According to this equation, the streamlines are cubic curves. 0 corresponds to one of the streamA value of the constant or C lines. ByEq. (2.9.29), we obtain two eqllationsforslu·.b a zero streamline: (2.9.30) y ~ 0, :i' + g' ~ M/(2"V) Hence. the obtained streamline is either a horizontal axis or a drcle of radius To = fM/(2,nV)J1/2 with its centre at the origin of coordinates. Assnming that the velocity V = 1, and the doublet moment M =- 2n, we obtain a streamline in the form of a circle
Ch. 2. KiMmlllics of II Fluid
1O~.
with a unit radins. If: we assume that this cirde is a ~olid Doundary surface, we can consider an incompr{'ssible now Ilear this surface as one flowing in a lateral direction over a rylinder of infinite length with a unit radius. The velocity potential of sllrh an ineompressible disturbed flow is determined by the first of Eqs. (2.fL2S) having the form lp :-'
X
11 -\-
11(x2
+ y2)]
Introducing the polar (',oordillates 8 and r we obtain ~ ~ ,(1 ~ II,') cos 0
(2.\.1.31) x;ro.~
0 - (x 2 -I-
y2)1/2,
(2.9.3q.
Differentiation yields the components of the ,'cJocity at a point. on an arbitrary streamline along the directionH of the radius rand of a normal .~ to it: V~ ocr/or = (1 - lIr 2) r,os e, V, ~ (II,) a~Ia9 = - (1 + II,') sin 9 (2.9.32)On a cylindrical sHrface in a now (a zero streamline) for which r .; 1. we flJld F r - 0 and y., -2 sin e. The above example illustrates the application of tile princ.iple of now superposition and the concept of a complex potential to the solution of a very simple problem on the now of an incompre.'lsible nuid over a hOlly,
3 Fundamentals
of Fluid Dynamics
3.t. Equations of Motion of a Viscous Fluid The fundamental equations of aerodynamics include equations of motion. forming its theoretical foundation. They relate quantities .determining motion such as the velocity, normal, and sh.ear stresses. The solution of the equations of motion allows oue to determine these unknown quantities. Let us consider the various kinds of equations of motion used in 'Studying gas flows. CIIrtesran Coordinates
We 511311 treat the motion of a fluid partida having the shape of an elemelliary parallelepiped with the dimensions dx, dy. and dz ~onstructed near point A with the coordinates x, y, and z. Let the velocity componenL<; at this point be Vx , Vv ' and Vr A fluid particle of mass flT (here .. "'""" d.r dy dz is the elementary \'olume) moves under the action of the mass (body) and surface forces. We shall denote the projections of the mass force by Xp"t, Yp"t, and Zp"t, .and those of the surface force by PxT, Py"t, and P:"t. The quantities Po;, P y' and P z arc the projections of the surface force vector related to unit volume. The equation of particle motion in the projection onto the x-axis has the form
whence dVxldl '" X
+ (lip) P,
(3.1.1)
where dV,/dt is tile total acceleration in the direction of the x-axis. We obtain similar eqnations in projections onto the y- and z-axes: dV,ldl ~ Y + (lip) P, (3.1.2) dV,ldl ~ Z + (lip) P, (3.1.3)
Ch. 3. Fundamental5 of Fluid Dvnamics
107
FI.3.U
Surface forces acting on a Buill particle
The surface force can he expressed in terms of the stresses acting 011 the faces of an elementary parallelepiped. The difference in the surfar,e forces in romparison witll an ideal (ill\'i~dd) now r.onsists in that not only narmnl. hilt al."o shear strcs.<;{'~ act 011 the fares of a partide. . Every surface forre a(·tiug on a face has three projeclions onto the coordinate axes (Fig. 3.1. t). A unit of snrface area of the leflhand face experiences a surfare fOI'('o \vllO~e projl'('.tirms will be de!lignated hy PU' 'lx:' and Txy. The quantity P.>:", is the normal stress, while T.u and 'l.\'y are the she;tr stresses. It cau be seen thl\t t.he lirst subscripl indicates the axis p4"rpendicuiar to Ihe face being con!:lidered. and the second one indir_ates t.he axis auto which the given stress is projected. The rear face perpendicular to lhe z-axis expel'iencO.<; the ~tress comlJOuenl.. PZf' 'l :_~. and '('z,,: the boUom fare perpelldicular to the Y-Ilxis expl"rien('cs the components Puu' '('y.>:. anci '(',f'
We shall consider normal stresses to be positive if they are lIiteded out of the clement being studied and. consequently, subject it to omnidirecLional tension as shown in Fig. 3.1.1. Pot'Jitivc shear stresses arc present if for the three faces interseding at initial point A these stresses arc oriented along directions opposite to the positivc directions of the coordinate axes, and for the other three facesin the positive direction of these axes. With this in view. let us can· sider the projerliolls of the surface forces onto the x-Ilxis. The lefthand face experienc.es tnc force dne to the normal stress -P:u: dy dz. and the right-hand one-the force Ip.>:", + (iJP",:rJi).E) drl dy tlz. 'fhe ret;ultant of these forces is therefore (iJP;x./iJ:r) d:r dy dz. The components of the forres produced by thl" sheal' stresses actiug on these faces are zero.
108
Pt.
r.
Theory. Aerodynamics of an Airfoil and a Wing
Account must be taken, h9W-e",erj of the shear stresses 'lzx and 'lyx' The rear face experiences the force- -Tzx dx dy, and the front onethe force ['In: + (i1T:z:Jaz) dzJ dx dy. The resultant of these forces is (iTrzx/az) dx dy dz. In a similar way. we find the resultant of the forces acting on the bottom and top faces. It is (a'llIx/ay) dx dy dz. Bence. the projection onto the x-axis of the surface force related to unit volume is Px :- apxx/()x
+ 8TlIx /8y
-I- 8'lzx/az
(3.1.4)
Similarly, the projections onto the other coordinate axes of the sllrfaee force related to unit volume are P y = 8'l:.:y/ax..!. apl/y/ay...;. iJTzy/aZ P z = a'lrz/ax+ rJTyz/ay -I- iJpzz/8z
1
(3.1.4')
A(',eording to the property of reciprocity of shear stresses, the values of those stresses acting along orthogonal faces equal each other. i. e. 'lz:" -= 'lxz. 'lZIl c-::: 'llI l • and 'llI r T rll • llence. of six shear stresses, three are independent. To determine the values of the shear stresses, we shall lise the hypothesis that stresses are proportional to the strains they produce. The application of this hypothesis is illustrated by Newton's formula for the shear stress appearing in the motion of a vise,Otls fluid relative to a solid wall. Dy this formula, 'ly.'I' -= Il (aVyf8x). Le. the stress is proportional to the half-speed E z """ 0.5 (av II/ax) of angle distortion in the direction of the z-axis, whence 'llIr =-: 21lEz (here Il is the dynamic. viscosity). This relation covers the general case of threedimensional motion when tho angular deformation in the direction of the z-axis is determined by the half-speed of distortion E z = '-,- 0.5 (8Vy/ax + aVr/ay). We shall write the other two values of the shear stress in the form T IIZ ----= 21lEx and Tzx -:: 21lEII' Consequently. -,---0
Tyo;;
'llIz
'In
=
21lEz =].1- (8VII /ax
+- avx/ay)}
= 2.u:%:-.=].I- (~VII/8Z: avz/ay) = 21lEII ~-=].I- (ovz/ax, av x/a:)
(3.1.5)
We shall use the ahove-mentioned proportionality hypothesis to establish relations for the normal stresses Pn, PIIII' and PZZ' Under the action of the stress Px.n a fluid particle experiences linear slrain or deformation in the direction of the x-axis. If the relative linear deformation is a;, then Prx -~ E~. where E is a proportionality factor or the modulus of longitudinal elasticity of the fluid. The normal stresses PIIII and Pzz cause the particlo to deform also in the directions of the y- and z-axes, which diminishes the deformation in the direction of the x-axis.
Ch. 3. Fundamentals 01 Fluid Oynemics
109
It is known from the course in the strength of materials that the decrease in the relative magnitude 6.G:T of this deformation for elastic bodies is proportional to the slim of the relative deformations in the directions of .II and Z lInder the action of the indicated stresses. Ac("'ordingly,
6.0x
,= -(11m) (Pu/E
+ PalE)
where m is a constant known as the coefficient of lateral Hnear defor· maUon. The total relative deformation in the direc-tion of tho x-axis is
e;
~ 8~ -1- &8,"" 1',.,1£ - 111m) (p"IE -1- p"IE)
e
(3.1.6)
a.
We can calculate the relative deformations y and along the y. :and z·axes in a similar way. The obtained expressions give us the normal st.resses:
Pn~E~.,(p,,+ p,,)/m PY!I """ E6 y + (Pu -+ pxx)/m PH =
I
(3.1.i)
Ea. + (Pu + P!lu)/m
The relative linear deformations of a particle along the directions Qf the coordinate axes determine its relative volume deformation.
Designating the- magnitude of this deformation by
0,
we obtain
(3.1.8) Summating Eqs. (:t1.7) and taking into account the expression for 0: we have PX.T -:- PUJI
Determining tho sum PUy ing it into (3.1.6), we find
+ PH
= mEfi/(m -
2)
(3.1.9)
+ PH from this expression and introduc(3.t.tO)
For our following transformations, we shall use the known relation between the shear modulus G, the modulus of longitudinal elasticity E, and the coefficient of lateral deformation m that is valid for elastic. media including a compres..<;ible fluid:
G
~
mE/12(m
+ I)]
(3.1.11)
Substituting this rolation into (3.1..1.0)J we obtain
Pxx= 2G8:c+
m~2 8
(3.1.12)
110
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
Let us introduce the symbol (3.1.13) "7r = (Pxx + PIIY + Pu)/3 Adding and subtracting on the right-hand side of (3.1.1Z), we have p:x:x=a+ZG6 x + m.2~2 e-(px';+PlIy·-!-pu)/3
a
Transformations involving Eqs. (3.1.9) and (3.1.11) yield p" - cr + G (28, - 28/3) (3.1.14) Similarly, p" _ + G (28,- 28/3); Pu - +G (28, - 2il/3) (3.1.14') It has heen established for an inviscid fluid that the pressure p at any point of the flow is identical for all the areas including this point, i.e. with a view to the adopted notation, in the case being considered we have p""" = PIIII = Pn = -p; consequently, P = - - (p" + P" + Pu)/3. Hence, when studying the motion of a viscous fluid, the pressure can be determined as the arithmetical mean, taken with the minus sign, of the three normal stresses r.orresponding to three mutually perpendicular areas. Accordingly, := -p. The theory of elasticity establishes t.he following relations for the shear stresses acting in a solid body: 'til'; = Gy yX ' 't IlZ = GYIIZ> 't zx = G'V:,; (3.1.15) where 'VIIX' yy:. and 'Vu are the angles of shear in the directions of the axes z, x, and y, respectively. A comparison of formulas (3.1.15) with the corresponding relations (3.1.5) for a viscous fluid reveals that these formulas can be obtained from one another if the shear modulus G is replaced by the dynamic viscosity Il, and the angles of shear y by the relevant values of the speeds of distortion e of the angles. In accordance with this analogy, when substituting Il for G in formulas (3.1.14) and have to (3.1.14'), the strains (relative elongations) e,;.6 11 , and be replaced with the relevant values of the rates of strain 9" = by = lJV,;llJx, 9 11 = lJVyllJy. 9: = aV:llJz, and the volume strain the rate of the volume strain 9=fJV:/lJx+lJV ll lay + lJVz/Jz =div V. The relevant substitutions in (3.1.14) and (3.1.14') yield:
a
a
a
e:.
e
P,;:r:=-p+v.(Z 0:Xx --}divV) pyy=-p+.v.(2 0:VY --}divV) PZI= -P+1l
(z
O~' --}div V)
(3.1.16)
Ch. 3, Fundlimenfliis 01 Fluid Dynllmics
111
The second terms on the right-hand sides of expressions ,(3.1.t6} determine the additional stresses due to viscositv. By using relations (3.1.4), (3.1.4'), (3.1.5), and (3.1.16), we can evaluate the quantities P x , PilI find Pl' For example, for P x from (3.1.4) we obtain
Px=-*+*[~(2 8Jxx -{-divV)J+*[~l( &::11 + a~x)J +*[~ (a~z + a:z%)]= -*+~liV. +t-~ div V + :~
(2 aJ.x --}diV v)
where
li
+*
(a:ux + 8:: ) +~ (a:sx +8;;)
is the Laplacian operator: li iP/ax'J. + a'J./ay1.
+ 82/az"
Particularly, liV.~
-eo
a2VxiaX~
+ iPVx /{Jy2 + a2v.",'oz'J.
In a similar way, we ran lind the expressions for P II and Pz' The relations for the stresses due to the surface forces in a fluid have been obtained here by generalizing the laws relating stressesand strains in solids for a fhlid having the property of elasticity and viscosity. We shall obtain the same relations proceeding from a number of hypothetic notions on the molecular forces acting in • fluid (see [9, 1\1). By compiling projections or the total accelef expressions found for P"" P y , and P r , we can obtain the equations of motion (3.1.1)(3.1.3) in the following form:
8~x+Vx 8:;~ +Vy a;/ .. it·: a;;",
-~.vdVx+f.+XdiVV-:+[~~
(2
==-x-+.-;;-
a;~x
+% ( a:;~ + Q;~, )+ ;~ (8:;T
-+diVV)
)J a;;" -r-Vr a~" =Y-f.* +vliVy-l"Y.*divv+f[*{2 8:; --}diVV) T
a;:~
a;~y ~,v,t a:~y +Vy
+*( a;:u + ~:")+ ~~I (iJ~y -[- iJ:Vx)J
I
(3.1.17)
~.
I. Theory. Aerodynamics of
q~Z +V"
aa:. +Vy a:u.
til"
Airfoil lind II Wing
+Vr a::~
=z-+.
~~
+\I~V:+i.~divV++r ~~ (2 a:~. -fdivV)
+ :~ ( :: + a:;" ) + :~ ( ~~. + a~u )] t
where " ~-:: J.l/p is the kinematic viscosity. The dillerelltial equations (3.1.17) form the theoroticalfoundation -of the gas d ),oamics of a viscous compressible fluid and are known as the Navier-Stokes f'qu8tions. I t is assumed in the equations that the dynamic viscosity 11 is a function of the coordinates x, y. and %, i.e. 11 = t (x, y, z). Presuming that /..L = const, the NavierStokes equations acquire the following form:
d:r x =x=+.*+'V~V",-!-f·*diVV
d:!J '=Y-*'*7v~Vu+T.*divV
a:r =Z--f,.. ~~ z
+\'~V~+~.~diV
(3.1.18)
V
When studying gas nows, the mass forces may be left out of ac~ount, and, therefore, we as..'iume that X = Y = Z -= O. In this -case, we ha\·e
~=-...!..~+v~V -C..~.~divVl dt p ih >: ' 3 QX dVy
=-.!.. . .it.+vl'.VII ..L~ • ....£....divV p ay . 3 iJy
d:'r
=
dt
-t'*+V~Vr+-i-''*
divV
I
(3.1.19)
J
For two-dimensional plane motion characterized by a varying -dynamic viscosity I' =F const, equations are obtained from (3.1.17) .as a result of simple transformations:
.!!i. at ~ _L.!'.E. p f J x+L...'C.. p f J x [~(2 ~-~diVV)] fJx3
1
+f·-;;(J.lez) fjf
1
fJ
1
fJ
8V
2
~~-..L+_._[~(2-2__3 divV)] fJt p iJll P 811 811
+f·-:X (I'ez)
1(3.1.20)
Ch. l. Fundamentals of Fluid Dynamics
113
When div V = 0, and 1.1. = canst, these equations are reduced to equations of motion of a viscous incompressible fluid. Let us consider the equation of motion of an ideal (in viscid) compressible Ouid. Assuming the coefficients ~ and v in (3.1.17) to be zero, we obtain:
8~x +Vx 8~x +V"iJ~:J:
-I' V"
8::
+V"
8:'11
+V:x;
~fI
8J,z ..i- V:x; 8'fzz
+V"
a: =x-f·* 1 z'"
8;'Y
+ Vy 8Ju' + V, 8~"
=Y-f·* =Z-f·*
t (3.1.17') J
These equations were first obtained by Leonard Euler, which is why they are called Euler equations. They arB the theoretical cornerstone of the science dealing with the motion of an ideal gas whose hypothetic properties are determined by the absence or negligibly small influence of viscosity. Vector Form
01 the Equations of Motion
By multiplying Eqs. (3.1.18) by the ullit vectors i, i, and k. respectively. and then summating them. we obtain an equation of motion in the vector form: liVid.
=G-
(lip) grad p
+ v f1 V + (v/3) grad div V
(3.1.21)
where the vector of the mass forces in Cartesian coordinates is
G = XI + Yj +Zk the pressure gradien t is grad p = (liplliz) I -I- (liplliy) j + (liplilz) k the vectors
av = aV;oi + aVyj + aVzk grad div V (Ii div VlOz) I + (Ii div V/liy)j + (Ii div V/Ii,)k II a fluid is incompressible, div V = 0, and. consequently, dVld, = G - (lip) grad p + v f1V (3.1.21') In the absence of mass forces, G = O. therefore dVld, = - (lip) grad p + vf1V + (v/3) grad dlv V
(3.1.21")
The vector of the total acceleration can also be expressed as dV/d. _ liViIi. + grad (V'12\ + curl V X V (3.1.22) 8-011U
114
Pt. I. Theory. Aerodynllmics of 111'1 Airfoil lind II Wing
With account taken of (3.1.21") and (3.1.22), we can write the equation of motion as
~: +grad ~-+ curl V xV= -fgrad p+ v6V +igrnd divV (3.1.22')
For an inviscid compressible nnid, the equation of motion is
~~ +grad'::;;" + curl V X V = --/;-grad P
(3.1.22")
Curvilinear C_rdlnates
Let us transform Eq. (3.1.22') using the concept ot generalized curvilinear coordinates qn' This allows us to go over r,omparatively simply to eqnations of motion containing a specific form of curvilinear orthogonal coordinates, as was done with respect to the continuityequation. Let us consider the transformation of separate terms in (3.1.22'). :For the second term on the left-hand side, and also for the first and third terms on the right-hand sides, with a view to (2.·1..18), W(J have the following expressions: ,
grad
~=
h
(grad';:
n-I
Li
3
n=
~ ~. a~~~2)
in
(3.1.23)
n=1
3
3
g"dp~."2; (gradp),i,~2; -hI '-.'P i, ,,~I
,,=1 3
3
,,=1
n-.I
graddivV=~ (graddivV)"i,,=h
n
-&-.
(3.1.24)
q"
a~~:v
in
(3.1.25)
For transformation of the vector product curl V )( V, it is necessary to Hnd the form of the exprc;.;sion of the vector eurl V in generalized coordinates. For this purpose, let us calculate the curl of both sides of Eq. (2.1.19): 3
curl V = L~ (V n curl i" + grad V nXin)
(3.1.26)
n=1
where (3.1.27) grad V" )( ill ,.. (grad Vn)Ji m - (grad V,,)mij Here the projections of the vectol' grad Vn onto the relevant eoordinate directions are determined from expres'ion (2.1.18) with V" substituted for <\'>. Introducing (3.1.27), and also the expressions for curi in from (2.1.22) into (3.1.26), we obtain the following ex pres-
Ch. 3. Fundllmental$ of Fluid Dynllmics
sion for the curl of the velocity: curlV:.-= hzlh3
>:
[1l(:;~3)
[a(;;:l) _ a(!~:'3) ]i2-! h:"2
-
11~
"I:;:.) }I-T-~ .- ()(:;~:\) :1 i3
[J(~~:")
(3.1.28}
Consequently. the erosr- pl'Oduri 3
cud V)(V=
L
n=t
(cud VXV)" '"
(31.29)
where the projection of tllis veetOI' 011 to a tangent to the corresponding coordinate curve is (curl VXV)n"'"
h~hi" [(IU~~")
_
iJ~;~J)
J (3.1.30)
Rerall that the reialioH as follows'
U('I\\'C{,H
j • ••
(he subscript!! m, j, and n is
2
3
3 1
1 2
The left-hand side of Eq. (:i.1.:!2) is the vcctor or the total alion W . dVldt thnt has t.he form
acceler~
,
\V="~l W"i"
(3.1.31)
where W" is the projcction of the fll'celE'l'atio!l H·ctor onto the direction of a tangent to the coordinntc line q". Each quantity W" ean be eonsidered as the slim of the relevant projections of tile vectol's aVld/, amI also of the vcctor~ (3.1.23) and (3.1.29) onto the indicated directions. Aerordillgly,
Wn~~ iJ~"
+ h:~n
.
iJ
+ ~: . ~~';, + h~;m
(~~~n)
-
h:;m •
~::
-
.
a(;;::n)
.
h~t :;~
(3.1.32}
Let u>; apply the Lap/adall operator to the vector V, and use the expression
,
.6,V= ~ l!Vnin=graddivV-cllrlcllriV ,,=1
(3.1.33)
where dF n arc the projeetions of Ihe vector along the coordinatelines qn'
Pt. I. Theory. Aerodynam!c$ of lin Airfoil and a Wing
116
The first vector on the right-hand side of (3.1.33) has been determined in the form of (3.1.25), while for calculating the second one Eq. (3.1.28) should be used. Taking the curl of both sidos of this equation. we obtain
, ,
curl curl V = ~ (curl curl V)"i n
._1
=~ "~hJ{<'I"J~:IVlJl
a[hm(~:;lV)ml}i"
(3.1.34)
.~I
where the corresponding projections of the vector curl V are found from (3.1.28). Having these data at hand, we can consider the transformation of the equations of motion as applied to specific forms of curvilinear orthogonal coordinates. C,IIJMlrle.' CooJCIIMfes
In accordance with (2.4.12), (2.4.16), and (2.4.25), we have ql = Z, q'l. = r, qs = y, hi = t, h'l. = t, ha = r, VI V~, Vz = Vrt Vs = V,. Consequently,
W = 8:t '+,Vs 1
8~s
+v,"8:rr +~.
a:"\~;
(grad
P)l =
ClplClx
Next we find (grad div Vh = CI diY V/ax Since div V is determined by formula (2.4.26), we have
8d:;V =i; ("~s + a;; ++. a:; +.!f) From (3.1.28), we have (curl Vh =
8Jzr - "::
(curl V)'I. =
+(0:V:c - r
~)
Introducing these expressions into (3,1.34), we obtain
(curl
,.urIV)I-+{'H~-~)] '[W;:-.~)]} ••
=
iJy
::ir - o;~: ++ Uat- ':: )_+.- (O;~: -r ::~;)
Ch. 3. Fund.mentals 01 Fluid Dynamics
117
Since (grad div V)l ~ iJ div Vlox in accordance with (3.1.33) we have (dVh = (grad div Vh - (curl curl Vh j
-7- aa2~:c +~.a;;z~++_
_ a;;.'C
a;;
\Vc tIm:! compile an equation or motiou in a projeetion onto th' x~axis
of a cylindrical coordinate system:
8~; '~-V:'~'¥'Vr~':~' D;y' ~=
-to :: (3.1.35)
We obtain the oLher two C(luat.iolls in projections onto the coordinate lines ramI y in a similar way:
~+ V:c D;: +Vr =
a;; -7-~' a:,: -
_.!. !.1.....1...\-. (.1.V p
Dr
'
1
~~
_.!..~_.:r..)-l_":::'. ddivV r
iiy
1'2
1'2
'3
Ii
or
.!2.+V !!!..1...V av? ¥L.!:L.~+ r,vy 8t ~a:·r8r·rfh' I'
(3.1.35')
= -f,:.*+v (flVy¥i_~. ai)~ -;r) + ir· Dt~~VV J In these equations, we have introduced a symbol for the Laplacian operator in cylindrical coordinates: .1.
=;;-t :,22+..!,. ~: ++-fr
We determine the divergence of t.he velocity by formula (2.4.26). For an axisymmctric Row. the equations of motion are simplified: aV:':.i-V iJt
.
:.:
DV~_:...V 8V:I::=_.-!._~-t--'\-'.1.V (Jz
.
r
(Jr
p
iJx.·
+....!... (JdivV .
3.%
r -i' V:. ~"";'Vr~:":'~ -..!....-~-:...vflVr ov iJt 0:': Dr p Dr' I
+-i-' adi: v
:.:
I II
1
(3.1.36)
J
where div V :.= iJV"liJx + iJVr/{Jr + Vrlr; fl = iJ 21iJ:x;'! + lPliJr" + (1Ir) alar. For a steady Row. one must assume in the equations that aV"Iat - aV,lat - av,tat _ 0
+
118
PI. I. Theory. Aerodynamics of an Airfoil and a Wing
Spherice' Coordinates
The spherical coordinates, Lame coefficients, and the projections of the velocity vector onLo the directions of the coordinate lines are related by formulas (2.4.13), (2.4.17), and (2.4.33): ql =-: r, (/2, ~ e, q3 .-" tp, hi -= 1. h z = r, h g ""=' r sin e. VI
. Vn V z
~
Vo, V 3
0...
V¢
According to these data, from (3.1.32) we find the projection of the acceleration onto the direction of the coordinate line r;
WI =
a~r + Vr
a;r
.l-..!! .~+r~;~a'~- ve~v; (3.1.37)
The projection of the pressure gradient is (3.1.38) (grad ph = 8pl8r Wilh a view to the expression for div V (2.4.34), we lilld the relation (grad djv V). ':=: v
ad;;
+, [~.a(V;:2) +rs:nu.~+rs:ntl. a~wJ (3.1.39) We shall lise this relation for determining the projections (11 Vh of the vector 11 V. To do this, we shall calculate the valne or (curl .curl Vh in (3.1.33). Fl·om (3.1.28), we have (curl
Vh =
+ (!:r) - a: J' [a
11
(curl
Vh =
rs:n
0 [~_
a(V¢~:in 0) ]
lntroducing these relations into (3.1.34), we obtain
'"
(3.1.40)
With a view to expres5ions (3.1.39) and (3.1.40), we find
(I1V)j = (grad div Vh -(curl curl V)j =I1V r _ 2~': (3.1.41) where
I1Vr=~·-:r (r2 a;r) + rls~lle'~ (sinS a:a')
+ Si~2 O. ~2:: ri
(3.1.42)
Ch. 3. Fund.!lment.!lls of Fluid Dynemic$
119
Taking into al'Coulit (3.1.37)-(3.1.3H) and (:U.!J1), we find the eqnation of motion ill a projeclion onto lhe toorriinal(' line r:
iJ;/~"J',u~;'i-~' _ r~~ rJ -- r!
°:0+
f.* + v
=
s~n U • J~~~.
_
2:: _
r:;~o' JJI~~
--h-. J~~o
(AVr
2Vllr~OlO) ~_
T' a d~; \.
(3.1.43)
We obtain the olher two equations in a similar' way:
~
iI,:·ro ..; ~. d~;~
;- V r
J1rl'o-:'~COlfl -
r2
__
+,.~
~~ n . o;~~
(AYo+-#r. o;j
~';~2 t! - r~ ~~~~AU' ~JI~~') + -1-.
O!';r ':-Yra~>
I'~. J;'~'
l'",(!·r-·· '"0 ("otO)
·:·v
r
-;-V
·1·
-+-.
1 '""' -pr sin
(AV~--r~~;~tu~ r2$~n9' ~I;
{J
(~~ \'
r!;~e'~
t
(3. t.43')
iJp
f)'
aif
r;~~:}~ iJnl~~)
,. ildiv \' ...,- 3r sin 0 '----aij:""
where tl](> Lapiaciull opel'alor in spherical coo['dinalc-s is 11
-~ ~ . .;.;. (r~
-:r) + r~
$:11 t:I •
-to (sin 0 -k) . r! ,'1111" o· ~ (a.1.4,'f)
For lwo-dimellsiollal l-lpatial gas nO\\'5 charal"terizNI uy a change in the parameters (\'elodty, pressUI'e, density. e\c.) in the riirection of only two {~oordilJale lines, let liS write the eqllation of motioll - a simpler form:
aV r
.
dt~~-
+v
--.;..~ _
~-
au .
21'; _ 21'ge~)tO) r • r
-I-...!..... &div V
~-i ~.~+ VrVO ... .~ ~-.-V at r ar r ali r -
+v
1
' \'(1 in', \"~ l"p ra, r-;:-'&o--,=-p'o,
V ar r
(CiV r _
_..!....21I
3
br
rp
(CiVo -:--'}. a~j -r2~~20) +*.bd~;\"
bO
J
(3_t.45)
HIO
Pt. I, Theory. Aerodynamics of an Airfoil and a Wing
where
Il=-J,.f, (r2
-!r) +
r<si1nll
.-:0 (sin e ~)
diVV"""~. U{~;r2) i ~.a(\'9a~inO)
(3.1.46) (3.1.47)
Equations of Two·Dlmenslonal Flow
of a Gas Hellr II Curved Surface
For the Case I eing ton..,idereu, the curvilinear coordinates. Lame coefficients, 311d Yelo('ily ("omponents are determined by formulas (2.4.40), (2.4.12), and (2.4.43), respeC'th·ely. We shall adopt the further condition that motion occurs near II. wall. and. consequently. y« R. Accordingly, q1 = X, 12 = y, q3 = y. hI = 1, h, = 1. h$ = r VI = dx/dt = V=, V, = V r, V, = 0 With a view to these data, the acceleration component is
W,
OV,lal
+ V, OV';O$ + V, aV,;ay
and the projection of the preSfmre gradient is (grad ph = fJplfJx
(3.1.48) (3.1.49)
In addition, (grad div Vh
=
(3.1.50)
iJ div V/iJx
where the divergence of the velocity has been determined by relation (2.4.44). Consequently, (graddiVV)I=-1;{+[O(:x"r),.
O(;r
l ]}
(3.1.51)
We use (3.1.28) to calculate the projections 01 the velocity curl vector: (3.1.52) (curl V), = fJV/fJx - iJV,,/fJy, (curl Vh = 0 Introducing these expressions into (3.1.34), we obtain
(curl curl V),=+.fu-[r (~_
U;y=)]
(3.1.53)
With a view to relations (3.1.51) and (3.1.53lt we have (.6.V), = (grad divV)I- (curl curl V),
=fz{+ [u(::r) (3.1.54)
Ch. 3. Fundamentals of Fluid Dynamics
Similarly, when considering the coordinate line
Q2,
+ v x aVylax + l'y av /ay = aplay
W z = aV,,,at
(3.1.55) (3.1.56) (:·U.57)
(grad ph (~Vh
(grad div Vh --: a div v/ay = (grad eli\' Vh - (nl1'l curl
12t
we have
vh
(::r) ]}-+.:z [r (a:: _":;T) J(3.1.58)
= -i;-{+ [8<:t) + a
Using the obtained relations, we ha\'e:
a~:t + Va:
"!: + V" a:V:t ~:~ -f'"*--: v(L\V).
I
V:t~+Vy~=- -+.*-:'v(~Vh
I J
7T'~
a~~
-t
1
a div \'
'Y
.
,.
8,1 vV
-:'T'---ay
} (3.1.59)
where (~Vh and (~Vh are determined by formulas (3.1.54) and (3.1.58), respectively, We have thus obtained various forms of the equations of Iliotion for a viscous liquid. Experience shows that this is needed because in some cases, when studying thc laws of interaction of gas flows with bodies in them, it is com'cuicnt to use one form of the equation, and in others, a different form. l.l. Equations of Energy and Diffusion of a Gas Diffusion Equation
The motion of a dissociating viscous fluid can be ill\'estigated with account taken of tIle influence of gas diffusion on this mot.ion. This is expressed. particularly, ill that diffusion i~ taken into consideration when deriving the equation of energy-one of the fundamental equations of gas dynamics. By dlft'usion is meant the levelling outoftheconcentration because of the molecular transfer of a substance. This is a thermodynamically irreversible process, and it is one of the reasons why a gas in motion loses mechanical energy. The diffusion oquation is an equation of the transfer of the i-th component in a gas mixture (it is a continuity equation for t.he same component).
122
Pt. I. Theory. Aerodynollmics of an Airfoil oIInd
011
Wing
To simplify onr investigation, we can assume that the intem,ity ·of thermal and pressure diffusion is negligibly small and determine the diffusion flux of the i-th component ill a direction n by the ·equation (3.2.1) where Ci is the concentration of the i-tll component, and Di is the diffusivity \.hat determines the diffusion flux when a concentration gradient is present. For a mixturo of gas components, we mnst take into eOllsideratioll the binary diffusivities corresponding to each pair of components, for example, to oxygen atoms and molecules, or to nitrogen atoms and molocules in the air. In approximate calculations, we ·can proceed from a certain value of the binary difiusivity TJ that is the same for all the pairs of component.s. Taking this into account, we have Qi.rI.n...." -pD oc;lon. Considering the directiOlls x, y, and z along which diffusion occurs, we have Qi.d.x = -pD oc;J(}.r, Qi.d.U .; -pD (}c;l(}y, Qi,d .• -: -pD (}c;loz (3.2.2) ·or in the vector form (3.2.3) Qi.d -, -pD grad Ci The diffusion of a substance is directed into a region with a reduced concentration, thorefore (}CiJiJn has a negative sign. Since the -right-hand side of (3.2.3) contains a minns sign, the quantitr Qi.d is positive. Let liS eOllsider the derivation of the diffusion equation in a erlindrical c,oordinate system assuming the motion to be steady. ,"Ve shall assume the flow to be three-dimensional and symmetric ahout the x-axis, i.e. such in which the velocity component Vl' .....:: O. Let liS separate an elementary volume of tho gas in the form of a ring with dimensions of dr and dx (Fig. 3.2.1) constructed near point P whose coordinates aro x and r and whose velocit}' components are V", and V r . \Ve shall assume that the substance diHusos only in a radial direction. ConsequenLly, the nux of the i-th component through the internal surface of tlle element is mr -= 2nrflV.ci dx ~ + QI.d2nr dx, where Ci and Qi.d are the c.oncentration and the diffusion flux of the i-th component, respectively, ovaluated per unit area. The flux of the i-tll component through the outer surface is m r ';' a;;r dr = mr -:- a (p~;rcil2n dr dx -\- a (~I;rlr) 2n dr dx
Consequently, the flow of the component into the volume being ·considered is 10 (pV,rc,)lad 2n dr dx +110 (Q',dr)lod 2" dr dx
v,
Fig. l.l.1 An elementary gas particle in an a:dsymmetric thr<'c-dim('nsional flow
DisregRrding the clirtll~ion nux of the suhslam'('
mx
--co
2:trV~c;rdr
and through the right-hand one m~ -:- (dm.,./(Ix) d.r m~
+ Ii) (p V~rc;)ldJ·l 2:1 dr dx
lIenco, the flow of til(' component inl_o t.he vol1lme is
2:t la (p V.,(cf)iO.rl dr d:r Since the amount of ga~ in the voilime does 1I0t change, tllC total flow of the component into the volume equals its outflow because of chemical reactions. If the rRte of formation of the i-th eomponent in a unit volume caused hy chemiC-al reaetions is (W ~h)" the consumption of the component in the elementary volume is 2:t (W dl);r >: X r dr dJ.:. Therefor£', the balance of mass of tllC i-th component in the volume being considered is
8 (pl'."rcJ/aJ.: -:.
a (pV rrc;).'8r
-= -8 (Q;.dr)i8r:.... (W~hLr (3.2.4)
This equation is known as the diffusion equatiun ill a cylindricnl coordinAte !"ystem. In A similar Wily, we can obt.ain a diffusion equation in the Cartesian coordinates x alld y for u pi nne flow:
+
rJ (pV"c;)/8x + rJ (pV vc;)J8y "'"' -8Q,.dJ8y (Welt); (3.2.5) where Qi.d is found from (3.2.3). If we consider a binary mixture of atoms and molecules. then L C; ,...., CA +- CM ..--' 1 and, consequently, QA.d .~ -QM.d . We determine the value of (Weh ); for a given reaction in a dissocialing gas by the formulas of chemical kinetics.
124
Pt, I, Theory, AerodynllmiC5 of lin Airfoil lind II Wing Energy EqueHon
Together with the equations of state, motion. and continuity, the energy equation belongs to the system of fundamental differential equations as a result of whose solution we completely determine the motion of a gas. Let us consider a system of rectangular Cartesian c.oordinates and compile an energy equation for a fluid particle in the form of an elementary parallelepiped. This equation expresses the law of energy c.onservation, according to which the change in the total energy of a particle, consisting of its kinetic. Rnd internal energy. during the time dt equals the work of the external forces applied to the particle plus the influx of heat from the surroundings. The kinetic energy of a particle of volume 't - dx dy ch is (p VI/2) 't, and its internal onergy is up't (u is the internal energy of a unit mass of the gas). Consequently. the change in the total energy during the time dt is
1t [( p;1 'r- pu) '(] dt
=
p't
*
(.x;.;. u ) dt
The work of the external mass (volume) forces in the displacement of the particle during the time dt can be represented in the form of the dot product G· V multiplied by the mass of the particle p't" and the time dt. The mass force vector is G Xi + Yj -:. Zk, consequen tly, -0":
(G·V) P' dt
~ (XV, -
YV, -;. ZV,) P' dt
Let us calculate the work of the surfac(' forces. First we shall consider the work done during the time dt by the forces induc.ed by the stresses acting on the right-hand and left-hand faces. The work done by the forces acting on the lelt-hand face equals the dot product 0,,' V multiplied by the area dy dz and the time dt. In the dot product, the vector of the surface forces is 0" =-= p,%,%i -,- 't.q,j + 't:uk The work done by the surface forces acting on the right-hand [ace is
[O.zV + 8
<:;V) dz] dy dz dt
Having in view that the forces fOT the left-hand and right-hand faces are directed oppositely, we must assume that the work done by these forces is opposite in sign, for example, positive for the right-hand and negative for the left-hand face. Accordingly, the work of all tbe surface forces applied to tbe left-hand and right-hand faces is
8<:t)-Cdt=[-!;(Puv,,+t:ruVu-i-'t"IV:] tdt
(3.2.6)
Ch. 3. Fundamentals 01 Fluid Dynamics
125
We obtain expressions for the work done by Lhe ~mrface forces acting on the lower and upper, and also on t.he front and rear fates, in a similar way: [a (au V)/fJyJ • dt and [a (a,V)/fJzl
'C
dt
Considering that the vectors of the surface rorces acting on the lower and rear faces are, respectively,
obtain tlte following expressions for LllC work: a
(::V)
Q(~;")
1:
dt =
['*
.dt= [-;;:
('C~ .•Y lI:'r PUyV ~ + 'ty~Vz)] .. dt
(3.2.6')
J -rdt
(3.2.6~)
('CuV.o:+'t~yV!l+PuVz)
In expressions (3.2.6), (3.2.6'), and (3.2.6"), the stre~ses arc determined by relations (3.1.5) and (3.1.16), respectively. The inl1ux of heat to the particle occurs owing to hea\. conduction, diffusion, and radiation. Let q", dy dz (where q", is the specific hent flow) be the amount of heat due to heat conduction or diffusion transferrer! to the particle through the left-hand face in nnit time. During the time dt, the heat flux qo: dy dz dt is supplied to the particle. The heat nux through the right-hand face is - Iqx ...l. + (dq."ld.r) dxl dy dz dt. The amount of heat transferred to the partic]" tnrough both. faces is - (fJq,,/fJx) 'C dl. We oMain similar expressions for tile faces perpendir.1I1ar to the y- and z-axes. Hence, the total heat flux transferred t.o the particle is - (aq:Jdx + fJq~/fJy '';-
+ agz/az) • dt.
If we consider the supply of heat caused by conduction, the specific heat nows, equal to the heat fluxes along the relevant coordinate directions through a unit area, are expressed by the Fourier law qT ..>: = -A. aT/ax, qT.u = -A. aTlfJy, qT"
= -A. fJTlfJz
(3.2.7)
With this taken into account, we have -(fJqT.x/aX
+ fJqT.ifJy + aqTjfJZ) 'C dt
= div (). grad T)
'f
dt (3.2.8)
where gcad T ~ (aT/ax)i + (aT/ay); + (aT/Oz)k. The energy supplied to a gAS particle at the expense of diRusion is qd.,.,=
f Qt,
rI. xiI>
qd. U=
f Qt.
d,
~i" qd, z = ~ Qf. rI. zit
where ii is the generalized enthalpy component of the gas mixture.
126
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
Hence. _ ( uqa; .\'
+ a~d; y + aq: / ) 't dt
=" _
(~
it
lJQ~zd. x
i
+h
aQ~!ld.1J
if
+ hit iJQ1at;) "dt= (- ~ i,divQ"d) "dt
i
i
i
Introducing into this equation the value of Qi,d from (3.2.3), we obtain - (Oqd. :iox i·iJqo. gloV+ Oqd. z/oz) 'rdt = ~ it div (pD grad e,)
't
dt
(3.2.9)
In addition to the energy transferred to a particle by conduction and diftusioll, il also receives heat owing to radiation, equal te> n dt (e is the heat flux due to radiation absorbed by unit volume). By equating the change in the energy of a particle during the time dt to the sum of the work done by the mass and surface forces and the influx of heat ber3use of c:ondllction, diffusion, and radiation, we oblain the energy equation:
PTt (~+ u) + "zgl' v-I
'----"p (XV;,: ,!. YVy +ZV1 )
+-tr (PnV;,:
'rXZ V2 ) -1'-;'; ("!I~V;,:+ PnVv + "VIV,)
+.;;. ('r;:tVx..L "zVVy+ pzzVz);- div (A grad T) + ~ i,dh'(pDgraric,l+e
(3.2.10)
i
Upon the motion of a chemically reacting mixture of gases, the energy equation expresses the condition of the heat balance including the heat that may appear as a result of chemical rcacLions. If we take into account, however, that upon the proceeding of reactions the generalized enthalpy of the gas mixture LCJii does not change, then upon int.roducing the generalized enthnlpy inte> Eq. (3.2.10), we cnn no longer take into account separately the release or absorption of heat because of chemical reactions. Introducing into (3.2.10) the expressions for the st.resses (3.1.5) and (3.1.16) and excluding the term taking into account the work done by the mass forces, after transformations we obt.ain an energy equation for a gas: p
1t (~-!. u)
= -
div (pV)_.. -} div (jlV div V)
+-f; (~~) +fu- (~
u:;) +-i; (~ u:!)
*
Ch. 3. Fund"mentals of Fluid Dyn"mics
2
-tx
f,!{VyCt7 Vzcy)I-i- 2
127
IfL (V."€l ;. V.f x)]
:- 2 "';;'I~ (V.~ey -; Vllc:rl/·;...div (Agraa T) +~ ijdiv(pDgradc/)+€
(3.2.11)'
Equation (3.2.11) sl10ws what causes the kinetic energy of a nuid to change. III addilion to conduction, {Jifrllsion, alld nuliatiotl. this energy changes at the expense of the work of comprC'ssion di,· (pV) and tho work of tho friction forces (the tcrJII~ of t.ho eqllation ("ontaining the dynamie viscosity pl. The di!'sipatioll of ell('I'gy i.'" 1ISSOciated with the losses of mechani(:l1I en('rgy ror on'('{'ollling the friction forces. j':nergy dissipation {'onsists in that the mcehanica\ energy in part tram;forms il'l'eYersihJy into heat. Accordingly. the friction forces arc ('alled dissipath·e. The terms on till' right-hand side of (3.2.11) ('ontaining 11 form the dissipative fUllction. For two-dimellsiollal plnllC' molioll of a \'i!'colls nuid. thC'· energy equation is p
~ (~-;. u) =
-div (pV)-i- div
(~lV div V)
+fx(~l~)~*(~W) 2 -if; (~lVllez) ._- 2
i; (fLV ",ell.1- div (Agrad T)
+ ~ i l div (pDgl'ad ci ) -: where grad T
V "'" V.ti (oT/B.d i
€
(0.2.12)1
+ V"/L di\' V '"'" iJV)iJx + (}r"ldy, + (oT/uy)j, grad C; -" (8c;:(h)i ~- (IJc;lay)j
Let us transform the energy eqllation (:~.2.12). To do this, we' multiply the first equation (3.1.20) hy V", the second olle hy V", and sum up the results. \Ve obtain
p
_ Vx~[~(2
-t- (~).~ - (V"~'.Vy*)
u:~~-+diVV)]'i-Vyi;[~(2 U~',_~ di\"V)]. -: 2
[l"o~ ~ (j.l€z) -I· VII
-;;. (Il€.) ]
We can show by simple tran.!'fol'mations thnt l'" up/ax VII iJp/ay = di" (pV) - P (Jiv V
+
(3.2.13)-
128
Pt. l. Theory. Aerodynemics 01 en Airfoil end eWing
where we find p div V by using the continuity equation p div V ~ -(pip) dpld" or p d'v V ~ P (dldt) (pip) -
dpldt
We transform the sum of the other two terms on the right-hand side of the equation to the form
V%.:x[~ (2 ;t--}diV v)] +V.. ';; [~
(2 ~ --} div V) J
~ -h (~~) +.:,- (~W)-2~[( '::)' +( '::
n
-{-diV (~V div V) ++~(divV)t For the last term in (3.2.13), we have the expression 2
[v" ;;'(fJo!,;z) l· V y ';;
(!-LBz)]
=2* (I-'V"e
l )
+~ (",V:t:ez) - 4",e~
Let us make the relevant substitution in (3.2.13) and subtract the obtained equation from (3.2.12). Having in view that i = = It + pIp, we obtain
p~=*+21-'{[(~r
+ea;y" r]-i- (divV)2+4e~}
-I- div ().grad T) + h i, div (pDgrad cI ) + e
(3.2.14)
In the absence of heat transfer by diffusion and radiation. we can write the energy equation in the form
p~
=%+21-'{[( 0;; r + ( 0;,;1 rJ --} (div V)Z+!ie~} +div{).gradT)
(3.2.15)
At low gas velocities. when the work of the friction forces is not great. we may disregard the dissipative terms. In addition. the work done by the pressure forces is also insignificant (dp/dt 1::J ~ 0). In the given case, instead of (3.2.15), we have dTld'
(Alpc,) d'v (grad T)
(3.2.16)
The quantity A/(pC p ) = a, called the thermal dillusivity,characterizes the intensity of mol ocular heat transfer.
Ch. 3. Fundllmenlotls or Fluid Dynllmics
129
3.3. System 01 Equations of Gas Dynamics. Initial and Boundary Conditions
The investigation of the motion of a gas, Le. the determination of the para.meters characterizing this motion for each point of space, con~ists in soiYing the relevant eqnations that relate these parameters to one another. All these equations are independent and form a system of equations of gas dynamies. We determine the number of independent equations by the number of unknown parameters of the gas being sought. Let us consider the motion of an ideal compressible gas. If the velocities of the Row are not high. we may ignore the change in the specific.' heats with the temperature and take 110 account of radiation. In this case. the gas How is a thermodynamically isolated system and is adiabatic. The unknown quantities for the now being considered arc the three "elocity component$ l'x, VII' and V z• and also the pressure p, dl>Jlsitr p, nnd temperilture T. Consequently. the system of equations of gas dynamics must include six independent- onc.'i. Among them arc the eqllation~ of motion, continllity. slate, IIlld ell(>rgr. which art' ('USlomllrily called the fundamental equations of gas dynamics, B(>fore compiling this s-y:-;(ern of equations, let us cOIl~idel' separately the energy eq1lation. III ac('onlnncc willi ollr m':-;umption on the nrliabatif: nature of (he Ilo\\", we \I'Hn:-;fol'lll til(' ('1I('rgy equation (:1.:!,14) tI!! follows: di
= dp/p
(3.3.1)
If we take into cOII$lideralioli the cquatiorl di -, c/• dT, aud also the ('xpressions el' - C p ..." Rand p -'" RpT. from which we can nnd
then (3.3.1) is reduced to the form dp/p = k drIp. Hence (3.3.2)
where A is a constant characteristic of the given conditions 01 gas flow. Equation (3.3.2), is known as the equation of an ndiabat (isentrope). Hence. in the case being considered, the energy equation coin('ides with that of an adiabat. Having the energy equatioll in t.his
130
Pt, I. Theory. Aerodynamics 01 an Airfoil and" Wing
form, we shall write all the equations of the system: 1
dV:\, _
8p
-p'Tz'
---cit-
dV lI ~=
1
ay
-p'Tu
~~:=-+,~, -*+pdivV=O p=RpT.
(3.3.3)
Aph-I=RT
Let 11S ('.onsider the system of equations for the more general case of the motion of an inviscid gas at high speeds when the specific heats change with the temperature, and dissoC'-iation and ionization may orcur in the gas. For generality, we shall retain the possibility of the heated gas radiating energy. ~ow the thermodynamic process in the gas flow will not be adiabatic. Accordingly, the quantity £ determining the radiation heat flllx remains on the right-hand side of energy equation (3.2.14). We shan note that the equation of state must be adopted in the form of (1.5.8) taking into ft(',count the change in the mean molar mass Ilm with the temperature and pressure. In accordance with the above. and also taking into ar,r.ount that the equations of motion and continuit.y do not change in form. we shall give the fundamental equations of the system: dV."\"
_
I
8p
8V"
1
8p,
(ii"'=--p'&%' d't-:'-p'Ty
d!'t%
:;;~
-+'*: ~+pdiVV=O
p=!;-pT,
f (3.3.4)
p-¥:-=*+e
We can see that in the given system in addition to the six unknown quantities indicated above (V", V y • V z • p, p, and T) three more have appeared: the enthalpy i, mean molar mass of the gas 11ru. and the heat flux e: produced by radiation. Besides these quantities. when studying the flow of a gas, we must also determine the entropy S and the speed of sound a. Hence. the total number of unknown parameters characterizing a gas flow and being additional1y sought is five. Therefore. we must add this number of independent relations for the additional unknowns to the system of hlndamental equations. These expressions can be written in the form of general relations determining the unknown quantities as functions of t.he pressure and temperature: i ~ /, (p, T) (3.:1.5) (3.3.6) S = t. (p, T) (3.3.7) ~m = t. (p, T) (3.3.8) • =t.(P, T) (3.3.9) • = t. (p, T)
Ch. 3. Fund~ment~l$ of Fluid Dynllmics
131
The linding of these functions is tlte subject of special branches of physics and thermodynamics. The solution of Eqs. (:1.3.1)-(3.3.0) determines the parameters of flow of an inviscid dissociating and ionizing gas with account taken of the radiation effect. Snch flow is studied by the uerodynamics of a radiating gas. Let us consider a more general clIse of flow chara(·tcrized hy the action of friction forces and heat transfer. \Ve shall assume that chemical reactions occur ill the gas. Therefore. Llw fllndamental equations of th(' sYiitcm (for simplirlcation we shall ("onsider twodimensional plane flow) will include two differential equations (3.1.20) of motiou of a yisrous compressible fluid \vith a varying dynamir viscosity (Il =t= cOllsI), and also l'ontinuity e(Iuation (2.4.1). These equations mUl'it be l'iupplcmented \vail equation of slate (1.5.8) relating to the general cuse of a dis~ociated and ionized gas, and with expression (3.2.11) that iii the ('nergy equation for a two-dimensional compressible gas rlow in which heat transfer by diffusion anc1 radiation occur. These equations describe the gelleral case of unsteady motioll and characterize the unsteady thermal processes orcufring ill a ga~ now. !fence, we have
~== _+.~++'~['I (2
iI;;. --}diV v)J
++~(Ile:)
~
=
-+.* r+'+v [~(2 a{~;1 ._+ div v)] +4·*(~eJ
(3.3.10)
47-, pdivV=O; fJ%=~
-'-2M
{[(~r .L(~r]-+{clh'VP· 4e~}
+div ("-grad T)+ ~ i, div (pD~rad cJ) :.
t·
This system mllst be slIpplelllelltf'd with relations n.3.5)-(:·I.~Ul), and: also with tbe gener,]} relations for the thermal conductivit.y
).
~!. (p, T)
"
~
(3.3.11)
the dynamic viscosity
!,
(3.3.12)
(p, T)
.md the specific heats
c,.
= /,
(p, T),
C1'
'-'
/9
(p, T)
(3.3.13)
182
PI. I. Theory. Aerodynamics of an Airfoil and a Wing
The last t\\'o quantities are not contained explicitly in Eqs. (3.a.10). but they fire RCvertheless used in solving them because when studying the flow of a gas its th(>rmociynamic characteristics are determined. Since the energy equation also t.akes into account heat transfer by diffusion. equation (3.2.5) has to be included additionally. It must be taken into accollnt simultaneollsly that the COIlcentration Cj in the energy and diffusion equations is a function of the pressure and temperature, and it can be written in the form of the general relation
c,
~
/" (p, T)
(3.3.14)
The abo\'e system of equations including the fundamental equations of gas dynamics and the corresponding number (according to the number of unknowns being sought) of additional relations is considered in the aerodynamics of a viscous gas and allows one to fmd the dist-ribution of the normal and shear streiises, and also the aerouynamic heat nnXeii from the heated gas to the wall over which it Hows. In spedflc ("aRCS, for which a definite schematization of the now process is possible, the above system is simplifled, and this facilitates the solution of the differential equations. When soh'jng the equations, it becomes necessary to in\'olve additional relations used for determining the characteristic parameters of motion. Among them are. for example, relations for determining the ~pE"cinc heats and the degree of dissociation depending on the pressure and tcmpcrature, 0.11(1 formulas for calculating the shear stress depcnding on lhc \'clocity. The solution of a system of gaswdynamic equations describing the flow onr a given snrfarc mustsatis[y defInite initial ami boundary conditions of this now. The initial conditions al'e determined by Lhe values of the gas parametcl'!'! for a certain instant and ha\'e sense, evidently, for unsteady motion. The boundary conditions arc superposed on Lhe solution of the problem both for steady and [or unsteady motion and must be obsen'ed at eury ins/ant of this motion. According to one of them, the solution must be SUeil that the parameters determined by it equal the values of the parameters for the undisturbed Dow at the boundary separating the disturbed and undisturbed now regions. The second boundary c.onditioll is determined by the nature of gas flow oyer the rele\'sllt :"urfnce. If the gas is in viscid and does not penetrate through such II surface. the flow is said to be without separation (a fre{' streamliop. now). In accordance with this condition, the normal velocity component at eac·h point of the surface is zero, while the vect.or of the total velocity coincides with the direction of a tangent to the surface.
Ch. 3. Fundamel'ltals of Fluid DYl'Wllmics
133
It is general knowledge that-the "ector grad F [nerc F (qt. q'J' q3) = = 0 i,!l the equation of the sllrfaC'c in the 00\..... and qh q'J' q3 are the generalizcII C'urvilinear C'ool'(linat~l coincides in dircction with a normal to the snrlaC'e. lIen(,f-, ror conditions of Row without sep· aralion. the dot prodnel of 111i~ vcC'tor and the "clority vector V is zero. Conseqllcmtly. the condition of Row withont separation can be written in a mathematical form ,IS follows: VgradF ·0 Taking into account that gradF
""-&. :~ i
l • -/;;.
;~
i'J
~-*. :~
13
(3.3.15)
the ('ondition of Row without separation can be wl'iUcn as
-&' :~ VI+i· :~ V2~-*' :~ V =O 3
(3.3.16)
For Cartcsian coorninate."', we haye grad F , . (iJF.'iJ,r) i l + (;)F.'iJy) i'J -:- (iJpiiJz) i.1 COII!lt'qllcntly. r.r (3.3.16) i'.,. ilFliJx ~- 1'" OFhJy --:- 1": flF/i)z ,... ()
(:1.3.17)
For two-dimensional plane flow 1""
V;"". -
aF/iJ.r; aFlog
(3.3.17')
If the equation of lhc surface is gh'en in q;lindrical ('oordiIl8tes. we haye
gradF-:-~il+*j'J-:-
+.*i
3
thererm'f', the ('omlilion of flow wilholll scpal"ulioll has Illc rorm (3.3.18) In a particulllr case. whNI a :'lUl'face of rc\'olnt.ioli is ill the flow, we obtain the equation Fx OF/O:r -;... {)F/i)r - 0 (:3.:3.18')
'·r
froll1 whieh we find the condition for the vclo('itr ratio:
f.; - ~~~:;
(3.3. "19)
Other bOlilldal'Y cOlldilioll~ ('an also he rormulated. They are determined for each spccinc problem, the boundary conditions ror
1.34
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
a viscous gas differing from the cOlHlitions for an ideal fluid. ParliClllarIy, witli'll studying the flo,v of a viscolls gas in a boundary layer, the solutions of the pertinent equations must satisfy the conditions on the sllrface of the body and at the edge of the boundary layer. According Lo experimental data, the gas partides adhere, as it were, to the sllrface, and therefore the velocity on it is zero. At the boundary layer edge, the velocity be{'~omcs the sallie as in free (inviscid) flow, and the shear stress equals zero.
3.4. Integrals of Motion for an Ideal Fluid The differential equations deriver! for the general case of motion of a gas are non-integrable ,in the finite form. Integrals of these equations can be obtained only for Lhe particular case of an ideal (inviscid) gas flow. The equation of motion of an ideal gas ill the vector form is 8V/fJt + grad (V2/2) + cud V x V = -(tip) grad p (3.4.1) This equation can be obtained from vector relation (3.1.22') in which the terms on the right-hand side taking into acconnt the influence of the viscosity should he taken equal to zero. In its form (3.4.1), the equation of motion was first obtained by the Rmisian scientist prof. I. Gromeka. With a view to the mass forces, Gromeka's equation becomes 8V/8t -;- grad (V2/2) eurI V X V ~ G - (lIp) grad p (3.4.2)
+
Let us assume that the unsteady now will he potential, hence curl V = 0 and V = grad 'P. In addition. let us assume that the mass forces have the potential U, therefore the vector G = -grad U
+
where grad U = (8U/ax)1 (aU/uyH --1_ (uUliIz)k. If a fluid has the property of barotropy characterited by an unambiguous relation between the pressure and density (this occurs, for example, in an adiabatic flow, for which p -:-:: Ar"), then the ratio dp/p equals the differential of a certain function P and, therefore, (lIp) grad p = grad P \Vith a view to this equation, expression (3.4..2) becomes 8 (grad rp)18t grad (1'1/2) = -grad L: - grad P
+
Substituting the quantity grad (8(r/8t) for the deri"ali\"e ,ve obtain grad (8rpI8t) grad (V212) = -grad U - grad P (3.4.~)
u (grad
+
Ch. 3. Fundementels 01 Fluid Dynemics
13~
Going o\'er from the relation for the gradients to one hetween the corre~pollding scalar fnne.tions. we rlild df{'ldt
+ V~/2 + P
where
-i- C -= C (t)
(:{.4.4)
Jdpip
p.=
(3.4.5)
Expression (3.4.4) is known as the Lagrange equation or integral. The right-hand side of (3.4.4) is i\ function that depends on the time, but docs not depend on the coordillates, i.e. is identical tor any point of t1 potential flow. The term:: on the left-hand sic/£' of (3.1.4) have a ~imple physical meaning: 1'2/2 is the kinelic energy, P "-:: ) dplp is tlle potential energy d\le to the pressure for a unit mass, and U is the potential energy due to tlte position of the fluid particles and related to their mass. To re\'eal the physical meaning of t.he Iirst term, let liS lise the expression for the potenlial function rJ(rlas -:-: T"s. where V" is the projection of the velocity "eclor onto a certain direction s. We can dctermine the function (f from the cOlUlition (f So and s are the coordinates of a
lix~d
-1
V..ds (where
and an arbitrarr point. respect-
i\·ely). The deriyathe lh(dl .~ .~ (dFs'r)l) d.~.
,.
Thl' local acccleration aV,,/at cau be considered as the projection of tllc inl'rtia force due to the presence of local acceleration and related to unit ma~s, and the product (iH's/dl) ds as the work done by this force on the se('tioll d.'1. Accordingly, the derivative a
+
+
If, parti('ularly, the y-axis is directed \'('rt.ically upward, then L' = gy, alld (3.4.6') Qlf'dt + ~rll/2 pIp gy =-= C (t)
+
+
136
PI. I. Theory. Aerodyn!!mics 01 lin Airfoil lind
II
Wing
Of major practical significance is the particular case of a steady potential [low for which aqJliJt = 0 and the function C (t) = tonst, Le. is independent of the time. In this case, Eq. (3.4.4) is reduced to t.he form
V 2 /2+
Jdplp--!·U=const
(3.4.7)
This partial form of the Lagrange equation is called the Euler equatiOD. It expresses the law according to which the total energy of a unit mass is a constant quantity for all pOints of the steady potential flow. Hence. the constant in the Euler equation is the same not only for t.he entire region of a flow, but also, unlike the function C (t) of the Lagrange integral, is independent of the time. The Euler equation for an incompressible fluid (p = canst) has the same form as (3.4.7), the only difference being that instead of dp/p it includes the ratio pip. Let us considC'T the more general case of a non-potential steady flow of a gas. The equation of this motion has the form curl V X V = -grad P - grad U (3.4.8) grad (V2/2)
J
+
or grad (V2/2
+ ) dplp +
U) "'... -curl V X V
(3.4.8')
The right-hand side of (3.4.8') equals zero if the vectors curl V and V are parallel, Le. provided that a vortex line and streamline coincide. In this case, we have
V'/2
+
Jdp/p
+U~
C,
(3.4.9)
This equation was ftrst obtained by I. Gromeka. The constant C1 is the same for the entire region where the condition of the coincidence Of the vortex lines and streamlines is observed. Such regions, to study which the Gromeka equation is used, appear, for example, in a flow past a finite-span wing. This flow is characterized by the formation of vortices virtually coinciding in direction near the wing with the streamlines. A flow does not always contain such regions, however. A flow is customarily characterized by the presence of vortex lines and streamlines not coinciding with one another. The family of vortex lines is given by Eq. (2.6.1), and that of the streamlines (pathlines), by Eq. (2.1.6). The flow being considered is described by Eq. (3k8'). Let liS take the vector of the arc in the form as -:--= dxi + dyj -++ dzk. belonging to a streamline or vortex line, and determine the dot product ds·grad (V2/2 + dplp + U) = -cis-(eurl V X V)
i
Ch. 3. Fundamentals 01 Fluid Dynamics
137
The left-hand side ill this equalion is tlJ(' 101,1\ differenlial of the trinomial ill pl'lrcntheses, consequently.
d CV2/2
+ ) dp/p
-:- L') = -ds· (curl V X V)
(3.4.10)
The product (Ourl V X V is A. veclor perpendicular to the \'f'('tors curl V and V. The dot product of this y('ctOI' aTld the \,prlor d'3 is zero in two cases: when the ('ector d~ COiIlCidl'S wi/It. 11'1' directioll oj a streamline (pathlinc) 01' {",hen (hi,) reef 01' coilleides Il"ilh the direction of a vortex. III these two ca,<:e~. the following '<:o\l1tion of tile equation of motion is valid:
P/~ -.:. \ dpip
+
C ....: C 2
(3.-1.11)
where the vallle of tlte cOI1..-\.alll C z clepc/HII< 011 what palhJine or vortex line is beillg considered, Relation (3.4.11) is known as the Bernoulli equation. It is obvious that for yariolls \'ortex lines passing through points on a given streamline, the conslaut is lhe :;iUlie as fOl' tile streamline. III eXfletly the same way, Ihe ('onsl.ants fire id~ntical for II rllllliiy of ~tr(>amlincs (pathlines) find the vortex through whose points a slrenmline pfl~~(lS. One must deflrly understand Ihe distinction between the Grollleka and Bernoulli equations con~idcred abo\'(·. TIley nrc both derived for a vortex (non-potential) nO\\', howen~r the iiI'S!. of them reJ1p(,\.f'. the fact that the totfll energy of a IIllit llli1~S 01' the IS in the entire region where the \'orlex liHl'S and while the second equation ('."'tabl [ghes Ihe law this energy is constant along fI giH'n streamline lIr L·or/c.! ingly. in the Gromeka equation. tlw COll;.:talll, is thp ."amp entire flow region being consi(ll:'red. wllere
138
Pt. 1. Theory. Aerodynllmics of lin Airfoil and a Wing
we shall write this equation ill the form
V 2 /2
+ pip + yg =
C"
(3.4.12)
When studying the motion of a gas. we may disregard the [nflu· ence of the mass forces. Consequently, we mllst assume that L' ,,= 0 in Eq. (3.4.11) and the other int.egrals. Particularly, instead of (3.4.12). we haye (3.4.13) Lel \lS consider the motion of an ideal compressible gas. In such a gas. heat transfer processes due to viscosity (heat conduction, diffusion) are absent. Assuming also that the gas does not radiate energy. we shall consider its adiabatic (isentropic) molion. Exam· ination of energy eq\lation (a,2.14) reveaL-, that Eq. (3.3.1) holds for s\lch fill inviscid gas in the ahsence of radia\io[] (€ - 0). Consequently. the Bernoulli integral is (3.4.14) In this form, the Bernoulli integral is an ene!'gy eqnation for an isentropic now. According La this rqlLalion. the slim of the kinetic energy and enthalpy of a gas partide is COlLstanL Assuming that i 0' cl,T cl,p/(pR), cp - CD c- R, and k - c1>'"c(., we lilld i ~ lk/(k - 1)1 pip (3.1,.15) Con~equently,
V'/2 V 2/2
+ [k/(k -
1)[ pip ~ C
+ kRT/(k -
1) --" C
(H.16) (3.4.16')
The Bernoulli equation for an ideal compressihle gas is the theoretical foundation for investigating the laws of isentropic flows of a gas. 3.5. Aerodynamic Similarity Concept 01 SlmUlrHy
The aerodynamic characteristics of craft or their individual elements can be determined twlh theoretically and experimentally. The theoretical approaches are hased on t.he llse of a system of equations of gas dyuamics that is solved as applied to a hody in a flow, the hody ha\Oing a given configuration and arhitrary absolute dimensious. \Vhen rUllning experiments intended for obtaining aerodynamic parameters that can he used directly for further ballistic calculations or for verifyiug the resnlts of theoretical investigations. it is not always possible to use a fnll-scale body because of its large size, and a smaller-size mod "I of the body has to be used. In tlds con-
Ch, 3, Fundamentals 01 Fluid Dynamics
139
Jl('ction, Ihe question appem's on Ihe poss.ihility of trans.fel'fing the l'xpl'rimental 1'('''-lIitS obtained 10 full-scllle ])o- llts and indicates dU'lracit'ri!UTnll that measurl.'Olent!< in a wind tUJlIl(>1 :riel(led II drag fo\"t'e which ill a('('ol'(lance with (1.:1.5) is Xm"~ c,,',modQmodSmnd' l\OW let us s('e wilen we ('all use the l'e,'-'11l1 ohlaillcd to determine the dl'flg rorre of a full-!>cal(' bod~' in a('('o\"llallre witll the formula Xf,; ..., (,..t~qtsSts in whid\ the Ilrag- ('oel1i('il'Jlt c.\',fs ror this hody is an unknown qlHlIItitr, while tile \"elocitr hl'afl qt~ and the rererence or dHiraeterislic Mea SfS an' gi\'(,Jl. FI'OIll the Iwo rormulHs for Xt~ lind X,"od' we oblain X,s '-' X mod (cx.fs:C.,·.'HOd) (lJ,sSf~'lJn'''dS'''''d) (3.5.1) A glance at this (!xpl'ession r('\'('als thai tlte experimental \"lIlue X mCld ran be used to e\"aluate the fllll-,"('il.le force X", only upou the ('quality of the aerodynamic {"oeffidput~ (,·.lIIool Hnd (\'.Is because the qUlIlIlities q"'(ldSmod and qtsSr~ are detCl'lIlined ullamhiguously by the given yaillt's of the wlo('ity h('uris and tILL' referellce areas. lIere both flows-the model and fl\ll-~cale on('s-haw the property of dynamic similarity. It consisls. in thi~ ('lISC ill Ihflt the preset fOt'ee characted~lic or one flow (the drag X ,urld) i!< lIsed to rInd the rhllr(lr leristi(' of till' olher flo\\" (thc force .Yr.) by simple cOHYersion, similar to tit(' IransitiOH from olle systf'm or units of mea~lll'emeHt to anoLher. The re(ll1irements whose satisraclion in the gin'll ('ase (,IlSlll'es the equality C.\"."".d - c.\"'~' ,11\(1 in Ihe gPlleral ra~e of other OilllC'llsiOllless aerodynamic coet"iicil'nls 100, are eslabli..-lted in the dimellsiOHal analysis allfl s.imilarity method pro(·('pding eilllCr from the physicAl natlll'c of the phenomenon bping stlldi('d or from the corresponding differential equation!> of aerodynamil's. Con..-iderLng the expression rOt' the ll(']'()(lyuamic coefiieiellt
(fx,,)1*
C'\·8= ) [iJeos(Il.T D )·;·(',."cos (3 ..'5.2) <s, Qhtnined from (1.3.2), we can !':'CE.' Ihat this coefficienl depends on dim('n~ionless gcomet!'ir parllnlell'I's. and also on (limensionless qlllllltities sudL as th(' pl'eSSlll'(' rQPllidont 11IIt! Jocal fricLion facioI'. Hellc(' it follows Ihat thc aerO(IYllllmic co('ffirienl..- fol' a fuJl-sclIle ohjed [lIHI an eXpl'fillll'lltal onl' with diffNent ahsolute dimell!: with 11 gi\'en configuration
p
140
PI. I. Theory. Aerodynllmics of an Airfoil and a Wing
of the body in the flow known \"alues of the angle of attack lind the sideslip ,mgle. as of the rudder and elevator angles, will be functiolls of the stream velocity l' the pressure poc, the density p "', lIH: dynamic yj~co!>ity .~oe. the specilic heats c" "" and C,,(IO of the gas, as \I'cll as of a certain characteristic (reference) linrar dimension of thf' hody L ConsequenLly. the drag coefficient will also depend all tltese parameters, and we can compile a functional relation for it in the form ex:...: t (V"',, p"", p"" fl=, c)lx" C,._. L). Since this coefficient is a dimensionless quantity, it mnsl also be a function of dimensionless parameters. From the general considerations or the dimensional method. it follows thnt the seven different argnments of the funclion C x call be reduced to three. The latter are dimensionless combhHltiolls compiled from V.", p"", poo, fix-, cl'''''' c,_ ,." and L because there arc fOllr independent units of measnrement, namely, mass, lcngth, time, ancl temperaturE'. These dimensionless combinations h,,\'E' the following form: II "" Vk~p<:.Jp"" '- V ",,/a ~ = = Moo-the :vrnch numiler for an Hlldisturbcd flow; lloop""LifLoo = = Reoo-thc Reynolds number bi\~ed on t.he parametl'rs of an undisturhed flow and the cilflracleristic linear dimension L; cp ...,Ie,."." = koc-the adiabatir exponent. In the expression for Jl,,,,, it is assumed that Vk""p""."p", = am is the spet'd of sOllnd in Lhe undisturbed flow. Indeed, in accordance with the gt'l)pral t'.\pre~sion for the speed of sOlmd a? ""'" dp.dp. and also with a dew to the adiflhatic nature of propagation of sonic disturbances in a gas, according to which p ==- Apk, we have 0 2 = = d (Ar~)/dp '----' kp'p. Ac('ordingly, the square of the spepd of sound in an l1udislurbed flow is a;' ".., k ...poo.p"". Hence, the ratio V .../Vk~p"",p= '----' All other dimcnsionlcss combinlltions except for M"", Re"", k"", formed from the seven paramet.ers indicated aho\·e or in general hom any quantities that can he determilled by them are fllnctiou!'; of the combination!'; Moe, Re"", and k,.,. Consequcntly, the drag coefficient is ex --= t (Moe, Re".., /c",,) (3.:i.;{} Similar expressions can he obtained for the olher IIcrodYll1l1nic coefIidents. It follows from these expressiolls that when tile nllm~ her." M "'" Re.., and the parAmeter k"", for a model and a flll1~scalc flows are equal, the aerodynamic coefficients for geometrically similar bodies arc the same. Hence, an important concillsion can he made in the dimensional analysis and ~imilarity method in accordance with which the necessary and sufficient condition for aeroclynamic similarity is I.hr constancy of the numerical \'alues of the clim(>nsionie:o;s comLinntion~ forming what is called a bns(>, i.e. a system of dimellsionless ql1antilies determining all the other parametPfs of a flow The~e (lirnensionless combinations are ("died Similarity eritNia. (10,
Ch, 3, FUfldamentab 01 Fluid Dynamics
141
The ~imilaritr critt'ria gi\'t~n above have a delillite phrsical meaning, In accordance with the e:o.:pression a:! = dp/dp, the speed of sound can be considered as a criteriou tlcpeuding 011 the property of compressibility, i.e. on the ability of a gas to change its densit}' with a change ill the pressure. Consequently, the Mach number is the similarity criterion that is used to characterize the influence oj compressibility 011 the flow of a gas, The Heynolds number is a parameter used to appraise the influence of ~'I$C'osily Oil a gas ill motion, while Lhe parameter k _ = cp ""fc r "" determines the features of a flow due to the thermodynamic properties of Q gas. Similarity (rU.ria Taking Account of
the VIKGsU, and Heat Conduction
In the more genernl conditions of flow characterized by the influence of a number of other physical and thermodynamic parametel's on the aerodYllamic properties of craft, the drnamic similm'ily criteriA are more complicated and diverse. To establish these criteria, we can usc a different approach of the dimensional analysis and similarity method based on lh(> use of the eqUAtion of motion of a "bcons Iwnl-conducting ga~, Lel us wrile these (!qtUltioll~ in the dimclI:-:iolli(>ss form, i.e, in a form such that the paramett'fs (vciociLr, presslIre, temperature. etc,) in the equations arc rclal('d to I'd('rt'llco panUl1l'lt'I'S. The illtter arc constants for f\ gh'en flo\\' alHl dl'lt'rmillt' iLs scal(>. \\'e shall take as references the paramnlers of lhl' (1'(>(' stream: it,;; Yeioci~y r ... , pressure p_, dCllsitr p_. lempf.'l'al1ll'C T...,. f[YIl
;; ~ .IL. Y ~ yiL. Z ~ ,no t
~
tit.
(3.5.4)
while those for the velocity, pressure, density, viscosity, and mass forces have the following form:
V.~VJV••
V,~I'~v•.
V.~V.IV •. p~plp•• }
p ~ pip.
;="'/""'"
v==v/""'"
X=XIH. Y=Ylg.
(~.5.5)
Z=Z/g
1+2
PI. I. Theory. Aerodynamics of an Airfoil and a Wing
Let us introduce dimensionless variables into the equation of motion (3.1.17) and the continuity equation (2.4.2). We shall use only the first equation of system (3.1.17) for trallsformation becau.sc the other two equations are compiled in a similar way. Let us write the indicated equations using dimensionless variables:
~.'iV.T+ t~
dl
V!(v /,
~+V cti\. "'c1l: U
iJfx),,~gX au'.;..y'iJi
_~ . ..!...~+ \'~v~ poeL
it c)x
+-'-[.i(2 P iJx
L~
{".1V +...!.3 ...!.. ax div V ;r.
iJT!"_..!..diVV)+~( av,:. +~)
+*
og
fJ.t;j
((l~;T
-+-
b~,
iJ,I
lJr
)]}
~.~+p""roe [(l{P~'Tl +iJ(P~ul +a(p~l)]=O h"
L
"l
()x
ag
bz
where div V '- afix/ax -I- aVuMi'· aV,/aJ. From the reference quantities in these equations, we can form dimensionless numbers charaeteriling the similarity of gas flows. These numbers, named after the scientists who were the first to obtain them, are given in the following form:
Sh .,--, V oot~/ L - the Strouhal number; Fr - V~/(gL) - the Fronde number;
M = V""la~-the Mach number; He = V ""p""L/~"" = V""U""" - the Reynolds number (the subscript "00" on M and Re has been omitted).
(3.5.6)
Here a .... = 'Vk ....RT"" is the speed of sound in an undisturbed now (k .... and T .... are the adiabatic exponent and the temperature of gas in the undisturbed flow, respectively). IntrodUCing these numbers into the equation of motion and the continuity equation. we obtain:
8
Sh-I u~~ _V%U~T+VJl u.~:r
in
ux
II
+V:
uV.:r =_p'
uz"
X
-k",,1M'+'~ +~ {VAV:r+4-.fz-diV V
Ch. 3. Fundilmilnillis of Fluid Dynllmics
143
...:..~[~ (2 8~~ -~div V) +.i (8~x -+- J~!I '1 P
ax
3
ax
ay
8y
a,x
I
+*( 8~~", ~ a:r~ )J}
(3.;J./)
Sh~+a(p~x) _1_a(p~!I) +ii(p~.) ,-,,0 ,1/ ux (ly ,h
(:{.;l.8)
Let us transform the energy equation (:1.2.14) in which we shall exclude terms taking into account radiation and diffusioll. \tVe introduce the dimcllsionless variablcs
'.i' '-"
TiT <>0,
c
p ': "'"-' C"ICI' ""
"'l -
A A",
(3.5.9)
where cp~ and A~ are the specilic heat and the heat conductivity of a gas in the 1Indisturbed flow, respectively. Having in view that di -_ c p dT and expanding tllt! total derivative dTJdt, we obtain after the corresponding substitutions:
P<>oC:~"'T~PCp*,
P.... C1JOO;.'<>oToo
pCp (Vx-f
l-VY~-i-Vz ~~ ).~ ~': .*+fl~,~~~ x {[(
U~[)'..;- (":n'J-fi'ldivV)" 4i',:) ...!.-
Let
liS
~"" div (i'grad T)
(3.5.10)-
introduce the dimensionless Prundtl nllmber Pr ,-= J.I. oocp ""iA<>o
(3.5.11)
by means of which we I>hall compare the relatiu eoect of the riscosity and heat conduction, or, in other words, appraise tlle relation he tween the heat flux due to skin friction and the molecular transfer of heal. Hence, taking into account that pooJp<>o= ilT.., = c,.,., (1 -1.k..,» X T <>0, we have:
:r -*)
PCp(Sh-'*~_V:<*-7-V" -r Vz koo - ' • .<0. _, ,\1'(1'00-" {[( a~,)" (al:, )'] dt· Ife rJ,x iJy
~Sh-'
1100
- {- ~ (eliv V)2 + 4~e~} + Ife'lpr div (Kgrad
T)
(3.5.10')
144
Pf. I. Theory. Aerodynamics of .!In Airfoil and a Wing
Let us converl several additional reilltions of the system of equa· :tions of gas u}'lIamics (see Sec. ::l.:~) to the dimensionless form:
p = pT ii"" = t. (p~p, T "'1) (1;"m~) i: = t. (p~p, T.T) (1J).~) ~=
'p
=
(3.5.12) (3.5.13) (3.5.14)
t, (p~p, T .1') (1t"~)
(3.5.15)
I. (p~i,
(3.5.16)
T ~1'i (l/cp~)
Assume that we are investigating two flows over geometrically similar surfaces. For such surfaces, the dimensionless coordinates of analogous points are identical, wbich is a necessary condition for the aerodynamic similarity of flows. To observe the sufficient condition for such similarity, we must ensure equality of the dimensionless values of the gas-dynamic parameters (velocity, pressure, density, etc.) at analogous points. Since the dimensionless variables are simnltaneotlsly sol1ltions of the system or equations (3.5.7), (3.5.8), (:1.5.10'), and (3.5.12)-(3.5.16), the indicated equality is evidentJ~' ohserved provided that the systems of dimensionless equations, allli also the dimeu.e:ioniess boundary and initial conditions for each flow are the same. ''''hell considering syst.ems of dimel1~ionless eqllations for two flows, we see that both these systems are identical if: (1) the similarity criteria are eljllfll:
1<'r l = Fr 2 , ReI = Re z , M 1 = M z. Sill
=
Sh 2
(3.5.17)
(2) the equality of the Prandtl numbers Pr l = Pr 2 is observed, i.e. (cp~"~/)'~), =
(cp~~"j),~),
(3.5.18)
and also the equality of the specific heat ratios for the t,,·o gas £Io\,'s (3.5.19)
(3) each of the equations (3.5.13)-(3.5.16) determines the depend.ences for the dimensionless variables -;:im, I, ~, or p on the relative quantities p and f. and also on the variables (3.5.17)-(3.5.19). Such relations do not exist for a dissociated gas because similarity criteria of the form of (3.5.17)-(3.5.19) or of some other form cannot be found. This is why the corresponding dimensionless equations (3.5.13)-(3.5.16) are not the same for a full-scale and a model flows, and complete dynamic similarity cannot be ensured. Two particular cases can be indicated when this similarity is ensured. The first is the flow of an undissociated gas for which the mean molar mass remains constant (f!ml = P-ma). while the specific
c
en.
3. FundallUlntals of Fluid Dvnamics
148
heals. heat conductivity, and viscosiLy var)' depending on the temperature according to a power law of the kind y = aTr.. In this case, Eqs. (3.5.14).(3.5.16) for the quantities 1. "ii. and p are replaced by the corresponding dependences only on the dhnension)rss temperature T = TIT .... The second case is the Do\\' of a gas at. low speeds when the parameters A., "', and c p do not depend on the temperat.ure. The corresponding values of these parameters are identical for the full-scale and model flows. For this case, the system of equations includes the dimensionless equations of Kavier-Stokes. continuity, energy, and also an equation of state. The boulldary conditions imposed on the solutions of the dimensionless equations gi\'e rise to additional similarity criteria. This does not relate to the condition of flow without separation. which does not introduce new similarity criteria. Indeed, this condition in the dimensionless (orm is
c
V. 8Flar + Y, aFiay + i', aF,a: = 0 and is the same both for a full-scale and for a model flow because of the geometrical similarit.y of the surfaces. Bul the temperature boundary condition according 10 which the solution for the temperature mu~t satisry the cquality T .... Til' (hcl'e T" is the temperature of the waUl introduce~ IlJl ndtlitional ~imilnl'ity criterion. In reality, it follows from the hOlindillT ('ourlilions for a fllllwscale and model sUl'faccs haYing the forlll T1 • (1',,"h Mltl 1"~ "" (1'w)~. rt'spccth-ely, lhnt the dimcn~ionle~s tt'lllpNatllr(lS arc TJ "'"' (T1,h and 1'2 = = (TW)2' From tht' cOIHlilion or ~illlilal"ilr. 1") '" 12, Iht'refore the equality (Twh ..... (Twh musi IJ~! nhsencl!. ifl'lIcl'. tiH:' boundary condition for the wuJlIC'lllpel'1lLlII'c 1:~'Hls to nu 1I11diliOilHi ~imilarity criterion: (3.5.20) Tho dimensionless gas-d:mamic yariablcs on the surface of a body in a Dow, as can he seen Crolll the sy~tem of dimen"ionl(>ss equations [provided that Eqs. (3.5.13)-(;~.3.Hi) determine il'm, [, ~, and Cp as a function of 11 depend on the dimensionless cool'dinates anu the time. and also on the similarity criteria (3.5.17)-(3.5.10). Particu· Jarly, the dimensionless pressure can he represent~d a~ the (ulldion pip. = CJll (F,., Re, M, Sk, p,., k.,
Tw ,
XI
y, Z, t)
(:~.5.21)
We can \l.SC the known pressure distrihuLion to dctC'l'lIIine the dimensionless drag force coerrident for a gh'ell inslanl: c:C" = X./(q_S) =
146
Pt. I. Theory. A.rodynMnies of .n Airfoil end a Wing
This expression determines the dependence of the aerodynamic drag coefficient on the dimensionless similarity criterion more completely than (3.5.3). But relation (3.5.22) does not reOeet all the features of fiow of a dissociated gas because it was obtained from the simplified equations (3.5.13)-(3.5.16). Consequently. formula (3.5.22) is less accurate for such a Row than for that of an uudissociated gas. and determines only partial similarity. The similarit.y criteria on which a dimensionless aerodynamic variable depends havo a definite physical meaning and characterize the real factors aRecting Lhe aerodynamic force. The Froude number Fr is a similarity criLerion taking into account. the infiuence of the mass force (force of gravity) on the drag. It can be seen from the equation of motion given in the dimensionless form t.hat the number Fr equals the ratio of the quant.ity V!oIL due to the infiuence of t.he inertia forces t.o the scaling of t.he mass forces g. The equality of the Frollde numbers Fr for a full-scale body and its geometrically similar model signifies that they have identical drag coefficients due t.o the influence of the force of gravity of the fluid. This similnrit.y criterion is of no siguirlcance when studying gas flows because the influence of the force of gravity of a gas on motion is negligibly small. But the importance of this criterion may be appreciable in hydrodynamics, particularly in the experimental investigation of the wave resistance of various navigable ves.c;els. When a body moves in n real fluid. the aerodynamic forces dt'pcnd on thE! viscosity. The viscous force is characterized by the Reynolds number Be that can be obtained as the ratio of the quantity V~/L describing the influence of inertia forces to the parameter v"" V "",IV' determining the influence of the viscosity. If the equality of the Reynolds numbers for two geometrically similar flows is observed. the condition of partial aerodynamic Similarity with account taken of the influence of viscosity is satisfied. In this condition. the friction factors for a full-scale and a model bodies are equal. The similariLy with respect to the Mach number is obtained from the ratio of the quantity V:"IL to the parameter p,,"/(p,,"L) that takes into account the influence of the pressure forces depending on the compressibility of tho gas. The partial similarity of two flows of a compressible gas flowing over geometrically similar hodi(>$ is observed when the Mach numbers are equal. When investigating unsteady flow, similarity with respect to tho 5trollhal nllm her is signilicant. It is obtained by comparing the iU('rtia forces and the forces due to the influence of tbe unstt'adiness, I.e. from the ratio of the quantities v:.,n and Veo.t"". Two. unsteady flows around n full-scale body and a model have partial aerodynamic Similarity with identical values or the Strollhal number. The similarity criteria with respect to the Prandtl number P,.
Ch. 3. Fund!lmenlals of Fluid Dynamics
147
and the ratio of the specific heals are due to dclinill' I'cquirements to the physical properties of the gases in the full-scale and model nows. Tho gases may differ, but their physical properties mllst k00 2 • The Praucltl observe the equalities Pr l = Pr2 and k""l number depeuds on the dynamic viscosity and tit(' heat conductivity, The d}'namic viscosity reflects the properties of II gas which the molecular lran~fer of the momenlum depends Oil, while the heat conducth'i!y chnracterizt·s the intensity of the molecular transfer of heaL. Consequently, the Prandtl Jlumber Pr "-, I1c poo Arin nrc contradictory, Lt,t II~ consider, for t'xampl<>, the R<'YllOlc[s, Froudt', all(1 Mach 1l11llllwrs, 1"01' th(' ohserv!lllce of simililrily with rN,pecL to the skin frirtioll f()i'(~es, ii, is pss('nti,,1 thut F 1 L" \"I::..o V/L~ \'2' If we a:::sume thal for 1\ fllll-~calt· 1111(1 mo(lel nows the coefficients \'1 . , \'2' then Ill(' spN'd of tll(' III {)(I I' I now 1"2 1'1 (LliL2)' Lt'. il is greater than the !lpI'('d of the full,scnl(' flo\\' the same llumlwr of times that the mod('l of the hocl~' ill th(' flo\\' is l('ss than the full-scale 011(', To enSllr<> similarity with r('specl to the forc('s of gra\'i[.y, it is necessary to ellsme I'quality of [,he Frolldc nllmber!l, i.e, V~ (LICI) '-" ,.- I'i (/J3g~), wlu:"'nce it follows that if experimt'lIls were rlln at idE'ntical values of g. tll('n tlw speed of the 1I101lei flow is 1-'2 = = F, V /,/1. 1 , We can see Lhat in the given case the speed V 2 for t.he smAlI-sizc model must be smaller lhan \/1 instead of grcater as in the previous eXllmple, lipoll equillity of the ,Mach numhers, we havc 'V,."a l :..~ 1"2.02' Assuming for simplilicatioll that a 2 ,.... a l , we oht.ain the condition for ('qnality of the speeds of the l1loc1('1 all(1 fulJ-selllc flows, II is n8tural that all these conditions for t1l(' speed CallIlot· be ohser\'ed simultaneously, th<>refore we cao (ollsi
148
PI. I. Theory. Aerodyntlmics of tin Airfoil tlnd a Wing
If at the same time the speeds are not great, then the influence
of the pressure forces due to compressibility of the gas is negligibly small, and, consequently, no account may he taken of the similarity criterion with respect to the Moch number. assuming Olat the aerodynamic coefIicient depends on the Reynolds number. The aerodynamic force. moment, or hoat flux from a gas to a surface is the result of the actiou of a moving gas on a body. Various processes occur simultaneously in the gas: skin friction. compression (or expansion), heating, a change in the physical properties, etc. Therefore, one must try to satisfy the maximum number of similarity criteria. Itor example. it is expedient that the equality of thfl Reynolds and Mach numbers for a full-scale and model flows be retained simultaneously, i.e. Rei = Re'l and Ml = M'l' This is especially important when studying aerodynamic forces, which for bodies with a large surface may consist of equivalent components depending on the friction and pressure due to compressibility. This condition can be ensured wheu running experiments ill variable density wind tunnels. If tests are boing performed in a gas flow in ..... llich the speed of sound is the same as in the full-scale flow (a 2 = a l ). it follows from the equality of the Mach numbers that V t = 1'1' Having this in view and using the equality Rei = Re 2 or, which is the same, VtP2L2/[.L'l = VIPILI/IlI' we obtain L2P2i!l-2 = L1Pl'f11' Assuming that 112 = !l-l' we lind that the density of the gas in the wiud tunnel Dow is P2 = Pl(L 1IL 2). Assuming that t.he temperature of the fullscale and model flows is the same (T2 = TI ) and using an equation of state. we obtain the condilion P2 = PI (LIIL,). Hence. to simultaneously ensure similarity with rcspecl to t.he {Ol'ces of skin friction and of pressure with account taken of compressibility, i.e. to observe the equalities Re l = Re 2 and Ml = M 2 • it is essential that the static pressure in the flow of a gas produced by a wind tunnel be greater than the pressure in the full-scale flow by the same number or times by which the model is smaller than the fullscale body. The design o{ a wind tunnel makes it possible within known limits Lo control tile static pressure in the model flow of a gas depending on tile sizc of the model. With a known approximation, the influence of heat transfer may noL be t.aken into account when determining the force interaction. lIere the aerodynamic coefficients will depend on the numbers Re • •V. and Sh. l{ in addition the tests are conducted in a gas for which k""2 -" 1.:- 1' we ha,-e c. ~
f
(Re, M, SI.)
(3.5.23)
~
f (//e,
(3.5.24)
For a steadr flow
c.
M)
Ch. 3. Fundamentals 01 Fluid Dynamics
149
3.6. Isentropic Gas Flows Conflgur~on
01 Gas Jet
Let us consider the steady motion of an ideal (in viscid) gas in a stream or jet with a small expansion and a slight curvlIture. Motion in such a jet can 1lC studied as one-dimensional, cha.racterized by the change in tbe parameters depending on one linear coordinate measured along the axis of the jet. For a steady flo\\', the parameters determining this now :lrc iclentical in each cross section at any instant. If the width of the jet is small in comparison with tJu.- radius of curvature of the ('('ntrl' line, Lhe lateral prc~lIre gradient may he disregarded. and one mar consider that the prC!ssure nt ench poiut of the jet cro~., section is til(' same. Thl' consideration of slIch olle-dimensional sl,elldy now of 1I ('Olllpr('ssiblp gflS IN\(ls to the simplest approximate solution of the equatiolls of gas dynamics. Condition (2.4.51) of n constant mass flow is retained along the jet, i.e. PIVISI = P2'V2S~ "-' PaVaS., ~ ... , or pFS = canst, where th(' subscripts 1, 2, 3 signify the COl're~ sp~nding paramcters of the gns at the control surfaces. Cross .!'lections of lhe channel with an area of SI> S2' ant} S3 have been chOSl'n as these slIrfflceg. Taking logarithms, wc obtain In p In V :- In S = = const. Differentiation of this l'xprl'ssion yields
,I
+
dpip
-;~
dvrV -;- dSIS
= 0
whence
(J.G.I) Le~
us use relation (3..1.11), by dilTcrentiation of which we have
IT. dV = -di
(3.6.:.!)
Substituting dp r for di. we fmd VdV ,-' -dp/p
(3.6.3)
Taking the speed of sount! fl :-..= V dpldp into accollut 1IIul llsing r~lalioll (3.0.3). we transform (;{.6.1) to the form dS;dV \\'hel'(' M
'-~
~
(Sir) (M' -
1)
(3.0..1)
V,'a is the }Iach Ilumber in the given cross section of
the jet. LeL lIS IlSS11lnC! (hot l.hl' vclocity along the jet grows (dV> 0), 1mt, remains subsonic and. CO[]~l·q1Lent.ly, M < 1. A glance at (:':1.0.4) rcyeals that for Illis caSt' the derivative dS/dV < O. This indicntes that the jet couverges downstream. COIl\'('rsciy, for a subsonic
1~O
Pt. I. Theory. Aarodynolllmics of an Airfoil and
0lil
Wing
V.p.~
p.r.d
v,;
''Fig. U.t
Parameters of
d
p~
to
Po '.
gas flowing over a body
Dow with a diminishing velocit.y (M < 1. dY < 0). the cross section increases. which is indicated br the inequality dS > 0 following from (3.6.4). Let us consider a supersonic Row (M> 1). If the velocity decreases, then. as can be seen from (3.6.4). the differential dS < 0, and. consequently. the jet converges. Conversely, when the velocity grows. the value of dS > O. Le. the jet diverges. Let us take a nozzle that first has the shape of a converging. and then of a diverging channel. In defmite conditions in the converging part of the nozzle. a subsonic now is accelerated, reaching the speed of sound in the narrowest cross section [here dS """ 0 and, as follows from (3.6.4), M = 1J, and then becomes supersonic. This is how nozzles are designed in rocket engines, gas turbines, and wind tunnels intended for obtaining supersonic flows. Flow V.locHy
Let us consider a gas jet Rowing oyer a surface (Fig. 3.6.1). We shall denote the free-stream parameters by Y .... P 00. P 00, T ,." L.,. and a .... and the parameters for the part of the jet in the disturbed region by the same symbols without subscripts. To lind the \'elocity in an arbitrary cross section of the jet. we shall lise Eq. (3.4.14) in which we shall determine the constant C according to the preset parameters of the frce stream: C = Y!.12 1~ (3.6.5)
+
Wit.h this in view. we have yIl2 i = V!./2 .:. too whence
+
V-VV:.-,.2(1~
I)
(3.6.G)
At the stagnation point. V:....: 0, consequently the enthalpy is t = io = V!..:·2 -:- i... (3.0.7)
Ch. 1. fynd~m.,nt.ls of Fluid Dyn~mics
151
Hence. t.he constant C as regards its physical meaning lcan be considE!red as the stagnation enthalpy. With a "iew to this valne of C. the "elocit r in t he jet is V~V2(i,-i)
(3.6.8)
The stagnation pressure Po and density Po correspond to the enthalpy i o• They are determined from the condition i o"'""
k:1'~'=- ~~~~. i ..
=.x;.+ k~l
.
~:
(3.6.9)
We can rewrite Eq. (3.6.9) as follows:
';:+ k~1 ·f:·
k~l .~
Sillce the flow is isentropic, we have pip' .~ p,lp\
(3.6.10) (3.6. \I)
consequently (3.6.12) Seeing that for conditions of stagnation the speed of sound is ao = VkPo/po. we fmd (3.6.13) Examination of (3.6.8) reveals that the velocitylalong the jet grows with diminishing of the enthalpy, and therefore more and morc or the heat is convcrted into kinetic energy. The maximum velocity is reached pro\'ided that the entbalpy i = 0, i.e. all the heat is spent to accelerate the gas. The \'alue of this velocitr is Vmu:= V2i;,
(3.6.14J
or with a view to (3.6.9):
VIPU=Vk~1·~=ao l/k~l
(3.6.15)
Accordingly, the velocity in an arbitrary section is V~Vm .. VI-iii,
(3.6.16)
V=Vmu. 'Vi_(pIPo)(II.-t)ill.
(3.6.17)
or In the narro'.... est. critical, cross section of the jet, the ,·elocity equals the local speed of sound. The latter is caUed critical nnd is
162
Pt. I. Theory. Aerodynillmics of ilIn Airfoil lind II Wing
designated by a*. The critical pressure p* and density p* correspond to the critical speed. It follows from the Bernoulli equation that for the critical section we have a· 2 , k
p.
-y -.- k=T' p* =
Po
k
-r=T 'P;-
Having in view that kp*/p* = a*z, the critical speed is a*=
V k~1 '*=00l / k!l
(3.6.18)
or, taking into account (3.6.15),
a* ~ Vron Y"(k:---C1")!"'(k"~c"1)
(3.6.19)
To determine the local ~ound speed a, we shall use Eq. (3.6.10). Performing thc substitutions a2 = kp/p and ~ = kpo/Po in it, we obtain (3.6.20) or a'1=
k-;i
a*'1_
k"2l
V2
(3.6.21) (3.6.22)
a2= k-;l (vfullJ.- V2)
Let us introduce the velocity ratio (relatiw ,'elocity) '}. = Via"'. Dividing Eq. (3.6.21) by VZ and having ill view that the ratio Via = = M, we can establish the relation between'}. and M: 1.' ~ I(k
+ 1)/2IM'/{1 + I(k -
1)/2IM')
(3.6.23)
Hence it follows that in the cross section of the jet where Vmu is reached, the number M = 00. We find the corresponding value of '}. = Amal< from (3.6.23) pro·vided that i.V -+ 00: 1.ruu~ Y(k '1)/(k
1)
(3.6.24)
Evidently, in the critical section where M = 1, we
1<;;; 1.<;;; Y;;;(k"+--'1"')/"'(k:--"1)
(3.6.2:,)
Pr.....,.., DensIty. lind , emp.rature It follows from (3.6.17) that the pressure ill an arbitral'Y cross section of the jet is (:1.6.26)
Ch. 3. Fundamentals of Fluid Dynamics
1~3
By (3.6.22), the difierence
1- l'I:x = (1 -- "-;-1 M~ f'
(3.6.27)
consequent! y, p=Po(1.-
k;1
;lf~)-II/(II-I)=Po .• (.b/)
(3.6.28)
where the function it of the arguDlPllt M is d~termined by the prt'SSure ratio plPo. For the conditions of the rT(,Il-~tream no\\", fOl'mula (3.G.2:8) yields: Po=poo
(1..:...-YM';,)II.lk-ll,./;.,..
(3.G.29}
Consequently I
p=poo {
~~',i::.~)).;',·~~ f.-III-I) =p,~ ;~..~~)
{3.6.30}
From the equation of an acliabat p ph ,... PO'P:' in which P isreplaced in accordancll wilh formulas (3.fS.2:G) And (3.6.28), we find a relation for the density p = Po ( '1 _. r~:J 1,lk-1) .-, Po ( 1 _;' "'-;- I 'l/~) - I;I/!_ I) ~ POE (;it') (3,G.3'1}
where the function e of the argument ,If is determined hy the fntiq. of the densities pIpo. Using the equation of slate pipo = (p.po) T, T L) (;-\.1).:-\2) and also Eqs. (3.6.26) and (3.6.28) fol' the pressure and l3.G.31)· for the density. we have
T=To
(1-
r~:x) ~~-]'o
(1+ k;-I
.1lrl.~·roT(.,,) (3Ji.~J3)
where the function 't of the argullll'llt .11 is rll"'t('rmineci by the mtio oC the temperatures TIT o. Tables of the gas-dynamic fnnctions :t (,It). e (.1I), ilud t (.U) for values of Lhe exponent k from 1.1 to 1.67 fll'C given ill 11:!!. We shall determine the corresponding fl'tations for llU' dlmsity P&and temperature To by means of expressions (:{.6.:~1) and (a.fi.:i3). For the conditions of a free-sLrNun now, lh('.o;c expl'('~~ion.s yield:
Po=p""
(1 + k;1
,"':or/{k-I)'--f:~~~"")
O.tl.34.) (3.G.o;,)
184
PI. I. Theory. Aerodynamics of an Airfoil and II Wing
III the critical cross section of the stream, M = 1. Consequently. f~om (a.(}.28), (3.6.31), and (3.6.33), we obtain the following formulas fo~ the c~itical yslues of the pressure p*, density p*. and tempe>rature 1'*: P* = Po ( k~t )"/{h_l) P*=Po
(k!t
= Port (1)
(3.636)
)t/(Il-t)=P08(l)
(3.6.37)
T·~T.( k~1
)=T.T(I)
(3.6.38)
where n (1). e (1), and 1" (1) are the yalues of the gas-dynamic functions at M = 1. The above formulas, suitable for ally speeds, can give us approximate relations for cases when the numbers M arc very large. A glance at (3.6.30) reveals that when M>- 1 a.nd M <10 >- 1. we have (3.6.39) Similar relations for the density and temperature have the form plp_ = (M .IM),,,h-.) (3.6.40) TiT _ - (M .IM)' (3.6.41) Using relation (3.6.27) for the velocity of the free-stream flo ...... ,
(3.6.27') for the conditions Moo >- t and M >- 1, we obtain the following .approximate formula for the local velocity:
(3.6.42) Flow of • O.s from • Reservoir
Formula (3.6.12) allows us to determine the velocity of a gas .flowing out through a noule from a reservoir (Fig. 3.6.2) in which the parameters arc determined by the conditions of st.agnation corresponding to a velocity of the gas in the reservoir of V ~ O. Such conditions are ensured in practice if the critical cross section of the nozzle is suffiCiently small in comparison with the cross:iectional dimension of the reservoir. In the critical section of the nozzle, which the value of the derivative dS!d~' = 0 corresponds
Ch. 3. Fund.ment.1s of Fluid DyMmic'
Fig. 3.6.1
Paramett'fs of
It
1I!1f~
gas nowiog from a res('n·oir
to, the Ma ch H1Imber, as can be seen from (a.!i.4), equals unity. i.e. Lit£' \'elocity in this section equals the local speed of sound:
V=a*=
V
k::J·
~:
"-'no
l/·I"-~ 1 ··'l /" 2:~~o
Such n speed is ensllred providc(1 t hat the pressure in t he reservoi r accordiug to (~.(l.3G) is not lower than thc vaillc
Po .• p*(¥)I,(I'-I)
(3.6.103)
If the nozzle communicatt>s with the utmospheric air whose prelSSure is p,. ,-: 1O~ Pa (this pressu.·e is call1'd the backpressure), now through the uonlc is possible whcll p* ;;;;:, 1O~ 1',1. Assuming that p* =- 10:' Pa and k = 1.1, we obtain the pre5sIIre needed to ensure the sp('('d of sound at the ant let of a eO[l\-erging nozzle:
1'0 = 10"( 1.4;-1 )1 ."(t.'-t)~".fI :-: J(lS Pa When the critical partuncters are re-uelle-d in the throat of a nozzle, a further reduction of thl' bnckpressure (pa < p*) no longer nfleets the \·;IIIIt'S of a* und 1'*, whi el. Ilepclld on the stAte of the gas in the reservoir. But Ill're conciitiolls (\fl' crl:'ated fot' the nppearance of a s uprrsonic specil of th e gAS in the diYergillg port of the TlO1.zil'. In Fig. ;Uj.2 . tllt'se c()J\d iHo[1~ of iscutropic motion are charaet('rhed br l' IIt'VPS 1 d(·termilling thr cilaugc ill tlt~ ralio " lpo and in the lIumber M
1156
PI. I. Theory. Aerodynllmics 01 lin Airfoil lind II Wing
Let us consider another set of possible condilions of isentropic now along a nozzle. Let us assume that the backpressHrc grows to the value p(:l > p(~l at ,,,hlCh the pressure p~~' at the exit reaches the same value as pi:). The subsonic flo,,, formed in thill case is characterized by curves 2 shown in Fig. 3.6.2. We find the veloci\.y ratio').. (M) of this flow by Eq. (3.6.23), choosing from the two solutions for').. the one that is less than unity. Upon a further increase ill the ll1lckpressure to the value pl~), conditions appear in which the pressllre p(~) at the exit will equal the backpressure, i.e. p(:) = p(~>, while the pre~sure in tl1('. throat will exceed the critical pressure p*; the rel~'vant speed in this croSs section is subsonic (1" < a*). Consequently, in the diverging part of the nozzle, the speed lowers, remaining suhsonic (curve 3 in Fig. 3.G,2). At pressHrf's at the ('xil smaller tILall p(~) and larger then p(~l, the isentropic uature of the flow is disturbed. At a certllin cross section of the !lolzle. a jump (di~contill\lily of the parmncters) occurs, the trnm;ition thl'ougl, which is attended by sharp deceleration of the flow. As a result of Ihis ciece\(»'atioll, having an irreLwsible adiabatic /lature, the supersonic. flow trallsjorms abruptly into a subsonic one. Till' change in the prcssure and in the number .}1 along the nozzle in these flow cOlulilions is characterized by curves 4 (Fig, 3,G.2). The surface of discontinuity of the parameters (shock) must be in a cross section with a 1Iach number M sucl) tltllt would correspond to the pressurc ps past this surface ensuring a gas pressnre of p~l' (p(Jl < plJ> < p
q
With a view to relations (iLG.19) and (3,(UH), we ftllt\
p: Po (1 -
VI~;:JI/(h-l)= Po (1- ;~! Po (1 _ ~=~
')..2)l!(1t-1i
. :':t
r/(l'-I)
Ch. 3. Fundamenteb of Fluid Dynamics
1a7
':~----i i
o.!
i
,~o
o
1.D~
Afler now det.ermining t.he rat.io Po i' from (3.6.37), relation (3.6.45) in the following form:
q(1. )~:)..( 1-
!~:
we obtain
A;1)'1(1t-1)( i.'~1 r:{II-Il=T
(3.6.46)
q
The function (A) is called till' reduced mas:; density. Taking formula (3.6.23) into ac·count, WI' can determine this function of the argulnl."nt M: q{.t1}"",,~J[~(I-
k.;1
jJf~)J-(l·';'I):[~tlt-:)J=~7
(3.6.46')
Tflhulated values of the function ~ withill till.' l'unge of "alues of k from 1.1 to 1.67 ill'e given in Ii:!]. Il foUo,,"s ft'om 111(>:;0 "elations t.hat the number A (or itI) in i\ cerl
q
Gscc=S*
(k~.'
f/(/-I)
l / k:~1
poro
where Pn i ... the density of I·he golS in the reservoir.
(3.G.Ii7)
HIS
pt.
I. Theory. Aerodynemics of en Airfoil end eWing
Upon the motion of an incompressible fluid. the mass flow rate equation in accordance with (2.4.51) and provided that p = const ' becomes VS = const (3.6.48) Kinematic equation (3.6.48) completely characterizes the OIlCdimensional now of an incompressible fluid. It shows that the flow velocity is inversely proport.ional to the cross-sectional area of the channel. The pressure ill steady flow is calc\tlated by the Bernoulli equation (:J.4.12) or (3.4.13).
4 Shock Wave Theory
4.1. Physical Nature
of Shock Wave Formation A fealure of ::;lljJN.'
g
flows is that deceleration is attcutled
by the formaLio!! of discontilluity !;urfaces ill them. When a gas passes through thest' slll'far('s, its parameters cilang{' abruptly: the \'clority ShOl'ply dimitli:-:hrs, [Iud the rll"('ssnre, temperature, nlld density grow. SUcil COlllillHily :-udaccs muviog 1'('lative to a gas arc often referred 10 flS shock W8\'PS, while the imnuJ\"flbJc discontinuity
surfaces arc ealled stationar)' shock waves oi' simply shocks. fit'low we cOllsider the condilioll:-: of gas nllw flfler stationary shock W8\"CS.
and we shall usc the term "shock",
(IS
a
r11iC'.
In the most general ca~, a shock 11<1$ a CHl'y('(1 sbapC', Figlll'C' ~.I.la shows schematically an attached curved shock fornH'd in a flow past a sharp-noscd hod~', and Fig. 4.1.1b-'1 detached cuned shock flPfll'al" ing ill front of a blunt smfil('(' in a ~upel'~onic flo\\,. LjlOn the supersonic flow past tI hody with ~t['aighl walls. an altached straight shock may appear (Fig. 1.4.1c). A glance at Fig. 4.1.1b reveals Illal the surfilce~ of sho('ks llIa~ ol'icnte(\ along a Ilorlllal to t.he free-stream velocity (the sho('k 8. n/2) or inclined at. an angle olllf'r than a rig-II! Olll' (8, < III the first case, Ihe shock is said 10 be normal, in Ihe oblique. An attached c1ll'\'cd shock ('.an ('viilf'nt 1\ bo a set of oblique shocks, and tariH'd shuck H'" ('onsisling mal shock ann a system of obliqw! ~11O('b. The form[llion of sltocks is (1m' 10 the ~pecille nature of propftgatiou of dist.llrbances in 11 sllpersonic gas now. B)· a di!~ at point () 0) (Fig. 4.1.~), 811('h di:oLurbanc(>s propagate ill a gal" at re!'lL in all (jir(>ctions at tlJC I"peC'd of sonnd a ill Ihe form of sphl'ricaJ waws in space or circnlM waves ill fI plane (Fig. 4.1.20). AI the instant I. thr. rndbls of a wave is r = (It. If fl. suhsonic glls fiow is
=
n-;--
160
PI. I. Theory. Aerodynamics of ~n Airfoil and a Wing
fig..U.t
,shocks; a-attaCMd curved shock;
b-d~!aclled
cur~d
sboC'k: e-attacMd straight .hock
(6)
(aJ
'fi,.4.t.l
Propaga tion of disturbance. in a gaB: a.-g.s at rest;
b-!lub~onll!
OOWI e-supt'rsonlc flow
incident on a source (V < a, Fig. 4.1.2b), the waves are carried downstream: the centre of a wavc travels at a speed of V < a. while tile wave propagates at the speed of sound. During a certain time t, the centre of tile ",,,ve tra\'els tile distance Vt, wlLilc the radius of the wave r = at, here at > Vt. Hellce, in a subsonic !low, disturbances also propagate up!;tream. In the particular case of a sOllic velocity (V = al, the front of spherical or circular disturbance waves is limited by a flAt vertical surFace or straight line tangent to a sphere or circle, respectively, and passing through point 0 because in this case the distance over which the centre of the wave travels dnring the time t equals its radiu!; at the same instant t. Let 11S assume that the free-stream velocity is supersonic (V> a). During' the time t. the centre of the wave travels the distance Vt, while a sound wave covers the distance at. Since at < Vt, [or all the spherical sound waves we can draw an enveloping conical snr[ace (Fig. 4.t.2c), Le. a disturbance cone (or !tfach con~). On a plane, disturbance lines (or Mach lint's) are envelopes of the family of circular waves. Tho di!'t\lrhancc~ arc the densest on the disturbance
Ch. 4. Shock Wave Theory
161
Fitl· U •1
OriglnatioD of a shock
cone or lines that are boundaries of a disturbed ann uodisturbed region because all the sound waves on this cone are in the same phase of oscillation-the compression phase. Such disturbed regionsr.onica. or plane waves confined within straight Mach lines, are called simple pressure waves or Maeb waves. The angle Il formed by the generatrix of a conical wave or line of disturbances i~ determined from the cond ition (Fig. 4.1.2c) that sin Il = at/(Vt), whence sin J.1
=
1/lW
(1,.1 .1)
The angle J.1 is called the Mach angle. A ~mpersonic flow carries all the sound disturbances downstream, limiting their propagation by the Mach cone or lines inclined at the angle ~L The front of a pres~ ure wave propagates at the same speed of ~o\lnd as Lhe front of a spherical (or circular) wave. This is why the projection of the freestream velocity onto a normal to the WII.\'C' front equals the speed of sound (Fig. 4.1.2c). In a simple pres~ure wave, a~ in a soun(1 one, the gas parameters (pressure, density, etc.) change by an infinitely small magnitude, which is indicated, particularly, by the relaLioli for the speed of sound a = 11 dp/dp known from physic~ . In the disturbed region. the velocity remains virtually the same a~ in the undislurbed flow. Therefore, a simple pressure wa ve can be con ~ idered as a shock (or shock wave) of all infinitely small strength, and we can assume for practkal purposes that the paramelers do not change when traversing it. This is why such a simple pressure wa.ve is al so called a weak sboek wave, while its front (Mach line) is called a. line of weak disturbances or a wavelet. It is natural to assume that the formation of a shock of a finite strength is associated with the superposition of simple pressure waves and, as a result, with their mutual amplification. Let us consider the process of formation of such a shock taking an oblique shock as an example. Let us assume that a supersonic flow initially travels along a level and smooth surface (Fig'. 4.1.3). We create artiflcially a 10rnl pressure increment at point A hy turning the flow through the inrlllitely small angle d~. This prodll r e~ a simple pres.q-
162
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
ure wa"e AR emerging from point A as from the source of clisturbam'c and inclined to the surface at the angle }1. If we again tUrn th(' flow slightly through the angle 6~, a new simple wav" AC is fOI'llH'd that emerges from the same point A. but iii higher tllfln the Jirst wave, But in II. supersonic flow, as shown above, waves ('an not propagate upstream. therefore the wave AC will drift downslream until it coincides with the first one, A more intensive wave is formed that if; considerably amplifled upon a furt.her turning of the flow. The shock of a finit.e strength formed in this way hali a speed of propagation higher than the speed of sound at whkh the simple prelisure wave travels. Therefore, the lihock of fmite strength nllliit deviate from the liimple wave AD to the }t'ft and occupy the new pOliition AD. Here it is kept in equilibrinm because the speed of its propagation equals the component of the free-stream "elocity along a normal to the shock front V sin a" where as is the angle of inclination of tht' shoc_k. It follows that the angle of inclination of a f;hock of finite strength is larger than that of the Mach line (cone), Le. 9s »!.
".1.
Genera. Equations
'or. Shock
We shall consider the more general rase when the ga.~ behind a shock, owing to substantial heating, experienr.es physicochemical transformations and changes its specific heat. Of major signific,anco whell f;tudying shockli behind which oscillations are generated and dissociation, ionization, and chemical reactions occur are the rates of the physicochemical transformations. Proref;ses hehind shock waves arc characterized by a fraction of the kinetic. energy of the moving gas virtually instantaneously transforming into the internal energy of the gas. In these conditions, we cannot ignore the fact that thermodynamic equilibrium is achieved after a certain time elapses only in conditions of such equilibrium do all the parameters experiencing discontinuities (the pressure, density, temperature) become time-independent. The analysis of these phenomena is a more involved problem and is associated primarily with studying of the mechanism of non-equilibrium processes, and with a knowledge, particularly, of the rates of c,hemical rear"tions in the air, The simplelit case is characterit.ed by an infinitely high rate of the physicochemical transformations and, consequently, by the inst.antaneous setting in of thermodynamic equilibrium. Such processes behind shock waves are pOlisible physically, which is confirmed by experimental studies. Let llS ronsider the basic theoretical relations allowing one to evaluate the equilibrium parameters behind a shock wave.
Ch.... Shock Wive Theory
169
Obll... Shock
A J;hock formed ill real conditions is charact('rized by a certain The parameters of t.he gas in sllch 1I ~hock change not inshntflllconsly. but during a certain time intel'\'ul. As .c:hown by theoretical alld C!xperimentlll in\'estigations, howe\'er, the thickness of a shock is \'ery small and i.e: of the order of the mean free path of the lllolerulct:. Fot· IItlllosphel'i(' condilions, calculations yielded the following vallle:-: of the thickness of a shock measured in the direction of the frce-l:::tl'enm .... t'locity: thid,:ues~.
:\Iach nnmb('r Moo . . . , ' . Thicklll's!!. nun
t.5 . . . . . , . 4.5xtO-·
1.2xH\'"~
to' l).7xto-' 1I.2xt()-&
For M oc- = 2. t.he thickness of a shock equals about four molecular free pnth:-:, and for M co .... 3-about three. Therefore, when studying a shocl< in an ideal fluid, this thickness may be disregarded and the shock represented in the form of a geometric discontinuity surface for the gnJ; paramcters. assuming these parameters to change inst.ant:meously. 0111' t.ask con~~ts in determining the unknown parameters of a gas hehind II shock aecording to the preset parameters c,harartflrizing 1he now of Ihe gas ahead of the ,c:hock. For lin oblique shock formed in a dis~ocitlting and ionizing gas, there tire nine lInknown parameters: the pres~ure Pz, density Pz. tempt"l'fl.lnr£' T:?" nlocity V'l , enthalpy f z. ('ntrop~' S2' .c:peed of .e:oulHl 1I2' the mean molar mas.<; ~ln\2' and the l!hock angle Us (or the flow de\'illiioll angle ~s). Consequently, it is lIecessary to compile ninc simultaneolls equntions. The parameters ahead of a shock wilt btl the known olle.<: in these equations. nam('ly. the pressure Pu density PI' \'Clarity V l' etc. Instead of the velocity V'2 behind a shock. wc ran dt"termine itl! eomponents along a normal V n'l and fl. tangent. 1'12 to 1.11(' !liloc}.;. This will in('rea~e the number of equatious needed to ten. Th~e (>qualiom; include the fundamentnl equations. of gas dYJlamir~ (of motion. continuity. energy, and state), a number of kinematic relations for the \'elodties, and also thermodynamic relations. t'haractcrizing the properties of a gas. Let us consider each equation of this system. Figlll'c 4.2.1 S}IOWS triangles of the flow velocities ahead of a shock (the parametcrs with the subscript 1) and behind it (the subscript 2), We flhall use Ih(' Jignrc to determine the following relAtions for these components: Vd = 1/ 2 COS (Os - 13.), Vn'l = Vl! sin (9. - 13,) (4.2.1) This yields the first equation of the system of simultaneous ones: Vn2IV'1'2 = tan (9, - 13,) (4.2.2).
164
pt. I. Theory. Aerodynamics of an Airfoil lind II Wing
The continuity equation (or the mass flow equation) is the second one. It determines the amount of fluid passing through unit surface of a shock in unit time: (4.2.3) here VD1 = VI sin 9. is the normal component of the velocity ahead of tho shock (Fig. 4.2.1). Let us use the equation of motion reduced to the form of an equation of momentum for the conditions of the passage through a shock. This is the third equation of the system. We shall obtain it by assuming that the change in the momentum of the fluid passing in unit time through unit surface area of the shock in the direction of a normal to this surface equals the impulse of the pressure forces: (4.2.4) Pl'VAt - P2VAI = pz - PI With a view to Eq. (4.2.3), we can write this equation in the form P1VD1 (V.I -
Vnz) = pz - PI
(4.2.4')
Equation (4.2.4) can also be written as (4.2.4') In this form, the equation expresses the law of momentum conservation when passing through a shock. If we consider the change in the momentum in a direction tangent to the shock surface, then, taking into account that the pressure gradient in this direction is zero, wo obtain the following relation: P1 VD1 Vn - PZVuZVd = 0 whence, when P1Vnl = PIIVnll, we have V'I"l = V"z (4.2.5) Equation (4.2.5) is the fourt.h one of the system. It indicates that the tangential components of the velocity when passing through
16~
Ch. 4. Shock Wave Theory
a shock do not change. A glance at Fig. 4.2.1 re\'eals that = VI cos at> therefore we can write (4.2.5) in the form
V'!:l
=
(10.2.5') We can write the equation of energy conservation in the form il
+ V: 12 =
i2
+ V!/2
Provided that
= V~I +
Vii' V: = ~2 + Vh. and V'l ,,.... 1'1'2 the equation of energy conservation can be tran~formed: + V~1/2 = i2 + ViJ2 (4.2.6) Dy combining the equations of state for the conditions ahead of the shock and behind it. we obtain the relation V~
'I
Pa - PI = R 2 P"T a - R 1P1 T1 or, taking into account that R = Ro./fJ.m. we ha\'e Pa - PI = Ro. (p2Ta/~m2 - pIT1/~lml) (4.2.7) We can write four equations of the system being considered that allow us to determine the enthalpy, entropy, mean molar mass, and speed of sound in a di~sociating gas in the form of general depl"ndences of these parameters on the pre~511re and temperature:
fa = II (Pt. T'J') S, (p" TiJ
(4.2.8) (4.2.9) (4.2.10) (4.2.11)
="
Vm2 = 13 (Pa. T 2) Q" = /, (Pa, T 2)
These functions are not expressed analytically in the explicit form and are determined by means of experimental in\'estigations or with the aid of rather involved c.alculations based on the solution of the relevant thermodynamic equations. The relations for these fnnctions are usually constructed in the form of graphs, and their values arc tabulated in special tables of thermodynamic functions for air at high temperatures (see [6. 7. 131). Let us express the basic parameters behind a shork wave in terms of the relative cbange in tbe normal components of the velocity, i.e. in terms of the quantity 6. Yn = 6.V n /V nl = (Vnl -
Vn2)1V nl
(4.2.12)
From (4.2.3), we find the ratio of the densities: p,lp, = 11(1 -
61'.)
(4.2.13)
pt. J. Theory. Aerodynamic~ of an Airfoil and a Wing
166
and from (lJ..2.4'). the ratio of the pressures: P2/Pl =-" 1
+ (flIV~/pJl
(4.2.14)
6.f'n
Introdudng the concept of the "normal component" of the number Ml as the ratio Mnl = Vn1/al or Mnl = iJIl sin 8s, and assumillg' that ahead of a shock the gas is not dissociated and the speed of sonnd for it is at :-::: Vk1P1/Pl' we obtain pzlpl = 1 + I.\M~1 6.Vn instead of (4.2.11.). We nnd the ratio of the enthalpies iz/i 1 from (4.2.6): i 2/i l = 1 + (1I2i 1) (V~l - V~I)
(4.2.15)
Let us write the diiIsrence of the velocity squares in the form V~l
-
V~z =
V~l
(1 -
Vnz/Vn1) (1
+ Vn:JV
n1)
= V:ll 6.Vn (2 - 6.Vn) Hence i,li, -
1
+ (~,/2) (
(4.2.16)
To cietermine the ratio of the temperatures T 21T1 , we shall use the equations of state for the cOltditions ahead of a shock and behind it, from which Substituting for the ratio of the densities equation their values from Eqs. (4.2.13) and we .obtain TJTl = (1 + klM~I6.Vn) (1 - 6.Vn)
j.1m2j.1ml
(4.2.17')
To determine the velocity behind a shock, we shall use the relations V: = Vi V~2 and V: = Vi + V~l -lrom which we find
+
V;/~ = 1 -
(V~l/V;)
6.Vn
(2 -
6.Vn )
(4.2.18)
Since then (4.2.18')
Let us find a relation for the flow deviation angle behind a shock. fly (4.2.2) and (4.2.5'), we have 6.Vn = 1 - tan (8, - ~8)/tan 9s
(4.2.19)
Ch. 4. Shock Wave Theory
167
Hence, having in view that ~s) =
tan (Ss -
(tan as - tan ~1I)!(1
+ tan Ss tan ~.)
we obtain
tan~s=tanas 6V~ (~+tan:te~)-I i-6Vn
i-Vu
(4.2.20)
The relative change in the velocity .1.Vn is determined in accordan('·e with (4.2.13) by the dimensionless density
<1.1',
=1-
p,/p,
(4.2.21)
This shows that the ratios of the pressures, temperatures, and enthalpies, and also the now deviation angle ~s can be written as functions of the relative den.<;ity P:a/Pl' In addition, we can introduce the values rill = VI siu a" and Mnl '-= MI sin 0, into tho formulas instead of the quantities Vnl and MOl' respectively. Iience, the solution of the problem on an oblique shock when its angle of inclination 0$ is known consists in flllding the ratio of the densitil'>s P2/Pl' or. whirh is the same, in determining the function .1. Vn' Tllis ftlllction is determined with the aid of expressions (~.2.16) and (4.3.17'), earll of which can be written in the form of a quadratic equatioll in .1.I'n. From the first quadratic equation, wo have
.1.Vn -, and from the second
i-Vi
2(t'&
il)/V~1
(4.2.22)
(4.2.23)
where
A=
k;:I~~1
B ... · k'~~1
(*. :::
-i)
(4.2.24)
The 5:igns of the square roots-minus in (4.2.22) and plus in (4.2.23)-were rhosen with a view to the fact that the velocity behind a shock i$ always lower than ahead of it, and, consequently, 41Vn < 1. The plus sign in (4.2.2.,) also indicaLes that physically the largl'>r of the two values of .1.Vn < 1 is real. Equations (1.2.22) and (4.2,23) are solved by the method of successive approximatioIl!'!. The problem of an oblique shock can be solved if the angle ~$ is given. Here the angle as is calculated with the aid of relation (il.:!.20), in accordance with which tan:!9 _ lanOI "
tan Ps
6j7~ 1-6Vn
+_'___ =0 1-6Vo
168
pt. I. Theory. Aerodynolllmjes of en Airfoil end
Tho solution 01 tan 8s =cotf3s
thi~
oil
Wing
equation yields
± V~--.--:""'r~:-_-_-,,-",-~-.]
[.!..
AV:
2
I-LW n
4 (1-6.Vn)~
(4.2.25)
t-6.V"
One of the l'(olutions (t.he plus sign before the root) del ermines the larger value of the angle as realized in a detached clln-ed sIwek. and the other (Lhe minus sign) determine~ the .'Illlallcr VahH!' realized in an attached shock with a lower strength. To fadli1.ate calculations, we can evaluate the angles as beforehand from the given ,'alues of ~s' and compile a corresponding table or plot a graph. With their aid. for a value of dVn which previoll.'\ly calculated values of the ratios PZIPl' P2./P1' etc. correspond to. we can find one of the angles es or ~s from the other known one. The velocity VI and the number HI of the free stream at which the parameters are realized in a shock with the gi\'en values of the angle O. (or ~s). and also ]-'111 (or Mill) are calculated by the formulas V l = "Vnl/sin e., MI = Mnl/sin Os (4.2.26) Norm.. Shock
The formulas for rakulating a normal shock can be obtained from the aho"e relations for an oblique one if we assume that as = = n/'2 and f3. '-'-' O. Accordingly, the \'elocity VIII = VI' and the number Mhl = MI' ~ow the basic relations acquire the following form: (4.2.27) Pa1Pl = 1 + kl~dV (4.2.28) izli l ,..- 1 -:- (V:12) (d ViiI) (2 - d V) (4.2.28) T.;.IT 1 ,-- (1 + k l M!4V) (1 - dV') ).lma~ml (4.2.30) l':Il': = t - dV(2 - dV) where the change in the relative velocity (4.2.31) av ~ aVIV, ~ (V, - V,)/V, i~ determined with the aid of expressions (4.2.22)-(4.2.2-1): llV=t-V1
2(i z
il)IV:
aV-A+VA'-B in which
A=
kl~:;-/
.
B-.
kl~Y
(if, ~:~ -1)
The relation between dV and tile relative by the formula
(4.2.32) (4.2.33) (4.2.34)
density is determined (10.2.35)
Ch. 4. Shock W"ve Theory
169
We have given the general relation::; for ~hork:-;. \"ow let 115 use them to analyse the nature of flow and the W,lyS of rakul<1tillg the parameters of a gas behind sllOck!> for constant specific heatg
varying specifte Mats. 4.3. Shock in the Flow 01 a Gas with Constant Specific Heats
Of major theoretical and practi(oal interest i..; till' problem 011 the flo,," of a gas behind a shock when t.he spedli<' hEllits c ,• and Cc are Cotl!->lnllt. Although such ;1 flow is considered to be a particular (idealized) case of the flow of a gas whose physi<'odlemical properties change to a greater or smaller extent when passing" through a shock, nevertheless t}le results obtained in solving thb problem make it possihle to comprehend the geHeral qualitafin: nature or a shock transition. The relations characterizing the change in the parameters of a gas when passing throngh a shock LIre obtained here in the expli("it form. They can al~o be used ror an approximate quanWatit-e C',,\imation of these parameters when the more. generill ("ase of varying ~pCt·ir.c heats is being treated. Tlll~ prohl!'1H bring considered also has an indcpcndellL signifIcance hC("ilHse ils :
The method of calculating an oblique shoek treated below is on the usc of II system of equations that is a particular case system (4.2.2)-(4.2.11). If in a shock transition the specific heats do nut change, it stHluld also be assumed that the meaH molar mass remains const.ant, Illld the speed of sound and the enthalpy depend only 011 tbe temperature. Equations (4.2.8), (4.2.10), and (4.2.11) aCl'ording}y ucquirc the following form: i2
Cp T2
(4.3.1)
)lm2 = )lml = )lm = COlist
(4.3.2}
a: '-'-' kRT2
(U.3}
110
PI. I. Theory. Aerodynamics of an Airfoil and a Wing
Instead of Eq. (4.2.9), it is necessary to wse tbe tbermodynamic eqllation for tbe entropy of a non-dissociating perfect gas: dS ~ di/T - dp/(pT) Having in view the equations di = cp dT and dp = d (RpT) = R T dp, we obtain dS = Cp d In T ....., R d In T - R d In p
= Rp dT
+
Uut CI• - R = Consequently,
and Rlcu ---= (c p
CD
dS =
Cc
(d In T -
-
(k -
cu)lcD --= k - 1 1) d In pI
Integrating this equation at k ..., const, we find S = Cu In (TlpLI) + const
(4.3.4)
It follows from the equation of state p = RpT that Tlp"'_l = p/(pAR), and In Ip/(pIlR)I
-----0
In (p/pA) -In R
Introducing the quantity In R into the constant on the righthand side of (4.:3.4), we obtain II relation for the entropy in the form (4.3.5) By applying this equation to the conditions ahead of a shock and behind it, and then determining the difference between the -entropies, we obtain instead of (1.2.9) the equation for the entropy used in the theory of an oblique shock: S2 - SI -'= CD In [(P2lpd (p~/p:)1 (4.3.6) Equation of st.ate (4.2.7) for the case of constant specific heats being considered is simplified: pz -
PI
=
R
(p ZT 2 -
P1Td
(4.3.7)
Equations (4.2.2)-(4.2.6) of the system are retained with no alterations. Formulas for Calcuilltfnt the Parameters of II Gas Behind II Shock
To calculaLe the density. pressure, and enthalpy, the formulas needed are (4.2.13), (4.2.t5) and (11.2.16), respectively. To determine the temperature, the formula needed is (4.2.17') with the assumption made that !lmz ---'--' !lm': TJTl -= (1
+ kM~laVn)
(1 -
aVn )
(4.3.8)
Ch. 4. Shock Weve Theory
171
The ullknowlI quantity in all these expres~iotls is the change in the relotive velocity 6.Vn . Let us uetermine it hy assuming that the shock angle 8 s and the free-slream Mach numher Ml ure known. We :::hall use Eq. (1.2.4') for this purpose. After didding it by PIV nl = P2Vnz, we obtain pz/(pzVnz ) - P1i(Pl/ j 'nl) rill f"z (1.:n)) Inserting a;,k and ai/It here instead of P Zip2 and PIIPI' respectively, and using Eq. (3.6.21) for the specr\ of sound, W(l nnd
~(k-~-l a*z_.k;-t l';)-~(~a*:!-~V;) ~-Vnl-Vn2
Introducing the WI.1HCS of V; -"""' l'n~ -i- V: alHI V;' Vnf -7- Vi and multiplying both sides of tile equation hy r"l V n2 . we obtain after simple transformations
~';~1 a*2(V"I-V nz )-- {;;~1 l'i(V'll--Vr~)
+
k;;/ lfnIVnz(Vnl- VIIZ)
-
Vnll'nz(V,'l-'" V"z)
Excluding the trivi
V'lIVnz"'-·a*Z-
:.=
0 corresponding
~:;~ Vi
(4.3.10)
This equation is used to determine the velocity behind a shock. It is called the basic equation of an oblique Shock, and allows us to fwd the relative change in the velocity 6V n = (Vnl- Vnz)/Vnl . For this pnrpose, let HS write Vnl V n2 =-= V~I (V,,2/Vnl) and usc the relations VOl = VI sin Bs and V"t = VI cos AS. After the relevant substitutions in (4.3.10), we obtain sinz 9s(1-
6i'n)=*- :~~
(1-sin Z 05)
where Al = l\!a*. Substituting Eq. (3.G.23) for Al and performing transformations, we lind
AVIl = k~1 (1- Misi1n1oJ Let
llS
introduce the symbol 6 ~ (k -
I)/(k
+ I)
with a view to which formula (4.3.11) becomes 6 Vn = (1 - 6) [1 - 1/(M: sin 2 89 )1
(4.3.11)
(4.3.12) (4.3.11')
172
Pt. I. Theory. Aerodynllmic5 of an Airfoil and a Wing
By inserting this quantity into (4.2.13). we can determine the density ratio: pJPt = M: sin 2 66 /(1 - is iSM: sin2 ( 5) (4.3.13)
+
To determine the pressure ratio Pilpl. we shall.use formula (4.2.15). We shall assume in it that MDt = MI sin 6... substitute expression (4.3.11') for AVD • and replace the quantity k in accordance with (4.3.12) by the value k ~ (1 0)1(1 - 6) (4.3.14)
+
The result is the parameter pJPt = (1
+ 6) M: sint: 6~ -
(4.3.1n)
6
that characterizes the shock strength. We may also use the following relation for this purpose: (p, - p,)/p,
~
iJ.p/p,
~
(I
+ 0) (M: sin' e, -
(4.3.15')
I)
Formula (4.3.15') can be used to obtain another parameter characterizing the shock strength. namely, the pressure coefficient Pt: = = (Pi - pt)lgt • where ql = [(1 6)1(1 - 6}1 PIM;12. By subtracting unity from both the left-hand and right-hand sides of (4.3.15') and relating the ex.pression obtained to gl' we have
+
p, ~ 12 (I
- O)IM:I (M: sin'
e. -
I)
(4.3.15")
Eliminating the quantity M~ sin 2 6s from Eqs. (4.3.15) and (4.3.13), we find the relation for the pressure-to-density ratios-the Hugoniot equation: (4.3.13') p,/p, ~ (p,lp, - 0)/(1 - Op,/p,) This relation is also known as the shock adiabal. Unlike the conventional equation of an adiabat (isentrope) of the form p = AI'''. it shows bow the parameters change in a transition through a shock wave. This transition process is attended by a growth in the entropy determined from Eq. (4.3.6). Hence, the process of the transition through a shock is non-isentropic. ]n accordance with the change in the indicated parameters, the temperature behind a shock grows. Its value is calculated by the equation of state
-k....: *."*!-= «i+ll) MfSiDI8j;-:!]S;:1l+llMf sin Osl l
(4.3.16)
The dependences of the ratios of the gas parameters behind a shock and ahead of it (PJpl' P2IPl' T"ITt ) on the quantity Mnt = = M _ sin as for k = 1.4 (6 = 1/6) are shown graphically in Fig. 4.3.1. and the Hugoniot equation (4.3.13'), in Fig. 4.3.2.
Ch. 4. Shock Wave Theory
113
'~LTl-_1--;. _
5
-'
5
-<--; -r
I/- -
J
,f
.
. -.:- : -,-
-.
,--'
:
-~;-I1.
J
"
.- '
--:
5 sMr.1
~I:pe~~':nces of the ratios of the gas parameters behind a shock and ahead of it on the value of Mnl = Moo sin 9$ for k = 1.4 (6 = t/8): ~-denslty
Fig.
ratlo P./P,; b_prcssurc raiio P./PI; c-leonperatUf(' ratio TilT.
,u.l
Shock adiabat (t) and trope (2)
isen·
Ch'=t,lt, II =1/6)
Tile special Corm of Lhe transition Lhrotlgh a ::;11ock, tliITering from an isentropic process, manifests ilsC'if in the different nature of the change in t.he gas parameters. It follows from (4.:\.15) and (4.3.1G) that at .iJf, __ 00, the pressure and temperature grow without a limit. It also follows from (4.3.13) that for t.lle same condition lUI __ 0;) the density tends to a certain dcflOitc value equal to (P2lp,).11 ,.. 00 = = 116. When Ie ,..." 1.1, this yields t.he value of 1/6 ....::: 6. It. follows from the formula p = ApI< or T = Bpk-' (A nnd R are constants) that in an isentropic process an infmite increaso in the Ilcnsit), and temperatllre corresponds to an infinite pressllI'c iocl'eilsn. A comparison allows us to Mrivc at the conclusion that wilh nn identical change in the pressure in bo~h a 5hock and isontropic proc(>.'!ses, the former is attenderl by greater heating of the gas, and this is just. what facilitates a certain drop in the density.
174
PI. 1. Theory. Aerodynemics of en Airfoil and eWing
Formula (4.:i.16) determines the raLio of t.he squares of the sound speed in accordance with t.he relation a!la~ = TzlT l (4.3.17) Using tllis relaLion, and also Eq. (4.2.18'), we can determine the number M2 behind a shock. Inserting expression (4.2.21) for LlVn into (/1.2.18'), we find V:/V~ = cos 2 as + (P l/p2)Z sin z 6s (4.3.18) Dividing Eq. (4.3.18) by (4.3.17), we obtain an expression for the ratio of the squares of the Mach numbers: M!/~ = (TIIT z ) [cosZas + (P l/p2)2 sin 2 as] (4.3.19) where TIlT'). and Pl/pz are found from formulas (4.3.113) and (4.3.13), respectively. The formula for calculating M 2 can be obtained in a sOme\Vllat different form with the aid of momentum equation (4.2.4"). Let liS writo it in the form
*
("1 + ~ VisinZas) =
1+
1;- V;sin2 (as-i-f}s)
Taking into account that kplp = [(1 + 6)/(1 - 6)] pIp, and determining P21Pl by the Hugoniot equation (4.3.13'), we obtain
1-;;2~~~~~J)
(1 -I- :~~ M; SioZ6s) = 1+ !~~ Misin2(9s-f}~)
After sub5tituting for we obtain the relation
M~
sin'). 6s its value found from (4.3.13), (4.3.19')
Let us determine the stagnation pressure for the conditions of a flow behind a shock. Considering the Dow of a gas behind a shock and ahead of it to he isentropic, we can compile the thermodynamic relations: p2/p~ = p~/p'~,
Pl/p~ = Po/p~
where Po and p;, Po and p~ are the stagnation pressure and density in the regions of the flow ahead of the shock and behind it, respectively. These relations yield the pressure recovery ratio across a shock wave: "0 = p~/po = (PiPl) (Pl/pz)" (p~/p()" Multiplying both sides of this equation by the ratio obtain
(Polp~)I>..
we
Ch, 4. Shock Wave Theory
17~
Let us use the energy equation y2/2'7 cpT = const and write it for the conditions ahead of and behind a shock: V:i~ + cpT, = V:12 + c"T 2 At pOints of stagnation, V, = V 2 = O. 1L follo\\'s from the energy equation that the temperatures at these points are identical, i.e. Tn = T~ or, which is the same, Po/Po = p~.(l~. Consequently. 14.:UO) (4.3.20')
w11ere P1ip2 and P2/Pl arc found by (4.3.15) and (4.3.13), respectively. By (3.6.28), the stagnation pressure is (4.:3.21)
*
Consequently, 1(1 -:. 6) Mi sin 2 9~_6J(6-1)/20 (.U, sin 9~)(J.r6)"o'
'(1_6)-(IT6)/26
.<
, (1·1·t=b. Ur
)(I~
6);26
(1+lt~{$ .Ui~inZ(l$rl
(,'1.;),22) 6)/2(:;
Let us determine the stagnation pressllre coefficient behind <'In oblique shock -
Po --
PG-Pl
ql
-.~ {III .: 6) -11~Ism . ~9 - (t+6)"I~
$-
61"
1)/26
'(M 1 sin9ap+6)/6(1_0)-'·II.Nm
x
(',-6 Ml),>;"'20
(f+~Mi.sin2as 6 )0
6)/20
} -1
(-'1.3.23)
An analysis of relation (4.3.~O) shows that behind a shock of finite strength the pressure ratio p~/p() is always less than unity. The stronger the shock, the largor arc the stagnation pressure losses and, consequently, the smaller is the ratio P~/po. When establishing the physical nature of these loss(!s, we cannot consider a shock as a discontinuity surface; we must take into account that a real compression process occurs in a layer with a small thickness of the order of the molecular free path. I t is exactly such a process
176
Pt. I. Theory. Aerodyn
Wing
of transition through a shock in a layer that is possible because the existence of two contacting regions with a fmite difference of their temperatures, pressures, and densities cannot be visualized. It is only a mathematical abstraction. The transition through a shock having a small thickness is characterized by so great velocity and temperature gradients that in the compression regions the influence of skin friction and heat conduction becomes quite substantial. It thus follows that the irreversible losses of the kinetic energy of gas in a transition through a shock are associated with the work done by the friction forces, and also with the heat conduction. The action of these dissipative forces and also heat transfer within the compression zone cause an increase in the entropy and a resulting reduction in the pressure in the flow behind the shock in comparison with an isentropic compression process. Obllqua Shock Anlla
The parameters behind an oblique shock are determined by the number Ml and the shock angle Os. Its magnitude, determined by the same number Ml and the flow deflection angle ~B' can be calculated by using Eq. (4.2.19). Substituting for .6.1'D ill it its value from (4.2.21), we obtain a working relation: tan 9$ltan (9 s - ~B) = P2/Pl (4.3.24) Determining Pl/P2 from (4.3.13), we find tan 9s/tan (as - ~s) = M~ sin2. es /(1 - 6 + 6M~ sin 2 9s) (4.3.25) On the other hand, this relation allows us to find the flow deflection angle ~s behind a shock according to the known shock angle. Figure 4.3.3 presents graphically the relation between the angles 9s and ~$ for various values of the number MI' In the region to the left of tho dashed Hne, which corresponds to the maximum values of the angle ~s (Le., ~s. mu), tho value of the anglo as diminishes with decreasing ~s, and to the right, conversely, it grows with decreasing ~s' This nature of the relation is due to the different form of a shock. In the first case, the change in the angle as corresponds to an attached curved shock ahead of a sharp-nosed body. As the body becomes blunter (the nose angle increases), the flow deflection angle grows, and, consequently, the shock angle increases. The maximum flow deflection angle ~s = ~s. rna;.: is determined only by the given value of MI' This angle is also called the critical How deHection (deviation) anglc ~cr. The points corresponding to its values are connected in Fig. 4.3.3 by a dashed line. Investigations show that a flow behind an attached shock is stable as long as in the entire region behind it the flow deflection angle is smaller than the critical one. This flow is accordingly referred to as a subcritical one.
Fig. 4.3.3
Change in the now dcneclioD angle various numbers Ml
~s
depending OD the shock angle
as
for
Upon a further growth of the nose angle. the angle ~6 may become crHical. According to Fig. 4.3.3. Us value grows with an increase in the number Ml . From Ihl:' physical yiewpoint, lhi~ is (>X'piaill(>d by an increase in the strength of a shock, a greatt>l' density behind it. and, as a result, hr the shock coming close (0 the> surface of the bOlly, which leads to dellection of the flow through a larger angle. At u still larger no,~e ungil:', t.he now ul'hind Illl attached shock becomes unstahle, as a resllit or which the shock 1II0Y('S away from the nose, Behind such it (ldnched shock, n new ~lIthle flow region appears. It is chul'aclcrizcli hr tieflt'clioll thl"Ough 1111 illig-Ie also less than the critical one. Bill unlikell suhcriti('ai now, 1I1i~ one i!l- called sup<,reritieal. This delillitioll correspond~ to the f'lct that the nose angle of the bod)' in the now exceeds the value III which a shock is still attached. A detached shock changes its shape absoiutcir. which can be seen especially clearly in the example of a flow over a sharp-nosed cone or wedge (Fig. 4.3.4). As long as the Row is subcritical. the shock is attached to the nose and the gcneratrix of its surface is straight. The now around thick wedges or cones may become supercritical, upon which the shock detaches ilnd acquires a curved shape. At the point of intersection of the shock surface with the flow axis, lhe shock angle as = n/2 and. consequently, the parameters change according to the Ill\\" or a normal shock, In practice, there is Il secl ion of such a normal shock near the nxis. WiLh an increase in the distance from the axis, thl' ::oiEoek allglc O~ in a(',corriance with Fig, 4,3.4 diminishes. remainililf 011 II eN'lain section inrger than the value that a subcritical flow corr('~p()lld!< 10, The change in the flow deflection angle is of Ihe oppo!
178
Pt. 1. Theory. Aerodynemics of en Airfoil end, eWing
FW~8~~~~--Silhonic 7'#fJiori
Fig....,..
Detached shock ahead of a sharp-nosed b,;,dy
At the apex or the detached wave bellind its straight part. ~s = 0, and then it grows. At a certain point on the surface, the angle ~s. becomes critical, and then it again diminishes together with the shock angle. A..."! follows from (<1.3.25), at the limit when ~s _ O. the angle as tends to the value 111::- siu- 1 (1IM l ). Hence, the shock angles at a preset number Ml yary within the interval ~Ll ~ as ~ ~ 1t/2. A shock of an infinitely small strength that is a simple pressure wave corresponds to the value Os = Ill' On a curved shock (see Fig. 4.3.3), we can find two points corresponding to two different angles as that at a given MI determine the same value of the angle ~s. This angle is calculated by formula (4.2.20), which after introducing into it tho value of ~ Yn from (4.3.11) acquires the form taops =cot 9s (M; sin 2 9,-1) [1 - ( 1~6 -sin z 9s ) Mn- 1 (4.3.26)
A strong shock with a subsonic flow behind it corresponds to a larger value 01 the angle as. and a weak sllOck with a supersonic flow behind it (if we exclude the vicinity of the shock with angles ~, close to the critical ones, where the flow may be subsonic) corresponds to a smaller value of the angle 9s. An oblique shock angle can be evaluated by using formula (4.2.25). Substituting the relative density from (4.2.2t) for LlVn • we obtain
t.ne.- cot
±
f!,[H*;--I)
1/+ (*- t)2 -tanZ~~ 1;-]
CO.3.2i)
On each curve shown in Fig. 4.3.3, we can indicate a point ('orresponding to the value of M 2 = 1 behind a curved shock. By connecting these points with a solid curve, we obtain the boundary of
Ch. 4. Shock Weve Theory
179
two flow conditions hehind snch ,I s-hol"k: to the left of thl' cune the Oow will be supersonic (M 2 > 1). aud to the right of it-subsonic
(M,
Hodogr.ph
In tlflditiOIi to llll anal~·t.icnl solution of lhe problem of deterlllinill(!" the Row pal'ameters bellilld an oblique shock, there is. a graphical method based on the concept of a hodograph . .r\ hodograph is a CUl"ve Iormillg t1w locus of the tips of the \"(~Iocity vectors in the plane behind a shock. Let us consider the equation of a hodogruph. Let point A (Fig. ·1.11.1) be the tip of the "elocity ,'ector V 2 lind 1.11' located. (~ollsequ('nll)", on Il hodograph cOIl~lrllcted in fI coordinate sp:.tem whos(' horizontal a.xis c!lincilles with the dir~ctioll of the \"1·loeit.~· VI ahead of tl shock. Hence, the inclination 01 the v{']ocH.}" vector \'2 i~ determill{'d h;\" the ang],· ~s. LI"t. lIS (]esigllate lhe "crliell( lind horizontal components of this "e(oeity by 1/' !lll(lll. r('!oi.p('cth·{']y. A glanc{' al Fig. 4.4 ..1 rc,,{'als that u and 16 can he expressed in terms of t.he normal V U2 and tangential V,. components of til{' "elocity V 2 10 lh{' plane of the shock as follows: u ~- Vt cos as. Vll2 sin Os. m = 1',. sin O~ - VD2 cos a, (1.4.1) Wl' dctl"rmillc the component. l'Jl2 from formula (4.3.10) in which a~SHme thal rIll :--: VI sill Os and r,. -' 1-'1 cos 0 •. Accordingly,
we
~:~: V~cos2fl.)/VI
Il:-'VjCos2 as-:-(a*z ...
(1.1.2)
Let us eliminate the angle as from this eqllation. For this purpose, w(' shall lise Eq~. (4.4.1). ~1111tiplyillg the first of them by cos e•• the second by sin as find summaUng them. we obtain
as + w sin as = V"' co~ as. we flDd (4.4.3) tan as = WI - u)!w Sllhstit.uting £or tan as its \"slue from (4.4.~) into the trigonometric u cos
Having in view that V1: = VI
relation cos2 as = (1 -7- tan 2 9,,)-I, we obtain
cos:! 9 s = 11 -:. (llJ - U)2/W2}-1 Introduction of this value into (4.4.2) yields
r, 1 . 111"1
+{a.2_~':~! V~[1-7- (
ul/u'12
l\;U rrl}/VI
180
Pt. I. Theory. Aerodynamics of en Airfoil arlCi a Wing
Flg.U.t To the derivation of the hIJdo· graph equation
Fill. 1.4.1
Strophoid (shock polar)
This expression, relating the variables wand u, is an eqoation or a hodograph. It is lIsually written in the form (Vj':'U)!
'(U-a*2/T'I)/
(k!l
V I "",- "V"'12
-u)
(1.1.1i)
Let us introduce the dimensionless quantities )'1' = ula*, A.u; = = wla*, }..l = V1/a*, and the parameter 6 = (k - 1)/(k -I- 1). The hodograph equation now becomes ~/(A, - A")' ~ (),"),' - 1)/[(1 - 6) 1.: -!- 1 - 1.,1..1 (4.4.4') Equation (11.1.1') graphicaUy depicts on the plane Aw , All a curve known as a strophoid (Fig. 1.1.. 2). Let us determine certain characteristic points of a strophoid. Particularly, let us calculate the coordinates of points A and D of intersection of the strophoid with the Examinatiun of (4.1.1') reveals that the condition ),10 is axis All'
Ch. 4. Shoc:k Weve Theory
181
satisfied if A.. = A1 or ;'u =.. I AI' The value Au = )'1 (Ielermines the coordinate of point A and makes iI possihle to obtain a solution corrpsponding to fl shock of I1n illlillit~I)' !<mall strC'ngth Ill'hind which the velority ,Io{'s not changl>. The value All ,-= LA) ,Ictermines the c.oordinatc of inler~ectiou point]) ('\osest to the origin ofroorilinates and is the solution for II normal shoek. )1 follows from thu r·onslrllctioll of n 5trophoid thal il~ Iwo hl'flllches 10 111£> righL of point A e;"\lellll 10 inrlllil)", asymptotically npproaching till' !
(M.5) When there is 1\ compression shock, Po > P~ and, .:onscquently, 8 2 - 8 1 > O. This conchlsioll cortl>sponds to the second law of thermodynami('~. ac('ording Lo which the cntrop)" of an i~olated sy!'!tem with ('omprl'ssion shocks increases. Let U!l !lOW considl'r the reverse l'ituatioll \\hetl n gus passes from fI statr characterized by Ihe slagnation pressure p~ (the parampll!rS wilh the subscript 2) into a stat.e with the stagnation pressure Po (the pnrameters with Ihl! subscript 1) through all expansion shock. In this case, by flnalogy wilh (4.·i.,;;), the challbJ{' in lIle entropy is SI -
S1.
. R III
(p~/po)
Benel'. whell the coudition p~ < Po is retailled. the entrop), should diminish. but this contradicts the second law of thermodynamics. It thus foJlows Ihat expansion 8hocks cannol appear, In accordance wilh Ihe above. Ihf' passage of a gas through a shock, which is adiabatic in its 1I111111'e becall:'m it uccurs in a thermally in~lIlaLed ~ysl('m, is 8n irreversible adiabatic non·isentropic process. Equation (4.:i,(i) cau he uscil to wrify tha.t a real process of an increase in the entropy (S2 - 8 1 > 0) corresponds to a supersonic
182
Pt. I. Theory. Aerodynamics 01 an Ai,foil and " Wing
flow [M 1 > 1 (a Ilormal shock). HI sill Os > 1 (all oblique shock)], while the physically impossible phenomenon of a decrease in the entropy (S2 - SI < 0), to a subsonic flo\\' (MI < 1 ilnd MI sin Os < < 1). Hence, shucks can appear only in a supersonic f/ow. We must note (hal the relations obtained for the change ill the entropy are valid when an irreversible process of transition through a shock attended by an isentropic flo\\<' of the gas Lolli aheild of a shod: behind it. It follo\\'s from the above that tlw branches of a strophoid extending to infinity have no physical meaning. The part of a stroplloid (to the left of point A, Fig. 1.4.2) having a physical meaning is called a shock polar. Such a curve is constructed for a given number Al (or M I ). Several curves constructed for various \'alues of AI form a family of shock polars allowing one to calculate grapllically the velocity of a flow behind a slwck and the flow deflection angle. Let us consider an attached shock ahead of a wedge-shaped surfAce with the half-angle ~s (sec Fig. 4.1.1c). To determine Lhe velocity behind such a shock, we construct a shock polar corresponding to the given: number Al (or MJ.. and draw a straight line from poillt 0 (Fig. 4.4.2) at the angle ~ . .::.?int of intersection N with the shock polar determines the vector ON whose magnitude shows the value of the velocity ratio A2 behind the shock. By formula (4.4.3), which \\'e "hall write in the form tan €Is = (AI - Au)':Aw (4.4.3') the angle ANG on the shock polar equals the shock angle €Is;.;. We must 110te that this angle can also be determined as the angle between the horizontal axis and a normal to tbe straight line connecting the tips of tho velocity vectors ahead of a shock and behind it (points A and N, respectively, in Fig. 4.1i.2) When considering the shock polar, we can arri\'C at the cOllclusion that a decrease in the angle ~s (point A' mo\'es along Llle curve toward point A) is attended by a decrease in the shock angle 9,;1'\. At the limit, when ~~ -+ 0, point N merges with point A. which physif'ally corresponds to the transformation of the shock wl'Ive into an infinitesimal shock wave, i.e. into a line of weak disturbances. The shock angle for such a shock €Is = III is deLermined as the anglc bet \\,pen the horizontal axis and the straight line perpendicular to a tangent to the slwck polar at point A (Fig. 1.4.2). A growth in the flow deflection angle (in J·"ig. 4.4.2 this corresponds to point N moving aWAr from A) leads to an increase in the shock angle and to a higher strength of the shock. A glance at the shock polar shows that at a certain angle ~5 a straight line drawn from point 0 will be t
Ch. 4. Shock Wave Theory
183
angle ~& > ~er' In the graph, solid line OH drawn from point 0 and nol intersecting the shock polar corresponds to this angle. Therefore, when ~s > ~tr, we eannot find a graphical solution for a slwck with the aid of a shock polar. This is due to the fact that the inequality ~s > ~or does not correspond to the assumptions (on the basis of which we obtained equations for a shock) consisting in that a shock is straight and sholtld be attached to a nose. Physi~ caUy-ill the given case of the wedge anglo ~s exceeding the eriticai deflectiou angle fie r-the compression shock detaches and becomes cUl'verl, The determination of the shape of such a cuned shock and of its distance to the bod~r is the task of 11 special problem of aerodynam· ics associated, particularly, with the conditions of supercritical flow past a wedge. If such a problem is not selved, then with the aid of a polar in the field of defmition from point D to A we can give only a qualitative appraisal of the change in tbe parameters in a re~ gion ahead of the surface in the flow, If, on the other hand, the shape of the shock is determined for presot fiow conditions (in addition to calculations, this can also be done with the aid of blowing in a wind tunnel), it is possible to establish quantitative eorrespondence between the points of a shock polar and the shock surface. Assume. for example, that we have set the angle Pa and points E and N on a shock polar (Fig. 4.1.2). The shock angle 6$N = LANG corresponds to point X, and the angle SsE = LAEK (EK ..L OB) to point E. If the configuration of the shock wave front is known, then by direct measurement \"e can lind on it a point N' with the wne angle SsN" and a point E' with the angle 6 5 1:;' (see Fig. 4.3.4). [n tht! ~aJLle way, we can find II. puillt C' 011 the shock that corresponds to the critical (maximum) denection angle Per. On a preset surface of a detached shock, point D on the shock polar corresponds to the shock apex (a normal shock), and terminal point A of the polar corresponds to the remotest part of the shock that has transformed into a line of weak disturbances. For an attached shock (Ps < ~ef)' we can indicate two solutions, as can be seen on the shock polar. One of them (point E) corresponds to a lower velocity behind the shock, and the other (point N), to 8 higher one. ObservaLions show that attached shoe-ks with a higher velocity behind them, i.e. shocks with a lower strength are possible physical b'. If we dra'.... on the graph the arc of a circle whose l'adius is unity (in the dimensional axes wand u this corresponds to a radius equal to the critical speed of sOltnd a*), we can determiue the regions of the now-subsonic and supersonic-which points on the shock polar to the left and right of the arc correspond to. In Fig. 4.3.4, the section of the flow corresponding to a subsonic velocity is hate·hed. A close look at the shock polar reveals lhat the yelocity is always
184
PI. I. Theory. Aerodyn./lmics of .!In Airfoil .!Inc! .!I Wing
subsonic behind a normal shock. On the other hand, behind an ob· lique (curved) shock, the velocity may be either sllper~onic (the rei· evant points on the shock polar are to the righl of point S) or subsonic (the points on the polar arc to th~ left of point S). Points on the polar between Sand C corre.spond to an attached shock behind which the velocities arc subsonic. Experimental investigations show that for wedge angles ~s less than the critical one ~cr or larger than L SOB, a shock remains attached, but it hecome!i curved. The theoretical values for the angle as and the gas velocity AZ on the entire section behind such a curved shock fOllnd on the polar accord· ing to the wedge angle B~ do not correspolld to the actnal values.
4.5. A Hormal Shock In the Flow of a Gas with Constant Speclfc Heats We shall obtain the corresponding relations for a normal shock from the condition that 6s ---: ,,/2 and. consequently, Vnl = VI and V n2 = V~. Basic eqnation (4.3.10) becomes (4.0.1) '(4.5.1') where Al = 1'1Ja* and A2 = V 2Ja*. We find the relative change in the velocity from (4.3.11'):
Ii iT ~ (V, -
V,)/V, ~ (1 -
6) (1 -
11M:)
(4.5.2)
We obtain the corresponding relation~ for the ratios of thE' densities, pressures, and temperatures from (4.3.13), (4.3.15), and (4.3.16): P2/PI = M:/(1 - 6 6M~) (4.5.3) P2iPI = (1 + 6) M: - 6 (4.5.4) T,fT, ~ [(1 6) M: - 6[ (1 - 6 6M:)fbP, (4.5.5)
+
+
+
By eliminating M~ from (4.5.3) and (4.5.4), we obtain an equation of a shock adiabat for a normal shock that in its appearance differs in no way from the similar equation for an oblique shock [see (4.3.13')[. Assuming in (4.3.19) that as = nl2, we find the relation for the Mach' number behind a normal shock: .V:iM: = (TilT 2) (P IJp2)2 (4.5.6) Let us consider the parameters of a gas at the point of stagnation (at the critical point) of a blunt surface behind a normal !ihock (Fig. 4.5.1). The pressure p~ at this point is determined by formula
Ch. 4. Shock W~ve Theory
18~
(4.3.20') in which the ralio." P2 PI allci P~ PI are found from (·'J.5.8}· and (1.5.4), respccliH·ly. With this in view. we han p~fpo =
[(1 -;- 6)
M~
61("-1)/~"
-
fM;/(t - 6 ...
6M~)JCh~)/~"
(4.:1.7)
Determining Po by (4.3.21), we fmd P;/Pl =
[(1 -;- 6) M; -
61(&-1)/~"
..tt(~'6)/6 (I -
6)-(14&J/~6
(4.5.8)
Knowing the absolute pre~surt! p~, we can determine I he dimE'lIsionless quantity = (p~ - PI) (h-the pressure coeflicient fit the stagnat.ion point. Taking into account that the velocity head is
Po
ql:"'"
kP~l/I
2(11-.'_°6 ) filM:
we obtain
Po=
(~.~6~~~ {1(1--· 6) M~ _61(6 - J)/~\ JI I (I i 61/6 x(1_6)-(1+6)/2~_
.. }
(1.'=-dl)
It is proved in Sec. 4.3 that the stagnalion temperature behind a shock does not change, i.e. T~ = To. Consequenlly. at the ;;tilgllation point. we have
T~'-' Td 1 + I~b
.un
(0.10)
Let us usc the expression i~ :..::... cpTo aIH) il = c,,TI 10 determine the enthalpy. Acconlingly. at the stagnation point. we haw
186
Pt. I. Theory. Aerodyn~mjcs of an Airfoil ~nd • Wing
4.6. A Shock at Hypersonic Velocities and Condanf Specific Heals
0'.
Gas
At hypersonic (very high) velocities, which values of MI sin O. >::> t correspond to, the dimensionless parameters of a gas behind .a shock are very close to their limiting values obtained at HI sin Os __ __ 00. It follows from (4.3.11') that for this condition, we have
(4.6.1) Consequently, the limiting ratio of the densities by (4.2.13) is
p,/p,' - 1- ~V.
- 6
(4.6.2)
Introducing this ..-alue into (4.3.27), we obtain the following ex;pression at. the limit when MI sin 8, __ 00:
tan e.= (cotJl,/U) [I-b ± ]f"'(1-'6"')';-4n;67ta"'n"'~"'.1
(4.6.3)
Let us lind the limiting value of the pressure coeffIcient. For the ·conditions directly behind a shock. as follows from (4.3.15~), when MI sin Os -- 00 and MI __ 00, we have
(4.6.4) We obtain the corresponding quanlity for the point of stagnation
f'om (4.3.23):
Po =
2 (1 -
6)(6-1)126
(1
+ 6)-{1.")!'.!" sin as 2
(4.6.5)
The ratio of the pressure coefficients is
(4.6.6)
Poipz
In the particular case when 6 = 1/13 (k '-' 1.4), the ratio = = 1.09. The limiting value of the llumber M2 can be found from (4.3.19), using relation (4.3.16) for TzIT 1 • A passage to the limit when MI sin 0, __ 00 and HI __ 00 yields
M:- {II [6 (I ~.6)J) (cot' e,+6')
(4.6.7)
To lind the limiting parameters behind a normal shock, we must assume that 9 s = :t/2 in the above relations. As a result, from (4.6.4) and (1.6.5) WC" h8\'e:
Po =
p, 2 (1 -
2 (1 - 6) 6)lll-il/26 (1 -i- 6)-(l+W26
(I,.G.·1') (4.6.;;')
en.
4. S~ock Wdve T~.eory
181
We C(ln :;ec that the ratio Po'I-;~ i:-: the .':'amea~ ror i1n olJliqut' Sllotl;. The limiting Milch lIumhel' behind OJ JI()l"m~d ~hock is
For 6;;;....; 1,'(j (k --: 1.4), the 11\11111)('1' M ~ - 1/"[7 :::::: O.3t\. The actual v(lllies of the dilnensionle-s5 parrlrnelf'I's behind a shock fit linitl:!. nlthough very large', Mnch I\Ulll!wr." depC'nd on M l · Let \I.':' considf!l' the corrl!sponding' worki1lg rrlation:-: fOl" lilt' r.a.':'e when !lUnched ;.;hock.'< originalr ahead or ;.;h'uder \\·('cigcs. ami the .sllOek angles ,"lrP thcrd"ol'C' low. A~"nlllillg ill (!I.~-t:!;;) Ihal tan s ~ .~ e~ nll(l tall ~J :::::: 0, - ~,. w(' oblai"
e
(e, -
(0, - ~,) 0, ~ (l - b ., 6M;O~)(,I1;O;) Introducing Ihe syrnbol K ~ Ml~~' art('r Il'all.'
*-~·t- ~t =u
(!t.6.8)
So\viug \.his eqllation for Os/~s and taking inlo arC'Ollllt that the COlutitioll ()s/~s > t is physically posl'Iihle. WI.' find
*,~ 2(1~&) ;-l/' ~(l~-b)~
;-
~~
(1.6.9)
IuspE'clioJl of Eq. (!t.(U») re\'eals that tilE' parameters determining the Dow in 11 shock at \'err high velocit leS arl' combined illto fUllcl iOlli'l1 groups snell that ther are a solution oj Ihe f!ro/dem oj a shock Jor a broad range 0/ number..: M,." and j'alues of the o/l{!le B, in the jorm oj a single curve. Eqllation ('l.G.Il) ii' :~" thE' qllalltitr K = M,I~s is;l similarity erilcrioil. This silllilal'il r n111sl lJE' I1l1derstooll ill tile sense that l'('gardl('s~ of tl((' ah.:·ol\Jtc \";1111("> of the (jU,Hltilies chal'acterizing h~'persouiC' nO\\·.~. \\"11('(1 Sill·1I rlo\\"s han> identical partlinett'I's K, the ratios of the angl('s e~;ll.,
Os/Bs
=
1/(1 -
0)
(/j .li.1U)
Let us consider the relation for 111(' pl'C~IIl'(, c.ot'flicit'llt. AI low values of 9 s , formula (L1.1;)'") ac((uirt's the forlll
P2
= 2 (1 -
6) (6: -- "I 'M;)
P2 =:2 (1 - 6)~: (Oi,~;·- 1 K~)
(·l.IL 11)
Introducing the value of 1 'K~ rrom (!di.8), W£I' ohtaill
p,
c"
~Il,e,
(1.G.H)
166
Pt. I. Theory.
Aerodyn~mi(5
of 8n Airfoil Md II Wing
S\lbstit\lt.illg for the angle Os the v"llI(' cle-termilled from (4JU1). we fmd (4.(;.12)
This formula $110\\'$ that K is nlso 0 similarity criterion for the ratio p2i~~. At the limit. when K -+ co, this nltio becomes P2"~~
=
2..(1 -
('1.£\.1:3)
&)
In accordance with (1.3.13), at small \·"Iue of Os, the df'nsily ratio is (4.H.14) where lhe parallleler Ks ,': in the followillg form:
Mle~
K""· 2(1"-6) [1
is dt'lermined with the aid of (/dUJ)
+ V'
1·
.4(1~})Z]
(4.{i.15)
We can see from ('1.:1.18) that at low volues of 9 s the second term all the right-Iwnd side may be ignored, and \n' can thus cOl1sider that 1'2 ~ 1'1. With this in vic\\", the ratio or the sqllares of the Mach numbers in accordance with (1.:·Un) is M;IM~ = T 1 1T 2 • Substituting for the ratio 1\:'T2 here its value from form\da (4.3.1fi) in which we assume that sin 9 s ~ es ' \\e ohtain e~Mi '-" K~,' {l(1 + 8) K~- 8] (1 - 6·:- &K~l} (ldi.16) When Ks
-+
00,
we have (4.fJ.1o')
0.
4.7. A Shock in a Flow a Gas with Varying Specific Heats and with Dissociation and Ionization
When solvillg a problem on a shock in a dissociated and ionized gas, the parameters o[ the air at an altitude H (the pressure PI' temperature T1 , density PI' speed of sound ai' etc.), and also the value of the llormal velocity componellt VIII (or the Humber Mill = :..... VIII/aIl arc taken as the initial data. HplIce, an oblique shock in the given case is treated as a normal one. Assuming ill a first approximation that the value of t!I. VII ~ {Ul-O.OJ, whidl corresponds to setting a relative clensity for a shock of PdJl '-'- (l - ~ Vn)-l ~ ~ 10-20, from (4.2.1;') we fwd the presstlre 1'2' aud from (4.2.16), the enthalpy i2 that is evidently close to the st3gnation C'nthalpy i~.
Ch. 4. Shock Wave Theory
189
By nexl usiug'lII j-S diagram (see Hi. 81). we determine the temperatlll'e T~. and then from Fig. I.;).i. Ihe mean molar mass 11m2' Instead of the diagram, Olll' 1ll1I:'; use suitahle ta!>les o[ the llH.'rmodyuamic fUlldioll."1 or air (Sl't' IiI). width will increase lht' accuracy of tlll~ calclllnli()n~.
By inserting the round \"alu\1's of 1'2' T z, and j.lm:! into the equation or state (1.5.81. m' Clln d('tl'l"llIinc the deJisily p~ ilnd
190
Pt. I. Theory. Aerodynamics 01 an Airfoil and a Wing
~':ti~·7~tf the air temperatures behind and ahead of a shock with account taken of dissociation and ionization: ~lId linu-T, = 2211 linrs_T, = 3:'11 X
K, dnshrd
lilA
" f---,----t-
"
~I:iit·~t the air dcns.ities behind
and ahead of a s.hock with account taken of (Iissociation and ionization: wild linu--T) = fW lInu-T,1 = 3aO K
I~ f--c,r~"""9~=+--c~
K. dnshrd
!I/P,
8DOr-,--,----"::H.::"~lm
6D"r--t---t--n"'Y-' II}:!
f ll·4,7·,
Ratio of the air pressures behind and ahead of a shock with account taken of dissociation and ionbation
"
2DOi---H:L-jf--"-=-1
on the physicochemical transformations of the air. The ratio P2iPI differs only slightly from the maximum value of P2iPl ""'" 1 + -i- klM~ll determined only by the conditions of the oncoming Ilow, but not by the change in the structure and physicochemical proporties of the air behind a shock wave.
Ch. 4. Shock Wave Theory
191·
The temperature behind a shock ill a dissociated gas is lower thall that with constant specific heats (Fig.1.7.l). The explanation is the loss of energy on the thermal dbsociatioll of L111' molecules. The lowering of the temperature due to this phenomenon CHII$eS the density to grow (Fig. 4.7.2). This greater "yielding" of the gas to compression reduces the space bf'tween a shock aud a surfa("e in the flow, thus diminishing the shock angle. And conversely, at the same angle as in a real heat.ed gas, the Jlow defleetion (tlw angle ~i) is greater than in a perfect gas (k = COllSl). The result is that in a heated gas a detached shoek fol'lus with a certaill delay in compal"ison with a cold gas. Particularly, tJH' wedge angle (critical angle) at which detachment or a sllOek hegins is larger ill a Ilea tell gas than in a cold one. Account of the infllH'IH~f' of dis:-;oeiutioll II',I(]." to fI certain illc["case ill t.he pressure bchilHJ a shork ill compnrison with fI gas haYillg constant specific heats (Fig. ·'1.7.3). This is explained by the increase ill t.he number of pal"tieles in the gas O,,·illg to dissociation and. by the increase in the losses of IduC'tic ('Ilergy wlll'n tiley eollide. But the drop in the te'mperatnre ill u di."sorialed gas caw'es an oppo!'ite effect, but smaller in magllitude. :\:; a l"1'SUit. thl' pressilre increases, although only slightly. The theory of a normal shock has an impol'tant practical application when determining the pacametC'l"s of a gas at lite point of stagnation. The procedure is as follows. According to the fOllnd values i z· and P2 with account t.aken of dissociiltion and ionization, we find the entropy 8 2 from an i-8 diagram or tablcs of thermodynamic functions for air. Considering the flo\\" behind a shock to be isentropic, we assume that the cntropr S~ at the stagllfllion point equals the value 8 2 behind the shock \\'a\·e. In addition, for t.his point We' cfln find the ent.halpy i~ = i l .!- 0.5V~. l\o\\" knowing 8~ and i~, we ran use an i-8 diagram or thermod)·namk tablcs to find the othf>r parameters, namely, p~, T~, p~, etc. The results of the calculations correspond to the preset flight altitude. A change in the altitude is at.t.ellllf!'d by a changc ill the conditions of flow past. a body and, consequently. in the paralllcters at t.he point of stagnation. This relation is shown graphically in Fig. 4.7.4. The curves allow us to determine the temperat.ure T~ and pressure p~ as a function of the veloci\.~' V, and the flight altitude H. The value of p~, in turn, can he used to ca!r.ulate the pressnre coefficient at the point of stagnat.ion: Po = O.'skM;'. (p~/p "" - 1). Its value grows somewhat in a di.!'50ciating flow. For exam piC', at an altitude of 10 km at M~ = ifi.7, its valne is Pi) '""'" ~.08, whereas without account taken of dissociation = 1.83 (for k =- cJ'.·c,. = = 1.4). The pressure coefficient can be seen to grow by abOHt 1R%_
Po
192
Pt. I. Theory. Aerodyntlmics of an Airfoil <'H1d
1I
Wing
Flg.UA
Pressure and temperature at the of stagnation
point
The results of calculating the density at the point of stagnation with account taken of dissociation and ionization for air, and also for pure oxygen and nitrogen. arc given in Fig. 4.7.5. An analysis of these results allows us to arrive at the following conclusion. At M"" = 18. when oxygen is already noticeably dissociated, the quantity P~!P_ reaches its maximum value. Upon a further increase in M "'" the gas completely dissociates, and tile density drops. Next a growth in M
Ch. 4. Shock Wa.ve Theory
193
At }1"" < D, what mainly occurs in the air is dis~ociation of oxygeu, and the drnsity curve for air is closer to the rrlevant curve for oxygen, At M", > 1:3. when the dissocialioll of the nitrogen becomes appreciable. the dependence of fl~lpoo on M"" for air reminds one of the similar relation for gaseolls nitrogcll bccause the iatter is the predominating component. in tile air. The C;llcnlation of the parameters of a gas behind a shock ,vith a vicw to the varying na~llre of l!le specifiC heats is Ilescribed in greater detail in [141. 4.8. Relaxation Phenomena Sectioll 1.7 deals with ways o[ calculating the parameters of a gas behind a shock with account taken of tile physicochemical transformations for conditions of equilibrium of the thermodynamic processes. In a more general case, however, these processes are characterized by non-pquilibrium, which has a delluite innuence 011 the gas flo\\' behind shocks. Non-Equilibrium Flows
It is known from thermodynamics that the ilssumption of thermodynamic equilibrium consist.s in agreement brt,veen the le,-els of the internal degrees of freedom and the parameters characterizing the state of a gas, For example, at comparati\'ely low temperatures (low velocities), equilibrium sets in between the temperature and the vibrational degree of freedom, which corresponds to equilibrium l.Jet,veen the temperature and till} specirlc heat. At high temperatur(}s (high velocities), when a gas diSSOCiates, the equilibrium state is reached as follows. As diSSOCiation develops, the probability of triple collisions grows (for a binary mixture of diatomic gases) because the numher of gas particles increas(}s. This leads to the acceleration of recombination and the retardation of the rate of dissociatioll. At a certaill instant and t(}mperature, the rates of the direct and reverse r(}actious become equal, and the gas arrives at an equilibrium state. The latter is characterized by a constant composition and agreement between the degree of dissociation, on the one hand, and the temperature and pressure, on t.he other. At still higher temperatures (very high velocities), equilibrium processes of excitation of the electron levels and ionizatiOIl can be considered. Upon a sudden change in the temperature in an equilibrium flow, the corresponding internal degrees of freedom also set in instantaneously; dissociation~and ionization call be considered as the manifestation of new degrees of freedom. Consequently, in these cases, there is'no delay in establishing the degrees of freedom, i.e. the time needed for achieving equilibrinm is zero.
194
Pt. I. Theory. Aerodynamics 01 an Airfoil and a Wing
In practice, an equilibrium flow is observed under supersonic Dows past bodies at Mach numbers of M CO> = 4-5 in conditions corresponding to altitudes of 10-15 km and less. The explallillion is that at the maximum temperatures of the order of 1000-1500 K appearing in these conditions, the main part of the intemal energy falls to the share of the translational and rotational degrees of freedom, which upon sudden changes in the temperature are established virtually instantaneously because only a few molecular collisions are needed to achieve equilibrium. This is wby the transl;ltionai and rotational degrees of freedom arc usually called "active" degrees. With an increase in the velocities and, conseqllently, in tlie temperatnres, a substantial part of the internal energy is spent on vibrations, then on dissociation, excitation of the electrolL levels, and ionization. Actnal processes are such that these energy levels set in more slowly than the translational and rotational ones beciluse a mtlch greater number of collisions is needed. For this reason, thE' vibrational and dissociative degrees o[ fl'C'edom are sometimes called "inert" degrees. Hence, the inert degrees arc featured by a delay in the achievement o[ equilibrium called relaxation. The time in which equilibrium sets in, i.e. correspondence between the temperature and the energy level is established, is callC'd the relaxation time. The relaxation time charactertzes the rate oj attenuation oj depa1'lures oj a gas state from an equilibrtum one, which in a general case manijests itself in the jorm of a change in the energy distribution limong the di/Jerent degrees 0/ freedom. Relaxation processes are determinl'd by whal degreE' of freedom is excited. If upon a sudden change in the tempf'rilture, vibrations appear, the corresponding nOll-equilibrillm process is callC'd vibrational relaxation. It is characterized by a lag in the specific heat when the temperature changes. If the temperature rise!', the specific heat grows because of the appearance of vibrations of the atoms in the molecules. The time during which vibrational motion renclles equilibrium is called the vibrational relaxation time. In a non-equilibrium dissociating gas upon a sudden change in the temperature, a delay occurs in the change in the degree of dis!'ociation. This phenomenon is called dissociative relaxation. Owing to the difference between the rates of formation of the aloms and their vanishing (the rate of dissociation is higher than that of recombination), a gradual increase in the degree of dissociation occurs. The equilibrium value of the degree of dissociation is achieved at the instant when the rates of the direct and reverse reactions become equal. The time needed to obtain an equilibrium degree of dissociation is called the dissociative relaxation time. Experimental data on the relaxation time for oxygen and nitrogen are given in Fig. 4.8.1.
Ch, 4, Shock. WlJve Theory
195
~I~p:;~ental (I, 2) curves of the vibrationaL relaxation time
and calculated (,1, 4) curves of the relaxation timc for the dissociation or oxygcn and nitrogpn, re-spectively, at atrnosphe-ric prcssure
At. tcmperaturcs approximaLely lip to 10 000 K, the \'ibrational and dissociative relaxation proces~ws are the nlilin ones, Tht' rl'iaxalion phenoJUpna a~sociateu wiLh the excitation of the elecLro" Il'yels o[ the Illoipcldes and nlomO', and also wiLh ionizAtion, may 1)(' 4ii~re gardc41 because a small fraction of the internal energy fallO' to Ihe shal'c ot' t\Jef':e 4legret's of freedom at the indicated t('mpf'l'atllres. A nOll-equililJl'illnl slate has a substantial influence on the Yal'iollS pl'ocess('s attending the flo\\' of a gns at \'Cry high \'e1ocities, l\'l'lic1(lady, vibrational and dissociativc relaxations change t1l(' parameters of a ga!; ill a trall!;itioll throllgh shock \\'n\'('s and in the fin\\' past bodi('s, This, ill turn, aHrcts the JlrOCe~Sl'S of friction, heat exchange, and abo the redistribution of thn pr('ssurp. The studying of nOll-cquilibrium flo\\'s consists ill the joiHt investigation of the motion of the fluid and of the chemical pro('(''!:ses occuning at Hnite ,,('Ioeities. Thi~ is ex:pl'essed formally in Ihnl an eqllatioll for the rate of chemical reactions is adcle{l to till' m:nal system of equal ions of gas dynamic'!:, EquilibrIum Processes
Equilibrium flows have been studied better than non-equilibrium ones both from the qualitative and the qnantitfltive ~t
t 96
PI. I. Theory. Aerodynemics of en Airfoil end- o!! Wing
ation phenomenon of another one. In more genoral case, overlapping .()f the regions of establishing equilibrium is observed. An approximate model of the process can be conceived on the basis -of the "freezing" principle. It is considered that a region of achievement of equilibrium relates to one or several degrees of freedom, while the others are not excited. The sequence of these regions of equilibrium can be represented by degrees of freedom with elevation of the temperature in the following order: translational, rotational, and vibrational degrees, dissociation, excitation of electron levels, and ionization. When considering, for example, the process of establishing equilibrium of vibrations, the first two degrees of freedom can be considered to be completely excited. This process occurs in conditions of "froztm" dbsociatioTL and ionization. Such a scheme is not suitable for certain gases, particularly nitrogen at high temperatures, because ionization begins before dissociation is completed. This is explained by the fact that the energies of dissociation and ionization of nitrogen differ from each other only one-and-a-half times. In this case, the regions of attainment of equilibrium overlap. A similar phenomenon is observed in the air at comparatively low temperatures. At temperatures exceeding 3200 K, tho relaxation time for the dissociation of oxygen is lower than the dUl:ation of setting in of nitrogen vibrations. Consequently, equilibrium in the dissociation of oxygen is established before the vibrations of the nitrogen are in equilibrium. Investigation of the flow of a non-equilibrium gas over bodies is facilitated if the characteristic time of this process is considerably lower than the relaxation time of one of the inert processes and conditions of a frozen flow occurring without the participation of this inert process appear. Particularly, if the duration of flo'''''' over a section of a surface is small in comparison with the time needed for chemical equilibrium to set in, but is commensurable with the vibrational relaxation time, a process with frozen dissociation and ionization may be considered on this section . •• laxation Effects In Shock Waves
In the transition of a gas through a shock wave, a portion of the kinetic energy is converted into the energy of active and inert degrees of freedom. For the active degrees (superscript "A")-translational and rotational-equilibrium sets in during the relaxation time tS commensurable with the time taken by the gas to pass through the thickness of the shock. Since this time is very small, we can consider in practice that the active degrees are excited instantaneously. At the same time, dissociation does not yet begin because during a small time interval the number of collisioll!i of the molecule~ is not
Ch. 4. Shock Wave Theory
197
m p,
.::
.
,; .---.'- . ,:" ----1-, ~Z
,-
;
p.
'
~
1""'-
~i~'o!:t~on
t;.-~t~
!t,x
.---,
proceSill in a shock wave:
(ll1e h.lclltd 't'glon delermlnl"8 the thlckni'S$ of Ihr shock way!'): J .nd l'-lront and ,ellr surtacfs 01 the shock WOYP, N'SpfC\i\")Y
large. Hence, directly behind a shock wave, the- temperature T~ is the same as in a gRS ' .... ith constant specific heats. In accordance wiLh this scheme. no inert degre('s of freedom (superscript. "I") are excited directly after a shock, Since these degrees have a fmite relaxation time tL considerably grt'aLer than t.he time t~ and the duration of transition through the thickness of a real shock, then the temperature lowers until the illl'l't degrees of freedom (first vibrations, thon dissociation. excitation of the electron 10\,eI5. and ionizaLion) reach equilibrium. In Fig. Ii.B.2, schematically ilhlstratillg a reluxation process, the relaxation time t~ measured from the instant of lite beginning of motion on the front surface of t.he shock wan' ahead of which the temperature of the undisturbed gas is 1\ corresponds to the temperature T;. The figure also shows the density p~ corresponding to thetemperat.ure T~, while the equilibrium state is established at a tC'lIlperature of T 2 < T;. A non-equilibrium process of gas no\\' behind 8 shock is altended by an increa,~e in the density to the e(luiliiJl'illlll valuc P2 > p;. Here a ~mall growth i1l the pres$lJre is ohscn(lod in comparison with an ideal gas, Simultaneously the degree 0/ dissoclatiOIl (and at vcry high temperatnres, the degree of ionization) groU's from zero to its equilibrium l.'aZue. Thc invesligation of the non-equilibrium now behind a shock consists in detetminin.a I_he length of the non-l'quilibrinm zone, or the relaxation length, and also in eSlimaling the non-equilibrium parameters. The system of equations describing such motion includes equaLions of the momentum, energy, state, and an equation for the rate of chemical reactions. The initial conditions upon its int<.>gratioll arc determined by the parameters direct1y behind the shock wa\'c, which are found with the assumption that dissociation is absent. At the end of the
198
Pt. I. Theory. Aerodynamics of
olin
Airfoil end
II.
Wing
:!ft"U:~! 01 noo-tquilibrium dissociation on the density and temperature behind a shock wave g~. ~)JIo.2,
T. _ 300
K.
p,-
relaxation length, these parameters reach equilibrium values corresponding to the equilibrium degree of dissociation. The results of sucll investigations of a non-equilibrium now of a gaseous mixture of oxygen and nitrogen behind a shock wave are given in Fig. 4.8.3. The solid curves were obtained on the assumption of instantaneous vibrational excitation, and the dashed ones in the absence of excitation. These results reveal that vibrations have an appreciablesigllirlcance in direct proximiLy to a shock. For example, without account of eXciLations, the temperature behind a shock is 12 000 K, while for a complotely exciLed state it is about 9800 K. i.e. considerably lower. At the end of the relaxation ZOne, vibrational excitation is of virtually no importance. Tnerefore in calculations of equilibrium dissociation, we can assume that the rates of vibrational excitation arc infinitely high, thus considering a gas before the beginning of dissociation to be completely excited. It is shown in Fig. 4.8.3 that the length of the non-equilibrium zone is comparatively small and is approximately 8-10 mm. The data of investigations of the relaxation in shock wan's can be used to determine the nature of non-equilibrium flow in the vicinity of a blunt nose in a hypersollic now. Here we find the length of the relaxation (non-equilibrium) zone and determine whether such a zone reaches the vicinity of the point of stagnation. In accordance with this, the stagnation parameters arc calculated, and with a view to their values. Similar parameters are evaluated for the peripheral sections of the surface in the flow. The length of the non-equilibrium zone is found as the relaxation length calculated by means of the expression XD
= O.5Vctn
(4.8.1)
where Yc is the velocity directly behind the normal part of the shock wave, and to is the relaxation time.
Ch. 4. Shock Wave Theory
The relaxation tim(' to ~
("all
199
be found hy the formula
2.1:) X 10-7 (1 -
'1. c HI/
(ll) r;1~ 'X~
(;\
. '1. r
»-·
CU';.:!.)
in which f (H) = tJooH"P"'I'; is a fUlictioll of the altitude (P''ll and p.,J:: are the densities of tl.e at.mo:-:-phere fit the altitude J[ aud at tlw Earth's surface. re.speclivcly). P'P"'1I Ithl' den ..;ily f' = 0.;;; (Pc -i- Pel. where pc and Pc are the densities bellin.1 the shock in the Illidissociated gas ,\litl for equilibrium di::;sociation. respective-
p -
l\'!.
. If the length XL) is smaller \.lUlH the distance .~u hetwel~n a shock wave and the point of :;tagnatioll, lhe non-eqlliliIH'iulH zone is ncar lI.e wave and does not include the body in lhe !low. COII:-;e'lI1l'lItiy, the condition for equilibrium in lhe vicinity of II blunled :;urface willlJe .1:0 < so' or O.5V{OtlJ < so. llence we cal\ abo lilld lhe inequality to < si(O.5Vc) in which lhe term 011 lhe right-ltaud st.lt' is lhe characteristic time t s• spent by a particle in the compression zone. If tso> > tu. a particle or the gas will have lime to reach till:' stalo of eq1lilibrium before it reilc.!ws the slll'[nce. It is not difficult lo sec rrom formula (1.8.2) llial the relaxalion time grows with increasing llight altitude II and, consequ('ntly, the length of the non-equilihrium zone increl'l.ses. At tlw same timc, til diminishes wHh an incre;lsc in Lhe intensity or a ("ompr{'~sioll sl.ock for very high supersonic velocities of the !low (here the rclati\'e density behind a shock ~ ,"" P.'P "'11 b~conU'::; larger). A non-equilibrium slale suhstantially aned:-:- 1ll!' distance between a shock wave and a body. With a completely lIolH'qnilibrium flow behind a shock (the degree of dissociation '"L '-" 0), lhis distance can be appreciably larger than in equilibrillm dissocialion (a -::: etl' > 0), It decreases in tlH' real cO\Hlitions of a gradualtrl.lllsit.ion from a nonequilibrium state to RIl equilibrium flow behind a shock wave, i.e. when the degree of dissociation ehRnges from 0 behind tile wave to the equilibrium value '"1. , ..:: etc at the ell~1 of the relaxation zone.
5 Method of Characteristics
5.1. Equlflons for !he velocity Potential and Stream Funcffon
An important place in aerodynamics is occupied by the method of characteristics, which allows one to calculate the disturbed flow of an ideal (inviscid) gas, This method make~ it possible to design correctly the contours of nozzles for supersonic wind tunnels, determine the parameters of supersonic flow over airfoils and craft bodies. The method of characteristics has been developed comprehensively for solving the system of equations of steady supersonic two-dimensional (plane or spatial axisymmetric) vortex and vortex-free gas flows. Investigations associated with the lISC of the method of characteristics for calculating the three-dimensional flow over bodies 8fC being performed on a broad scale. Below we consider the method of characteristics and its application to problems on supersonic twodimensional flows. Equations for a two-dimensional plane steady Dow of an illViscid gas are obtained from (3.1.20) provided that 1.1. = 0, 8V,/8t = = 8V,/8t = 0 anel hAve the following form: V
oV", '"
V:c
oz
~~. _..!..~ }
+l' g
8y
P
iJz
(;"'d.1)
0:; + V, ~:' =, _~. ~~
For a two-dimensional axisymmetric flow, the equations of motion obtained from (3.1.36) for similar conditions (v = 0, 8V,/8t = = 8V,IOt = 0) can be wriLten in the form
'V~=-..!.'..!f....}
Vx
DV", ilz
V:c
o~r + Vr :~r
I
r
p
or
=
iJz
(5.1.2)
_~.*
The conLinuity equlltions for plune and axisymmetric Dows .having
r('~pcctively the form of (2.4.5) Rnd (2.4.32) can be generalIzed as
Ch. 5. Method 01 Characteri~tic~
20t
follows: a (pV""yt),()x -j d (rVyye),'uy
~-
0
(5.t.3}
When e = 0, this equalion coincides with the continuity equation for a two-dimensional plane now in the Carlesian coor(linates x and y. If e = 1. we have a continuity equation for a two-dimensional axisymmetric Dow in the cylindrical coordinates y (r), I. Accordingly,. for both kinds of flow, we may consider that the equatiolls of mot ion Fd.1) arc written in a gent'ralized form. Having determined the partial derivative!> ill continuity equation (5.1.3), we obtain
(v~ ~~
+ V" !~
)ye+pyt ( a;~x.1
{}~1I) ~pVyf:yF-l
.. 0
(5.1.4}
V'le can replace the partial derivative ap./oJ: wiLh lhe expression Dp/ax = (ap/Elp) aplfJx, in which Elp/fJp = La:!, while the derivative &p/&x is found from (5.1.1) in the form
-r (Vo. o~:
_;
Vy (.~;Ix)
\\'jth this in view, we have
;~ '-- -%(Vx
{}fJi:x -:
Vy
(5.1.5)
An expression fo!" the del'ivat ivl' rlp/oy is way:
~';
,;, -7 (V
x
rJ~;"
f()lItHI
+ V~, ;',~> )
similar
(5.1.6)
Substitution of the vnluC's of th('!ie d(>ri\"{\ti\"t.~ into (:U.4) yields
This cqualioll is thC' fundam(>ntal differential equation of gas dynamics for a two-dimensional (plane or spatial axis~ mmelric) steady tlow which the velocity cOlilponelll~ and mllst satisfy. Since this equation reJatr:i the \"('Ioritie~, it is {liso 1'C'Cerrcd to a~ the fundamental kinematic {'(Iuation. If a flow ii' potential, tllell
r ..
l:x'-- dq,u,t·, VII :."': oq uy, dVx/{Jy thcrc!ort·, Ey. F).l.7) (Vi-a2)
~"j:~
.j
l:U!l
2V.Y'J
ry
= 8J'v·lIJ' = (P!p.'ih ay
he transformed as follows:
(:'.1.8)·
PI. I. Theory. Aerodynamics of an Airfoil and. .e. Wing
202
where
Equation (5.1.8) is the fundamental differential equation of gas dynamics for a two-dimensional potential stead)' flow and is called an equation for the vcloeity potential. Hence, unlike (5.1.7), Eq. (5.1.8) is used only for studying vortex-free gas flows. If a two-dimensional gas now is a vortex one, the stream function 1Jl lias to be used for studying it. The velocity components expressed in terms of the function 1Jl have the form of (2.5.5). Replacing p in accordance with formula (3.G.31) in which the stagnation density PI! along a given streamline is assumed to be constant, expressions (2.5.5) can be written as follows: Vol = y-t (1 _
f1)-1/(4-1)
o1jJloy,
VII = _y-. (1 _ y2)
_1/(4_1)
o¢/ox (5.1.9)
where j7 = VIV,u:\~. The cRlculaLion of a vortex gas flow consists in solving a differelltial equation for the stream function "p. To obtoin tllis equation, let tiS rUffprentiate (5.1.9) with respect to y and to x:
"~;'
= _ey-e-l (1_V"2tli(4-1l : :
(1- yttl/(A-Il-1
":~"
.. _ y-e
a~2 . ::
-j_
+y-e~
y-e (1- VZ)-I/(k-I)
~ (1- V2tl/(h-1)-1 ~~2
.
~~
~:~
_ y-e (1- Vit1(1t-1l
~
T,1king into account expression (3.0.22) for the square of the speed of sound, and also relations (5.1.9), we can write the expressions obtained for the derivatives oV,)oy and oV,/ox as follows:
J:;
'--co
_+ v.,.,-
~:~. "~2 +y-e(l_V2)-,/(h-1l ~:~
":;"~, :a~. a~2 _y-e(1_Y2)-I/(h-1l ~:~
(5.1.10)
(5.1.11)
We shall determine the derivatives oV2loy and oV'llox in Eqs. (5.1.10) and (5.1.11) with the aid of relations (5.1.9). Combining these relations, we obtain (5.1.12)
Ch. S. Method of Chllrllcteristics
203
By difierent.iating (5.1.12) with respect to x anti y, we lind the relevant relations: 2 :~ . ~:~. -t- 2 ~,!
2
~;
.
.a~:fJ~
~. ~ t
_ P y 2f
yte
. ~:~ _, ~~2 y2"
a~~~y .:- 2 ~!
2eV2 y 2e -1 (1 _
~~:
--.-
Y2)2/(/'-t) _ 1'2y2e
V2)~/(~-L)
(t -
('I _ V2)10l/(I!-I)-I
(~~2
(1 ._-
V?)'~I(I!-I)
~ 0 - i12.)2/(I<-I)-1 a%y2
We transform these relations with a view to Ell. p.6.22) for the sqllflre of the speed of sound and to relations (5.1.\)):
-- Vr, !:~)
+ Vx a~2:y - VI/
=
4-. ~~~ ('1 _. '" f· aiJ~~2
Y1.)IIi"-t)
a!2:y -i' l' x !;~.
'{ (1 -
~) + y~-e V2(1 -
('1 - -;.)
(:i.1.13)
(1- V~)t/(I'-I) j72)l/lll-l)
(5.1.1-1)
Dl'termining the derivatives OV'l-IOy and (JV~iOx from these expression~ and inserting them into (5.1.10) and (5.1.11), respecth'ely, we obtain
f(1-V2)1/(~-t)(1-~) iJ~:x
~.:)
;,: ( 1 -
- !/_e 1'2 (1-f (1 -
-Zy:_eVx(I--ln)II(I'-1)
-r :~~ [.- V" a'I,2!y T V ~ ~:~ 1'2)1/1/'-1) ]
+ -} (1 - ~: ) ::~
~: ) 0;;, -.: ~:: (-Vy !:~..:-1',. :;~y)-}(1- ~:) ::~ Y2)t/(l'-IJ
(1-
Subtraction of the firsL equation from the second one yields a relation for a vortex:
a;: _ o~/ .;- V",
:;~y
)
=
(1- 1'2t
a2~vt'7 l / Ch -t)
-7
(-Vy ::~ (1- V2
t 1/{h- l ) ~:~
204
Pt. I. Theory. Aerodynamics of en Airfoil end
~
Wing
A vortex can also be determined with the aid of Eq. (3.1.22') transformed for the conditions of the steady flow of an ideal gas and having the form (5.1.16) grad (V'12) + curl V X V ~ - (lip) grad p
By applying the grad operation to the energy equation (3.4.14), we have: . grad (V'12) ~ -grad i (5.1.17) Consequently, Eq. (5.1.16) can be written in the form curl V
XV~
grad i - (lip) grad p
(5.1.18)
Vortex supersonic gas flows call be characterized thermodynamically by the change in the entropy when going over from one streamline to another. This is why it is con venient to introduce into the calculations a parameter that reflects this change in the entropy as a feature of vortex flows. In accordance ''lith the second law of thermodynamics T dS
= di
- dp/p
or in the vector form, T grad S = grad i -
(1/p) grad p
(5.1.19)
Combining (.'i.1.18) and (5.1.19), we obtain curl V
X
V -= T grad S
(5.1.20}
Let us calculate the cross product curl V X V, having ill view that by (2.2.12) the vector curl V = (OV,)8x - 8V,/ay) i3 (here i3 is a unit vector), and, therefore, the projections (curl V)x = O. Let us use a third-order determinant: = (curl V)y
=
curl VXV
I ci i~ (C~~l Vh I V", V y
0
in which the projection (curl V)3 = aV1l 18x - OV,)8y
'2013
Ch. 5. Method 01 Ch
Calculations yield curl YXV = -
Vy ( 0;; _ 0:; )i, :_ Vx ( V~~tl
_
u;;"" ) i2 (5.1.21)
Accordingly, for tho projection of the vector curl V X V onto a normal n to a streamline, we obtain the relation (curl V
X
V)~
= (curl V
X
+ (curl V X V)~ = V' (aV,la. - aVJay)'
V)~
(5.1.22)
Examination of (5.1.20) reveals that this projection can also be written in the form (curl \' X V)/l. = T dSldn or, with a view t.o (5.1.22)
V (aY,lax - aY.:ay) = T aSld"
(5.1.23)
Since a Z = kRT, then, taking (3.6.22) into account, we fl"!ld an expression for the temperature: Tc.""",
;~ ,_ k~1 .*(V~lax-V2.)= k~t. V~nJ! (1--V2)
Introducing this relation into (5.t.23), we obtain
o~,:
_o~:
.=
~. 1~1~~~
(1- V~)
~:
(5.1.24)
By introducing the vortex according to this expression into (5.1.15), we lind a differential equation for tile sLream function:
(Vi-a 2 ) ~:~ +2V:'Yllo!~o~ L(V~_a2) ~~l;
+ yt'-e a2V
%
(1- V2 ),/(h-1)
= k;;1 • V;;'J! (1- V2).\/(.\-O ye (02 _
V~) ~:
(5.1.2:1)
5.2. The Cauchy Problem
Equations (5.-1.8) for the velocity potential and (5.1.25) for the stream function are inhomogeneous nonvlinear second-order partial differential equations. The soilltion~ of these equations cp = cp (x, y) and W= W(x, y) are depicted geometrically by integral surface~ in a space determined by the coordinate systems x, g, cp or x, y, 1p. In these systems, the plane x, y is considered as the basic one and is called the pbysical plane or tile planc of independent varlableA.
206
pt.
I. Theory. Aerodynamic, of In Airfoil Ind
I,
Wing
r!'ii!f'~urve AB; the required function and its first derivatives witil respect to:e and y are on it
known
Tho Cauchy problem consists in finding snch solutions in the vicinity of all initial curve y = y (x) satisfying the additional conditions set Oil this curve. The values of the required function (jl (x, g)
~i~~alx~~?d~~i~r~!~e -fll~:~ d::!v~~i:\~~ ~1~S (t;'~hY~: ~~~t:~e ~~~Jlit~~:~!~ From the geometric viewpoint, the Cauchy problem consists in finding nn integral surface in the space x, y, cp (or x, y, 'lJl) that passes through a preset spatial curve. It is just the projection of this curve onto the plnno g, x that is the initial curve y = y (x) on this plane. The solution of the Cauchy problem as applied to the investigation of supersonic gas flows and the development of the corresponding method of characteristics arc t.he results of work performed by the Sovict scientist, prof. F. Frankl. To consider the Cauchy problem, let us write Eqs. (5.1.8) and (5.1.25) in the general form Au
+ 2Bs + Ct + H =
0
(5.2.1)
:~~~ed up:rtT:i J~~i~;t~~:s:=\:;V v~Ol::rV~f ~,d ~,=a:dv Jo:qt"a'l' :~: coefficients of the corresponding second partial derivatives, and the quantity H is determined by the free terms in Eqs. (5.1.8) and (5.1.25). We shall find a solution of Eq. (5.2.t) in the vicinity of the initial curve AB (Fig. 5.2.t) in the form of a series. At a point M (x o, Yo). the required function is
~ (x"
y,)
~ ~ (x, y)+ ~
+, [(dx)' ::~
n=f
+f (L\X}fl-J 6y 8.t:~~~8v + X
m::'·,.'
n
(~~1) (6x)n-2 (6y2)
+ ... + (dy)' ~:n
(5.2.2)
Ch. S. Method of Characteristics
207
where IP (x, y) is the value of this function at the given point A (x, y)on the initial curve, fix = ;:j';1J - x, fUnl fly = YIJ - y. The function 'If may be used ill (5.2.2) instead of IP. Series (5.2.2) yields the required ~olution if values of Lh(' [HIl('Lions IP (or 'If) and also their derivatives of any order exist on a gin'n curve and arc known. Since the flrst derh'alives on this curve [\\'(' shall denote them by p = (fx (or 'fr) and q "-' lPu (or 1f,)} arc gin'lI. our task consists in fmding the second derivatives on it, and ul!'o clt'rh'utives of a higher order. lIcmce. the solution of tlu' Cau(,hy prohlem is associat~d wilh flUding of the conditions in which til(' higlwr (l('ri\·.:I.' tives 011 the givl'1l curvl' ('UIl b(' delermilled. shalilimiL (lul'~('IYCS to determination of thc second derinth"es. Since these cic'rh-ilLiYes are three in numbel' (u, s. rille! t), we ha"'(' to compih· Ihl' !'ame lIumber of indl'Jll'II
"'c
+ +
Hence, we call write the systcm of equations for determining the second d('rivnti\'('s ill the form
A,,+2B8-' Ct-; II =.0 } dxu·, dys-:
O.t-dp~O
O.u-~dxs+dyf-dfJ=-
(fi.2.3)
0
This system of eql\Cltion~ is !;ol\'l'd rOI' the unknowns u. s, and t with the aid of clcterminallt~. H we introduce the symbols 11 and a l" 6~, h, for the principal and partial determinants, respectinly, we have whero
211 C\ dy 0 ; dx dy
\. - II "'...
dp dq
l -III I'
2B C \ ) dy dx
-II C\ \ A 2B dp 0 ; 8/ = dx dy dq dy 0 dx
0 dy
Ui .2A')
dp
dq ) It follows from these relations tlHl.t if the principal deterDlinant 6. does not equal 7.ero on the initial curve AB. t.he second derivatives u. 8, and t arc calculated unambiguously.
208
PI. I. Theory. Aerodynamics of an Airfoil and a Wing
Let us assume t.hat t.he curve has been cilosen so that t.he principal det.erminant on it is zero, i.e . .1. = O. Hence A (dy/h)' - 2B (dy/h) + C ~ 0 (5.2.5) I t is known from mathematics that when the principal determinant .1. of the system of equations (5.2.3) is zero on the curve given by Eq. (5.2.5), the second derivatives u, s, and t (5.2.4) are either determined ambiguously, or in general cannot be determined in terms .of q', p, and q. Let us consider quadratic equation (5.2.5). Solving it for the derivative dy/dx, we obtain (dy/h),., ~ y: .• ~ (I/A) (B ± V B' - AC) (5.2.6) This equation determines the slope of a tangent at each point of the initial curve on which the principal determinant .1. = O. It is not difficult to see that (5.2.6) is a differential equation of two families of real curves if B3 - AC > O. Such curves, at each point of which the principal determinant of system (5.2.3) is zero, are called characteristics, and Eq. (5.2.5) is called a characteristic one. From the above, there follows a condition in which the unambiguous determination of the second derivatives on the initial curve is possible: no arc element of this curve should coincide with the characteristics. The same condition .1. 'ji::O holds for the unambiguous determination of tile higher derivatives in series (5.2.2). Consequently, if .1. 'ji::O, all the coenicients of series (5.2.2) are determined unambiguously according to the data on the initial curve. Consequently, the condition .1. =I=- 0 is necessary and sufficient t.o solve the Cauchy problem. This problem has a fundamental significance in the theory of partial dinerential equations. and formula (5.2.2) can be used to calculate the flow of a gas. Dut from the viewpoint of the physical applications, particularly of the calculation of supersonic gas flows, of greater interest is the problem of determining the solution according to the characteristics, i.o. the method of characteristic'S. This method can be obtained from an analysis of the Cauchy problem and consists in the following. Let us assume that the initial curve An coincides with one of tho characteristics. and not only the principal determinant of the system (5.2.3) equals zero along it. but also the partial determinants .1." = d, = .1., = O. It. can be proved here that if, lor example. the determinants !J. and A, equal zero, i.e. (5.2.5') Ay" - 2By' + C ~ 0 (5.2.7) A (y'q' - p') - 2Bq' - H = 0 where p' = dpldx. q' = dq/dx. then the equality to zero of the other determinants is satisfied automatically.
Ch. S. Method or Cherecteristics
209
Ar.cording to the theor~' of s)'stems of algebraic equations, the equality to zero of the principal and all the partial determinants signifies that solutions of system (5.2.3), although they are ambiguous. can exi.,t. If one of the solutions. for example, that for u, is fmite. then the 801lltions for sand t are also tinite. 5.3. Characteristics CompatlbllHy Conditions
Equations (5.2.5) and (5.2.7) (1 by differential eqllation (j.LH) ('orrc."lpolld to points of the CUf\'e~ determined hy differl:'nlial eqlHlt-ion (5.1.25). Hence, II definite point 011 a duu·ac_teristic ill the plane p, q corresponds to <'Ilch point on a characteristic in the plane x, y. This corr4:'spolldence ran evidently he estaulished in different ways depending 011 the preset boundary cOJldition~. and. as will be shown below, it is just this correspondencc that makes it possible to use characteristics for r,alclliating gas flows. Accordingly, a feature of charaderistks is that the initial conditions cannot be sci arbitrarily along them, wbile they can along a ntf\'e that is not n rharactcristk. Detennlnatlon of QaracterlsHcs
Kind of Characteristics. A close look at (5.2.6) reveals thal the roots of quadratic characteristic equation (5.2.5) may be real (equnl or not equal in magnitude) and also complex c-onjugate. The difference between the root!! is determined by the expres..c;ion B" - AC = = 6. When 6> O. Eq. (5.2.5) gives two
210
Pt. I. Theory. Aerodynamics of an Airfoil and eWing
characteristics; the value of 6 = 0 determines two identical roots. which corresponds to two coinciding families of characteristics, i.e. actually to one characteristic; when 6 < O. the roots of the equation are a pair of imaginary characteristics. Since the roots of a characteristic equation depend on the coefficients A, B, and C of diflerential equation (;").2.1), it is cllstomary practice to establish the type of these equations depending Oll the kind of clulra("teristics. When B > D, Rq. (fi.2.1) is a hypp.rboHc one, when 6 =-= 0, a parabolic, and when 6 < D, an eUiptic one. The cocflidents A, B, and C in Bqs. (5.1.8) for the velocity potential and (5.1.25) for the stream function are determined identically: A =
Vi -
aZ ,
lJ "'"' V:.:1'II'
C
= v; -
aZ
(5.3.1)
Consequently, 6 ~ B' - AC - .' (V' - .,)
(5.3.2)
where V = V Vi + V~ is the total velocity. Therefore. for the regiorls of a gas fiow with snpersonic velocities (V> a). hyperbolic equations are employed, and for regiolls of a flow with subsonic velocities (V < a), elliptic ones. At the boundary of these regions, the velocity equals that of sound (V = a), and the equations will be parabolic. Characteristics in a Physical Plane. Characteristics ill the plane x. yare det.ermined from the solution of the differential equation (5.2.6) in which the plUR !-lign corresponds to the characteristics of the first family, aud the minus sign to those of the second ono. The quantity "-I = dyldx directly calculated by expression (5.2.6) and the plus sign in this expression determine the angular coefficient of the characteristic of the ftrst family, and '-2 = dyldx and the minus sign-that of the characteristic of the second family. Both these characteristics are customarily referred to as conjugate. With a view to expressions (5.3.1) and (5.3.2), we obtain the following expression for the characteristirs in tbe plane x, y:
"-1.2 =
dyldx
=
[1/(V~ -
aZ)) (VcYv,v ± aVV! - aZ)
(5.3.3)
The characteristics in the plane x, .'I have a defmite physical meaning that can be determined if we find the angle ~ between the velocity vector V at a point A of the fiow (Fig. 5.3.1) and the direction of tile charact.eristic at this point. The angle is determined with the aid of Eq. (5.3.3) if we usc the local system of coordinates Xl' 1h with its origin at point A and with the Xl-axis coinciding with the direction of the vector V. With such a choice of the coordinate axes, Vx = V. VII = 0, aud. consequently, "-1,1 =
tan Pl.2
:....c:
±
(M2 -
1)-1/ 2
Ch. S. Method of Characieristic5
211
f~' SJ;~
explanation of the phYsieal meaning of a characteristic:
It 1I11l~ follows thaL ~L ill the ;Ual"ll angle. We ha\'E' there-rore ('stablished 1111 importilllt property of characteristirs con!"islillg in that at every point belonging to a rharacteri!"tir, the angle between a tangenl to it and the velocity v('rtor at this point equals the .1lach angle. Consequentl)', a rharacteristir i!" a line of w('ak di~turbances (or a Mach Hne) haYing the shape or a eurve in the general case. The defmition of a characteristic as a Mach line has a direct application t.o a t.wo-dimensional plane supersonic flow. If we have to do with a two-dimcn~i()na.l spatial (axisymmetric) sup(>I"sonic now, the l\Iach lines (characteristics) should be considered as the generatrices of a surface of revolution enveloping the Mach cones issuing from ve-rtice~ at the points of disturbance (on the characteristics). A surface confming a certain region of disturbance is called a wave surface or three·dimensional Mach wave. 'Ve alre
212
PI. I. Theory. Aerodynllmics 01 lin Airfoil lind
II
Wing
to detl'rmine their ciirectiolls at tlds poillt uy calculu.tillg the Mach angle by the formula Il = ±sin _1 (11M). We determine ..he angular coefficients of the characteristic's in the coordinates x, y (Fig. 5.3.1) from the eqnation A102 = dyldx = tan (~ ± Il)
(5.3.4)
where ~ is the angle of inclination of the velocity vector to the x-axis; the pills sign relates to the characteristic of the first family, and the minus sign to that. of the second family. Equation (5.3.11.) is a differential equation for the characteristics in a physical plane. Characteristics in the Plane p,g. If we substitute for y' in Eq. (5.2.7) the first root of the characteristic equation (5.2.5) equal to y' -= AI' the equation obtained, namely,
A (Ad - p') - 2l1q' - H
~
0
(5.3.5)
is the first family of characteristics in the plane p, q. A similar substitution for y' of the second root,l/' = 1..2 yields an equation for the sE'cond family of characteristic's in the same plane:
A (A,q' - p') - 2Bq' - H
~
0
(5.3.6)
Equations (5.3.5) and (5.3.6) for the characteristics can be transformed by using the property of the roots of quadratic equation (5.2.5) acc,ordiug to which Al
+ A2
=
2BI A
(5.3.7)
By considering the lirstfamily of characteristics and by introducing the relation A Al - 28 = -A2A obtained from (5.3.7) into (5.3.5), we compile the equation
A (A,q'
+ p') + f/
~
0
(5.3.8)
Similarly, for a characteristic of the second family, we have
(5.3.9) With a view to expression (5.3.4), Eqs. (5.3.8) and (5.3.9) can be \vritten in the form
(5.3.10) where the minus sign relates to characteristics of the first family, and the pillS sign to those of the second one. ~~quation (5.3.10) determines the conjugated cbaractedstics in the plane p, q.
Ch. S. Method 01 Chllrlleter;5tie5
213
Ortbogon;lIIlty of Chllrllc1crlstlu
If we rcplan~ the differentials in the r(lu
(5.3.4') where Io. Yo arc the coordinateg of a lixed point, Al is all angular coeffitient calculated from the parameters of the gas at tltis point. and x, 11 are the runlling coordinfltes. Let lIS compile an ('quaLion for an element of a chm'actel'hitic of the second fllmily in the plane p, q in acrordallr.e witl, (5.::U)): A
"1 (q -
1u)
+A
(p -
Po) -1- H (x -
xo) = 0
(5.3.no)
where P'I' qll al'e tbe \',II11e:" of the functions p alltl (I at point. Xo, Yo of the physical plane, the angular cocHidcnt Al and also the values of A and H are calculated according to the parameter~ of the gas at tJIi~ point.. find p and q Me rnnlling cool'tlinates. Examination of Eqs. (5.3.4') and {,").;-Ul') f{'v('als that the inclination of a straight line in the plane;1:, y i~ df'lermined hy the allglllar cocfficient AI' and in the plane p, q by Ihe ,-\lIgular coefficient -1:'1.. 1 ' It rail be proved similarly lhat fill element of a rharacteristir or tlte sccond family in the plane x. y has the angular coefficient 1.. 2 , and an element of n d,araneristie of the first lomily in the' plane P. q-the 811gular coeHicient -1/;'2' It Hl\I;.: follow:" that the characteristics of different familie.9 iTt the two planes are perpendicular to ellck other. This properly makel'l it pos~ible to determine the> direcLion of the chflracteristics in the plane p. 1 if the direction or. the conjll~ate characteristics in the phYl'lirlll pl
214
PI. I. Theory. Aerodynllmics of lin Airfoil lind
II
Wing
'~<
0'
//
"';1'"
Flg.,s.U
Property of orthogonality of cttaracterlstics
tile
of the fir ... t family is constructed in a similar way in plane p. g perpendicular to ... traight line PM and at a distance of 6 z from point P' (Fig. 5.3.2). The property of orthogonality of characteristics helonging to different families manifests itself in the ("aile of a potential flow for the planes x. y and Vo Vv (the hodograph plane), and also ill the casE' of fI vortex flow, for which the plane p. q is replaced with tile same hodograph plane V.t , V v' Trllnsformatlon of the Equations
for Characteristics In a Hodograph
Let us transform Eq. (5.3.10) to a form such that it will determine the characteristics in a hodograph where the wlocity components
J;\~' d~&::eenW:t~~;r~!~~!::i~~! n~~r~~)d~:: ~I~' ;~~~~;.~s t~~~: t\~!Sc~ll~~?:~~ the derivatives dp/dx and dqldx:
_ ye d:; (1- V~)j/(k-I)
+ :~1 ~!
Vv (1- V2)I/tk -Il-1 V
~~
=-5;- (~~) =VxlOy~-1 *(1_V1)IJ(II-J>-:~ye X (l_ll2)'/(h-l)_
(5.3.11)
d1::o:
:~1 V:o:(1-vzPJ(l<-u-IV :~
(5.3.12)
Ch~r~cteristics
Ch. S. Method of
215
A glance at Eq. (5.1.25) reveals that H=f.ye-ta2Vx(1_V2),/(A-O_
~ y
~"i~x
k2~1
}
~~ ; A-'Vi-az
(1_V2)1'/(I<-\)Y"'(a 2 _VZ)
~q
After introdueing (:J.3.11)-(5.3.la) into (5.3.10) and replacing the quontiti('s dy/dx with the angular coefficients 1. 1•2 of the characteristics. auti the flillctions Ian (~+ ~l) with the corresponding values of A2.1' we ol)taill eye-I AI .2 (1- V 2)I/(h-1) (V XA2.1 - V y ) 1- ye (1- y2)1/(~-1J
(d:~:r A~.,- d:~,/) +- J.~~et -.-
~~
(1_V2)1/(I<-t>-IV
(Vy-I'X A2.1)
Fi~tl~ [ey~-la~Vx(1-jT2.)I/(h-I>-- k;;.1
~~~;x
(1_YZ)k/(I,-uye(a2_V2)
~~
]
,-,0
After cancelling quantities where possible and inLroducing the dimensionless variables
Vx
-= V.JV.nnx' fy = l'yfVmu.
V
= V/Vmu ' VIa -=
M
we ean wriLe this equation in thf' rorm
f A1 .2tV')'2.1- Vy)!
(d~'(~.I_d;:)
r!I-Vx1<_~ .•
21'
if
e
i\
'--;;:-=t.~ dI-Y·1·-','~!a~
- k;t _1:1'
·(1_V2)
t~~;i/~~ .~_=O
(5.3.14)
lly illtroc.lucillg Lhe polar aug-Ie B, we obtain tlJ(~ following expressions for tl\e projections or the velodLy vecLor:
v." LeL
liS
=
Vcos~,
Vy
"""
Vsin~
(5.3.15)
differentiate thege expressions with respect to x: dVxldx ," (dl'/d.r) cos ~ - I' sin ~ d~/dx elV yld.t = (dV/d.r) sin ~ -7- V cos ~ d~/d.r
I
(5.3.16)
introducing (5.3."15) and (S.:l.Hi) inLo (5.3.14) an<j taking into account that tan" ~l = (lll~ - 1)-l and sin~ ~l "= M-", we have
*(
k2~21' sinf\~;·\:sc"s~ +A2.,cos~--sin~)
216
Pt. I. Theory. Aerodynllmics of lin Airlo,,,;'-',,,"",,,'--W=ing'---_ _ __
- V :~ 1 :-0
(A 2•1 sin ~ -j
C()~~S~~Sin2IlJ-
cos~) +
k;l .
;v
f V [A
1•2
(AM cos ~ - sin ~)
l'_C:~~t~~in2,"" .~=O
(1-V2)
V 2 /Vin .. x) .." a2JV~Qlt =
l(k - 1)/21 (1 -
Having in view that fh sin 2 fl, we find
e4
4+d~ (~2.1:~.~~o:-~oS~?:~n11l '<
[A1.2
(A2.1 cos
~ Si~~~~~:i~2 '""
+ :~ - (sin21l C()Si~~i~::.cos~
C()S2
~) ]
sinBl
~:
tan 2 /-t =0
(5.3.17)
Performing the substitutions Au -_. tan (~ =F fl) and A1.2 = ± /-t). we transform the individual expressions contained in (5.3.17) to the following form: = tan (~
:::~~::~+:~:~11 AI.:!
~t~nll
(A2.1C!lS~ s~~~)~~~1~21l __ CIJS2~)
(sin21l- CIlS2
~~i~:'1 CIlS 1:\_ .. sin 1:1)
=
(5.3.18)
::)~f/::)
-"'
±C()SS~~3~ 11)
15.3.19)
(5.3.20)
With a view to (5.3.18)-(5.8.20), Eq. (5.:t17) for the characteristics of the first and second families, respectively, acquire the form d: _ tan /-t d~-- e~. sin :):~~ ~
~- :~ . co:~;~:_t 11) . :~
d:-
~~n 11
= 0
(5.3.21)
-I--tanlld~-e~.sin~:~~~~~nll -
:~
.
co:i;~~~t)
.
!~
= 0
(5.3.22)
The entropy gradient dS/dn can be calculated ac('ording to the vahle of the derivative of the stagnation-pressure dp~/dn. For this purpose, we shall usc relations (-1.3.6) and (4.3.20) from which we obtain the following formula for the difference of the entropies: S2 - S1
;0.'
-c u (k - 1) In
(p~/PQ)
(5.3.23)
Since c.., (k - 1) = R, then by calculating the derivative with respect to n am] designating dS:!Jdn by dSldn, we fine!
-}. !: ~ _:~ .~o
(.5.3.24)
Ch. 5. Method of Cherederi5tics
217
For our furthE'r tran~rorml\tiolls, we shall introdll('.e the new va~~ II
v
r
(t)
= a{cot ll
dV
v
(5.3.25)·
that is an angle. We shall express the ratio dl//V ill the form V/a*), and ('ot Il with the lIid of (3.6.23) in the form
d)./')....
(A
eot~~VM'-I~V(''-I)/(I- :~:: .,)
(5.3.26~
Consequently. >•
.. ~ i\.
Vr("_l)/'(l_~").!!:k I 2
(5.3.27}
,.
Integration of (.'i.3.27) yields
(·),...,.V ~:'~!
=
tall-IV'4:=_:'+:-:-'=]'~'~1
'-8}.2
=
~I VI=]'~' ,. __k-j,l k_··_';"2
- tlln- I .,
Having in view lh'lt A1II8:" = Vmula* = lain
O)~
11 (k
(5.3.28)·
:-1)j(k
1). we olr
V :~! lan- V A~~::=-\~ -tan-1 V :~: . A~~:x~A! J
(5.3.29). Substituting M for).. ill (5.3,28) in accol'dance with
(1)=",/ !~.~!
tan-I
(5.~.2(j),
we have
V !,~_: (M~-1) -tllll- 111 M:!.-1 (5.3.30}
Examination of Eqs. (5.:~.2\J) and (5.3.30) reveal~ that tlIe
218
Pt. I. Theory. Aerody""mics of "" Airfoil "nd " Wing
TolJle 5.3.1
"I 1.00 1.10 1.20
1.30 1.40 1.50 1.60 1.70 ·l.SO 1.90 2,00 2,'10 2.20
2.3<' 2.40 2.50 2.00 2.70 2.8) 2.90 3.no .3. to 3.2] 3,30
3.40 3.50 3.60 3,70 3,80 3.90
w,
d"
I ., d" Ii ... I
0.000
L3J6 3.558 6.170 8.987 l1.9C5 14.861 17.810 21.725 23.586
26.380 29.097 31.732 34.282 36.746 39.t24 41.415 43.621 45.746 47.790 49.757 51.650 53.470 55.222 56.907 58.530 60.091 61.595 83.044 64.440
~J.OOO
II
4,00
55.381 4.10 56.443 I 4.20 50.285 . 4.30
45.585 ~ 4.40 4LSI0 !4.50
38.682 4.&1 36.1'132 4.70 33.749 4.80 3t. 757 ' 4.90 31l.(}UO i 5.00 28.437 5,'10 27.036 5.20 25.7'11 5.30
24.624
5.4.0
23.578 5.50 22.620 5.60 21.738 5.70 20.925 5,80 2(1.1,1 5.90 19.471 6,00 18.819 : 6.10 18.210 t7.G40 17.1t5 6.40 16.602 6.50 16.128 6.60 15.681 6.70 15.258 6.SO 14.857 6 90 1 •
I ~:~ I
w,
I . . d,. II.. I
d,"
65.785 67.(182 68.333 69.541 70.706 71.832 72.919 73.970 74.986 75.969 76.92..) 77.841 78.732 79.596 8<).433 81.245 82.032 82.796 83.537 84.256 84.955 85.635 86.296 86.937 87.561 88.168 88.759 89.335 89.895 9.1.441
14.478 14.117 13.774 13.448 13.137
12.814
11
7.00 7.10 7.20 7.30 7.40
I 7,50
12.556 7.60 12.284 ; 7.70 12.025 I 7.80 11. 776 7. 9;) 11.537 8.00 1
~~:~~ j ~:!~ 10.876 10.672 10.476 10.287 10.104 9.928 9.758 9.594 9.4.35 9.282 9.133 8.989
! 8.SO
•• SO •. 00 9 20
1 •.• 40
'J,60 I 9.SO 10.00 110.20 10.40 to. 60 to.80 •. 850 11.00 8.715 8.584 8.457 j11.60 8.333II 1.80 112 .00 I
i!::~
w,
d.,
90.973 91.491 91.997 92.490 92.970 93.440 93.898 94.345 94.781 95.2.:8 95.625 96.430 97.200 97.936 98,1)4.2 99.318 99,967 100.589 101.188 t01..763 102.316 102.849 t(iS.362 1(13.857 114.335 104.796 105.241 105.671 trI6.087 t06.489 HI6.879
I •. d., 8.213 8.097 7.984 7.873 7.776 7.662
7.5M 7.462 7.366 7.272 7.181 7.005 6.837 6.617 6.525 6.379
'.240 6.107 5.979 5.857 5.739 5.626 5.518 5.413 5.313 5.216 5.133 5.032 4,945 4.861 4.780
InlrolJu<"ing the angle (tj into (5.;3.21) and (5.3.22), we obtain the following equation for the characteristic's:
d(w=F~)-e-T. :.~~il5~~n:)
±
:~
.
~:~t(P;~~
.#n-=O (5.0.31)
Equation (5.3.Hl) corresponds to th(' mo~t general case of super· .sonic two-dim('nsionai (plane or spatial) vortex (non-isentropic)
now of a gas,
Ch, 5, Method of CharacleristiC$
219
Equlltlons (OJ ChllJlldeJisHcs In a Hodograph lor Pllrtfcular Cllses 01 GIIS flow
The form of Eq, (5,2,5) for charaderislics in a pllysical piane is the same for all ('ases of gas flow if til(' latter is Sl1pf>['soni('. and twodimensional. Bllt in a hooograph. the eqllations for the ("haracteristks difl'er and depend Oll the kilul of now. If a two-dimensional flow is \'ortex-free, then 1\("{'ortiing to (5,1.2:1) at all points of :::pa("e occllpiefl hy tin' gas, the entropy is com;tant (dSldn = 0), and, therefore, the efjuation for the characteristics 3equircs a simpler form:
d(w=F~)-€f' ~:'~~]1s~~)
,0
(5.3.32)
For a plane lion-isentropic now (€ -= 0). we have
In the simplest ease of a plane \'ort('x-free flo\\' (dS'dn -: 0), wc have d (00 'tO~) ~ 0
(5.:1.:1-1)
lntcgl'ation of (5.3.34) yields ~ - ± (r) -+- const. Introducing lixcd values of tho angles ~l atl(l ~2 instead of the ('onsLanl. where ~l (',01'respollds to t.he pIll," sign of ro, and ~2 to the minus sign, we lind lUI equation for the ('haractE'ristics in the form
- ± ro Introducing instead of
B= ±
t,l
---j-
relation
(V:g tan-
-tlln-ll;/
~~.!
Bl.:!
(3.:l..15)
(~).:{.2U), I }-/
we oblnin
i.::::-\1
. i.Lzx-:....\2) ·.:.·~I.Z
(5.3.36)
Hence. unlike Eq. (;),2 .•'» lor ('hnracLt'rislics in a php:icai plane and Eqs. (.'i.3,31), (!i.:1.~{2) or (!i.;{,3:~) for rharadl'l'islics in a hoclograph ill a difie]'('utial Io]'[\}. the ('ol'l'eSpollfHng er]uation (5.;~.:J()) for the charact.eristics of a pin!}£' isentropk flow has an explicit form. GeomClt'('ally, this eqllation deline's two families of ('urves-('haracteristics in a ring whose iuncr radiu!> is ").. ---.., 1 anci whose outer one is i. nlax = [(k ._. I)!(k -- 1)11/~ (Fig. 5.:1.3). The integl'
220
Pt. I. Theory. Aerodynamics of an Airfoil and II Wing A,
Fig. S.l.l Epicycloids-characteTistics of lL plane supersonic flow: l-charaelprisHc or the first lamn),; 2-charactnistlc of the Sf'cond ra· mU)'
and can be obtained as the path of a point on a circle of radius 0.5 (Amax - 1) travelling along the inner (or outer) cirrle of the ring (Fig. 5.3.3). The angle ~ of inclination of the velodty vector and the speed ratio A are the polar coordinates of points on an epic.ycloid. By analysing a graph depicting a network of epicycloios, we can condude that angles ~ larger in their magnitude, i.e. a more considerable deflection of the flow from its initial direction, correspond to an increase in the velocity due to expansion of the flow. A smaller deflection of the flow is observed at lower velocities. The angle of inclination of the velocity vector is neterJllined directly by the angle w whose physical meaning can be established from (5.3.35). Assume that the integration constants ~I.2 . .." O. This signifLCs that expansion of the flow begins when ~ --= 0 and M = 1In accorciance with this, the quantity ~ = ±w (M) is the angle of deflection of the flow upon its isentropic expansion from the point where M 1 to a state characterized by an arbitrary number M > 1 equal to the upper limit when evaillating integral (5.3.27). Hence, there is a difference hetween the angle of deflection of a flow with the numher M > 1 from a certain initial direction and the angle ~ .- ±w determining the complete turning of the flow upon its expansion from a state characterizeci by the number M -: 1. The angle of flow deflection in an arbitrary cross sectioll can be determined as follows. Let lLS aSSHme that we know the initial number Ml > 1 which as a result o[ expansion of the flow increases and rea('hes the yalue M2 > MI. The angles WI and 002 of {Iefle('.tion of the flow from its direction al a point with the number M --' 1 correspond to the numbers Ml and M 2 . The angles WI and 002 can be determined graphic-ally with the aid of all epicyc.ioid from expression (5.3.30) or Table 5.3.1. We lise their values to lind the allglrl of inclination of the velocity vectors. Considering, partieularly, a characteristic of the lirst familr, we obtain ~I -= WI (M 1) ami ~2· 002 (M 2). Conse-
Ch. 5. Method 01 Characteristics
221
quently, the angle of deflection from the initial direction is ,,~ ~ ~, -~,
"00, (M,) -
00, (M,)
(5.3.37)
The calClllati()ll~ can hI' performed in tllC oppo1'itc sequence. determining the local number M'}. according to the known angle d~ of
00, (M,)
~ ,,~
+ 00, (M,)
"2
(5.3.38)
We lind tbe corresponding values of or M2 from Fig. 5.3.3 or Table 5.3.1 according to tllC value of (0)2' Of interest is the calculation of the ultimate Row angle or the angle of deflection of the flow needed to obtain the maximum \'elocity l'max' Assume that deviation begins from the initial lIumber M '--" 1. In this case, we lind the ultimate flow angle by formuli\ (5.3.::l0) in which we must adopt a value of M corresponding \0 V ma " equal to infinitr: (5.3.39) for k .-' 1.1, the value of W ma " - U.72{b: -' "1:~{J.1(jo. Consequent Iy. a supergQIlic flow cannot turn through an angle larger than wmax • and theoretically part of space remains unfilled with the gllS. H the initial Ilumuer M from which the deflection begills is larger than uuity, the angle of this (Icflec-lion measured from the direction at. M = 1 will be OJ (M), while the ultimAte flow angle corresponding to tills Humber is ~,,,,,~oom,,-w(M)' (~/2)1V(krl)/(k I) Ij-.o(M) (5.3.10) for k . 1.4. the angle ~lIlax - J3U.4(r - (0) PI). For hypersonir velocities, the calculation of the function (0) and, ('oJlseqllently, of the angles of deflection is simplified. Indeed. for M» I. the terms iu (5.3.30) can he represented in the follo ..... ing form '''itll all at'cllri\{'Y within quantities of a higher order of inflllitesimal:
V~~~ tall-'l/~~~ (M2_"1)~V~~! (T-~V:=!) tan-I VM"--l ~ a/2-1/M. Accordingly, (5.3.41)
The formula corresponding to (!J.3.3i) acquires the form
~~=~2-~1=
- k~1 (;2 - ~t)
(U3n
222
Pt. I. Theory.
Aerodyn~mic$
0/ an Airfoil and
~
Wing
All the above relations have been found for a perfect gas. At very low pressures. however. a gas is no longer perfect. This is why the calculated ultimate now angles are not realized and have only a theoretical significance. 5.... Outline of Solution of Gas-Dynamic Problems A"ordlng to the Method 01 Characteristics
The determination of the parameters of a disturbed snpersonic flow is as)':ociated with the solution of a system of eqnations for the ('hnracteristies in a physical plane and in a hodograph if the initial conditions in some way or other are set in the form of Cauchy's conditions. In the general case of II two-dimensional non-isentropic flow. this system has the form: for characteristi('s of the first family dy - dx t.n (~ -;- ~,) (5.4.1) d (oo - ~) - e (dx/y) l (dx/kR) (dS/dn) c - 0 (5.4.2)
+
for characteristics of the second family dy - dx t.n (~ - ~,) d (oo -;- ~) - e (dx/y) m - (dx!kR) (dS/dn) t = 0 where the coefficients are 1 = sin ~ sin )licos (~
+ 11),
si11 2
C .-
~~
(5.4.3) (5.4.4)
cos J1lcos
m = sin ~ sin !-lIco.<; (~ - )l), t = sin 2 j.l cos ,lIeos (~ For a two-dimensional isentropic flow. we have dS/dn - 0 therefore the system of equations is simplified: for characteristics of the first family dy ~ dx t,n (~ + ~); d (oo - ~) - E (dx/g) l
~
(~
+ j.l)
j.l)
(5.4.5) (5.4.6) (5.4.7)
0
(5.4.8)
for characteristics of the second family dy ~ dx t.n (~ - ~); d (w + ~) - e (dr/y) m - 0 (5.4.9) For a plane flow, we assume that
t
= 0 in the equations, for a
SP~1:~~I:i1~s:~C~:;d!~:~:ih~ ~eii):dn~f:~~;a~;:!fsCt~c~ ~~s!:~d p~!g: lem on the flow over a body t:onsisls of the solution of three particular problems.
Ch. 5. Method of Characteridics
223
(6)
~~~e:~':ion or the different families:
(I"-physical
pl~ne:
velocity at the intersection point of two characteristics of
tl_I)lane or hodograph
The Jirst problem is assoeiated wiLh the determination of thevclarity and other parameter." at the poillt of iElIl'r~eclioll of characterist.irs or tliffereut fnUlilie!'; i!lsuing 11'0111 two rio!le poillt~_ Assume that we arc delermining the parameter,s at point C (velocity Ve. IlIliub£.>l' .J/r,. flow uefleciioll angle ~e. entropy Sc. etc.) at. the inter~ectioll of clements of charfl('.tcristirs 01' the lil'st anti gccond families drawll from points A and JJ (Fig. 5.4.10). At these points~ whirh arc on diffl'rent strealillille~. we know the velocities VAt Vo• and other parameters, including the cntropie.<; S.". and So. All the ralrulatioJls arc based on the lise of Eqs. (5.4.1)-(5.4.4) for the ('haracteristit's. which in linite dirrcrenrcs havo tli£.> form: for the fir!!t family 6Yn ..." 6XB tau (~s '7- ~In) (5.1.10, ..l.Ws - .1.~1l - e (6xs/YIl) ls -:' (l\:rs'kR) (6S/.1.II) CD ~.= 0 (5.4.11)· for the
~e('oncl
family .1.YA . - 6.T,. tall (~,. -
~I,.)
(S.1.12}
6w,. ...:, .1.~,. - f (.1.J·,.ly,.) til,. - (1lr,.lkll) (6SI.1.II)I,. ·0 O"iA.13) wherr 6yo ..." Yc - Yn- 6JIl'-O; .I'e -.t"Jj. 600a -= (ole - (0)11' } t:J.YA ._' Yc -
1IA.
&~B ~ ~c 6XA:~ Xc - .TA_
-
~B
6(1),.
&~. ~~" -~.
We -
roA.
(5.4.14)
'224
PI. l. Theory. Aerodynamics 01 .n Airloil and a Wing
In . , sin ~B sin I-'u/cos (~B
-I-
I-'B)
+
sin~ J1B cos Jln/cos (~B Jln) (~A - I-'..d tA :-: sin!! JlA cos /-lA/COS (~A - /-lA) CD -
mA .:.
}
(5.4.15)
sin PA sin I-'A/c.os
Equations (5.4.10H5.4.13) were derived assuming that the caeffi·cients I, m, c, and t remain constant upon motion along characteristic elements BC and AC and equal their values at the initial points B and A. The change in the entropy per unit length of a normal b.S/b.n is ealculated as follows. It is shown in Fig. 5.4.1a that the distance between point.s B and A is b.n ::::::: (AC) sin IlA (BC) sin j.1n
+
where AC ....:; (xc -
XA)/COS (~A
JlA),
Introducing the notation e "-' (xc - XA) sill JlA {'os (PB -\- J.lB).
XB)/COS (~n
Be -- (xc -
I
.= (xc -
-
xo) sin ,...8 cos X X (~A -
we obtain IlSllln ~ (S. - SB)
,OS (~B -'- ~D) COS (~. -
Ils)
~A)I(f
+ e)
~.)
(5.1.16)
The entropy aL point C is determined from the relation Sc=b.Sn-l-SB =
!! (BC)sinJ.ls+Sn= (SA/~SeR)/ +Sn
(5.4.17)
The entropy gradient may be replaced with the stagnation pressure gradient in accordance with (5.3.24): t
AS
1
Apo
7f'Afl= -P;;'""'Kn=" -
(PO,A-PO.S) cos (f}n+lls) CtlS(f}A-IlA) (j·:·e) II
Po.
(5.4.18)
The sLagnation pressure at point C is (5.4.19) Po, c ...: (Pl!. A - Po, n) fl(f -I- e) + Po, B To rilld the coordinates Xc and Yc of point C, we have to solve the simultaneous equations (5.4.10) and (5.4.12) for the elements of the eharacteris1ks in a physical plane: Yc - YB = (xc - xB) tan (Pe -1- J.l8); Yc - YA = (xc XA) tan (~A - J.l.A)
A graphical solution of these equations is shown in Fig. 5.4.1a, We use the found value of Xc to determine the differences b.xo '-"
Ch. 5. Method 01
=Xc
and
'-::L'c
CI1~r,!lcle"stics
22ri
in Eqs. (3.4.11) and (5k13). The incremenl~ 6.008, .1.UlA. 6.~o, flnd 6.~.\ Hre tilc ullJ..:no\\'ll~ in these equations, but the latter are onl:.' two ill numher. The numbet· of unkuO\\'rls can be reduced to two in lluordance with the number of equations in the system. For this purpose, we compile tlte obviolls relations .1.wA ,- Ule - UlA, -- 6.~lB -+ Wn - ~JA \ (3.4.20) 6.~A ~B - ~A - 6.~B + [31J - ~A -Xll
u,tA
-XA
."0
t
With account taken of tllese relations, Eq_ (5.4.13) is tran:::Iormed as follows:
~W.TW.-wA·' ~~nT~B-~A-' A:~' mA- ':/,'. !! 'A-O
(5.4.21) By !'olving this equfltioll simultaneou:::ly wilh (5.4,11) for the \"al'iflbie 6.~II' we obtain
ll~n=+ [k~l . ~~.
tLixAtA :·.1.ZoClI) :-8
(~:A
- ~:·l.) -(W'-"'A)-(~n-~A)]
\-Ve use the found value oI function W by (5A11):
6.~fl to
rnA
(5.1.22)
evaluate the inrn>lllellt of the (5.4.23)
Xow we can cakulRte the angles for point C: ~c = Li~B
+ i3B;
We = 6. Ws
+ (us
(5.4.24)
Ar('ording to the found \'alu!'! of (U(;, we determine the number M(; and the :Uar.h lingle Ile at point C from Table ;i~l.1. A grapbiclll solution of the ~ystE'rn of pqu.1lioIlS for characteristic .., in a hodograph a5 a result of which thf' angle ~c and the nllmher AC (Mel are detet'mined i5shown in fig. 5.4.1b. where fi'e' and A'C' are elements of the charnclerisLics of tIle fm;t and serond families corrc!>ponding to element:; of tlte eonjngate characteristic.
226
Pt. I. Theory. Aerodynllmics of en Airfoil .nd .. Wing
(oj A,
(al " V,
~~'~~~:ion of the velocity at the int.ersection point of a characteristic with a surface in a flow: a-pbysical plane; b-planp of bodogrl. ph
at point C in the first approximation, i.e. from the values: fl1' ,= (flA ~~' ~ (~A
+ Ilcl/2, IllJ' + ~c)/2. ~~'
::-=
(fiB
~ (~.
+
+
Ilc)/2
}
~c) /2
(5.1.25)
We fm<] the more precise coordinates .cc and Yc of point C from the equations 6YB = AXB tall (~il ' +- IllJ ' ); 6YA = 6z A tan (~A2' - fl.~')
(5.4.26)
The se~oDd problem consist:!' in calculating the velocity at the point of interscction of a characteristic with a surface in a now. Assume that point H is located at the inter5ection with linear element DB of a characteristic of the second family drawn [rom pOint D near a surfacc (Fig. 5.4.2). The velocity at this point (both its magnitude and direction), the entropy, and the coordinates of the point are known. The velocity at point R is determined directly with the aid of Eq. (5.1.1) for a characteristic of the second family. If we relate this equation to the conditions along characteristic element DB and express it in fmite differences, we have 6wI) .~ -6~D -7- e (6xo/Yo) mo - (6xo/kR) (6S/6n) to (5.4.27) where 6wo = WB - WO, 6~o = ~B - ~o' 6xo = XB mo = sin ~o sin flo/COS (~o - flo) to = sin2 flo cos flO/COS Wo - flo)
Xo
} (5.4.28)
Ch. $. Melhod of Ch~r~deri5tics
227
The coordinates XB and YB of point II ;1fP drt0fmincd by the "imllltaneous soIntioll of thc equation for (\ clwra('teristic of the l'f.'conri fnmily and the eqllation of the wall contour: YB - Yo -, (rn - Xfj) tall (~l) - /.'-0); Yo "--' f (xD) (5.1.30) A graphical sohllion of these equations is ~ho\\"l1 in Fig. 5.·1.2a. 'Ve lISe the fOlllld ('oordinates .Tn and Yn Lo evaluate the angleo ~s from the eqllation tun ~!.1 = (dyld.c)Jj (5.4.31)
The entropy S8 (or the stagnation pressure P~. B) at point B is considered to be " known qllantity and equal to its \'slue all die streamline ("uindding with a tnngellt. to t.he surfa('e. The entropy is con~idcred 1.0 be constant alOllg au clement of the tangent. Vpon ca\cllinlillg the innelllcnj. 4wv by (5.4.27). we can liltd the angle (r)R· - AWD -L. WJ). Hlld theu detel'mine the number i.,s from Table 5_3_ ·1. Ilo\\" the velocity at point H is determined is shown graphically in Fig. 5..'i.2b, whert> element Dr fl' of it characteristic of the ~econd famiJy ill 0. hodograph ("orrl'spond~ to all element of n ('.hafnrteri~tic of the same family in n physicnl pI nne. The third problem consist~ in calculating the YP\ocity at the interl'prtioll of the charartE'ristics with a ."hod, alld in determilling the rhl'lnge in t.he inclinatiOIl of the shock at. this point. Since a characteristic in its nature is a line of weak di.~lllfhan('e.", Ihis illtersectioll physically ('orrc!'iponds 10 the inlcraclion of a wpal, w(\\-e \\"lth a compression shark. Assume that closely arrangpd expan"ioll wa\"('~, ""hi('\l rharacteril'tics of the first family ('orrespolld to. fnll at points J and H on ('omprc~sioll shock A· of a preset shape!J ..,.t tl") (Fig. 5.4.3a). As a rcsult, the streng!11 dimillishL's. as dol'S the inclination angle of the shock_ Sin('c points.r nnd Jl :lr(' SOllrces of (Iistllrbanccs. expansion wa\"C~ appear, AJI(I ('h~facteristics of thc .~ecOl\d bmily ran he drawn through these point". Olle of stich rharartl'fistics, pa.<:sing through point. J. interser.ts the neighbouring conjugate ('hnrart('ristic at point P (-ailed lhe nodaJ point of the cbarac~ teristies. To determine the rhaugc in tIll' illl"linl'ltion of tbt'! shark oud in the ,·elority behind it, olle nlll~t u~e the properties of the C'hornC'tcris-tics .r F and FH pH.~sing through nodal point F. and also the rcla~ tions for calcllhllillg" a compl"f!SSiOll shock. Since length JH of the shock h, »mall. this scrtiOH lTlay be a!'>sumed to be linear. The angle of inclinl'llion of thc »ilo('k on this section and the corresponding pa-
228
pt, I. Theory. Aerodynamic.s of an Airfoil and a Wing
(6)
(•.1
~y
rameters of t.he gas are approximately equal Lo their values at inLersection point II of clement FH of a first family characteristic with the shock. One or the unknown parameters is the angle of indinatioll ~H of the velocity vector at this point. which can be written as ~H = = A~p + ~p, where A~Ji' ~H - ~F. and ~F is the known angle of inclination of the velocity vector at point F. To evaluate the second unknown (the number MH at the same point) we shall use the formula MH ~ MJ -I- (dMld~)JA~JH (5.4.32) in whicb (dMld~)J is a derivative calculated with the aid of the relevant expressions for a compression shock according to the known parameters at point J, and the quantity A~JH determined by the change in the angle ~ along a shock element equals the dirt:erence ~H - ~J. We may assume that this quantity approximately equals the change in the angle ~ along the element FH of a characteristic of the lirst family A~PH --' L1~p PH - PF' The derivative dMldP is evaluated as a result of difterentiation of (4.3.19'), 00<
dd~
=
~1&1
=
-M2
+ 2 (f~6)
[cOt(9~-P5) ( ::: -1) ~,) f.-)]
s;n' (a, -
+. (
(5.4.33)
We shall calculate the derivath'c d (P2/Pl)/dP. on the right-hand side by differentiating (4.3.13):
d:s (Tr-)=2cote~(1-~*)i7-' ::
(5.4.34)
Ch, 5. Method of CharaderjstiC$
229
We determine the derivative de_\.'d~s as follows. We differentiate
(4.3.24):
d~S (~)-~~{ :~; [sine5Ico~As - "Csi"""'(e';-'-~;~),"'"CCs(mll,----c~".) + sin (0" Bs/cos (0;: tis)} This relation ('au be transformed ::;omcwhat by using
().~.2~):
d~s (~) =~. tan05eos~(OS ~s) X
{~:: [cos:(;~;o:~~)_~ ]+Tt}
(5.4.35)
Equating the right-hand sides of (5.4.34) and (5.4.35) and solving the equation obtained for the derivative dO,/d~~, we obtain:
~~: .~~ [2cog2 (6s -Ps) (1-l)~) (5.4.36) Let IlS de-rive an expre!'sioll for the change in the number.V when trave-lling along the characteristics from point F to point H, Le. for the quantity llMp - M'H - M}I. To do this, we sh",n s\lb.~tittlte expression (5.1.32) h('re for MH: ll,"~. = 111J -:- (dMJd~h ll~F - .uF (S.,i.~7) \Ve can go over from the nllmber M to the rUllction
(0)
determined
by relation (5.:1.30):
(5.-'1.38) where (dwld~)J ~ (dMid~"
(5.4 ..1U)
(dwidM),
The derivative dw/dill is determined as n result of differentiating
(5.3.30): (5."-40)
Equation (5.-1.11) for a characteristic of the first family applied along clement FH yields ~"'F
-
~~,
-
e (AXFiy,.) I, -'- (AxFlkR) (!!'sIAn)
'F ~ 0
(5.4.41)
where 6,Wp=WH-WF,
ll~F=~ll-PP.
IF = sin ~}' sin IlF/cos (~F -:- JAF) CF =
sin 2 J-tF cos
).tp/cos (tiF
'i-IlF)
llXP=Zll-XP} (5.4.~2)
230
pt. I. Theory. Aerodynllmks of lin Airfoil lind II Wing
The distance along a normal to a streamline between points F and H is 6.n= (FH) sin I-LF= c::~~~~I-lF sinllF
(5.4.43)
COIl!<eqllently, the entropy gradient in (5.4.-'11) is I1S/6.n . . . .: (SH - SF) cos (~F
!:
= -
P:' F
(po.
+ IlF)/(:rH
-
xF) sin Jl.F
H(~:O.~)):~:~;+~f»
(S.4.4'i) (5.4.45)
Po,
where H i~ found ac(',ording to the nllmber Mn from the shock theory. The entropy SF or the st.agnation pre~~;ure P~.F at point F can be adopted approximately eq1lal to the corresponding values at point J on the shock, i.e. SF ~ SJ and P;,. t' ~ J. Solving Eqs. (5.4.38) and (!j./i.41) for 6.~F' we obtain
Po.
I1~F:~[( !;L-1rl(WF-WJ+f: tl:t~'
IF-
~~;. !~
Ct·)
(5.4.46)
Illserting the value of 6.~F into (5.4.38), we call find 6.wp, ('~alcu late the angle Wn ....; 6.Wt' + Wt" and determine more precisely the number M H . By (,alculating the angle ~n ~...: 6.~F ~F' we lise the values of this angle, and also of the preset llumber Moo to lind the shock angle 8s. H at point H and, consequently, to determine the shape of tbe shock more accurately on section JH. If necessary, til" calculations can be performed in a second approximation, adopting instead of the parameters at point J their average values between points J ami H. Particularly, instead of the angles WJ and ~J' we take the relevant average values of 0.5 (wJ -;- WH) aud 0.5 (~J ~ ~n)· Figure 5.4.3b shows how the problem is solved graphically. Point H' 011 a hodograph, corresponding to point H on a physieai plane, is determined as a result of the intersection of element F' H' of a first family characteristic ,,,ith a shock polar constructed for the given free-stream number Moo. The vector 0' H' determines the velocity AH at point H.
+
5.5. Applications of the Method of Characteristics to the Solution of the Problem on Shaping the Nozzles of Supersonic Wind Tunnels
The method of characteristics allows liS to solve one of the most important problems of gas dynamics associated with the determination of the shape of a wind tnnnel nozzle intended for producing a two-dimensional plane parallel supersonic flow at a preset velocity.
Ch. 5. Method of Cho!lro!lcteris/ics
231
e:<··~lt;::~J 7
ff:iz~:'~r sup~rsonic :p~~t:ferwall; t
5
4. J
tunnl'\;
nod .1 -;iM walls; J-t'xit 5"CtiOO; "-bottom
'Ilo... II:
,,-critical
s~cliQn;
7-
FitJ·U.l
Unshape>d two·dimensional supersonic nozzle with a radial now The nozr-Ie ensuriug sur-Ii a now is a mouthpiece whose l'lidc walls are nat. while its top and boltom walls have a spedaUy shaped contour (Fig. ;).5.t). Tn addition to detCl'mining tile shape of its curved contour, the design of a nozzle inc.ludes calculation of the pllrameters of tile ga9 ill the reeeiver (Lhe p;uameters of stagnation) and in the critical section, and also ill'l area S*. The parameters of the gas at tile nozzle exit are Ilsually pl'escl. namely, lht' nllmber Me.;, tile pressure p"". the area of the ('xit section S ~ lb, and the temperature of the gas in til(' receiver To. The area of til(' <:Titleal nozzle section is found from flow I'ale (,([lIalion (;l.G.fa!a) which we sllall write in the form p""V""S ... P*Il*S*.lIence S* - {r.",V",/r*a"') S -= It follow!' from (:~.(j.li(),) thai Ihe parameter q is rletcrrnilled IJY the preset number M", nl the noule exit. According Lo this value of Moe and Ihe PI'csslLre p"", at the exit, and by lIsing formula (:-)'G.:.J.G), we c.an lind the pressure Po in the recei\'e'r necdCll to ensure the pre~ct lIum\)Ci' M xo at the exit. Sext the anglf! 2" of an ullshaped nou.lc is liet (Fig. ;).J.~). E-xpcrimental illve:
qS.
232
Pt. I. Theory. Aerodynllmics of lin Airfoil and
II
Wing
FIg. S.S.)
Construction of a:shaped supersonic two·dimensional nozzle
termined by the quantity r* =- 8* [360/(2n·2y·1)1, while the distance to the exit section r A ~ 8 [360/(2n·21'·1)1. In the region of the nonle confmed by two cirrles of radii r* and rA, i.e. beyond the limits of the critical section, the gas flow is supersonic.. Shaping of tile nozzle consists in replacing straight wall BC in this region with a curved contour ensuring the gradual transition of the radial flow into a plane parallel flow at the eTit at a preset velocity. For this purpose, let u.s draw through point 0 a number of closely arranged liJlPs and determine the velocities (the numbers M) on these lines at their intersection points AI' ...• An with a characteristic of on(> of the families AAn (let us consider it to be a characteristic of the sl;lcond family) emerging from point A on all arc of radius rA (Fig. 5.5.3). Point Al is at the intersection of ray r 1 = = OA I with the element of characteristic AA I drawn at an angle of ~"" -- _Sill- I (11M ",,). We fwd the number MI at point Al \vitl! thp aid of the expression 91 = 8*IS1 • Since 8 1 = 2ar1 ,1 (21'/360), and S* -= :!nr*·1 (2yl360), we have ih = r*lrI . Employing (3.6.·'.6), \ve can lind Al and the corre~ponding number MI' We determine in a similar way the coordinates of intersection point A z of the adjacent ray rz = 0.1 2 with the element of character· istic AIAz inclined to straight line OA I at the angle ~(l = = _sin- l (1IM I ), the number M2 at point A2\ and so on. As a result, wo can construct a characterist.ic of the secolJd family in the form of broken line AA1 ... An_tAn int('r~ect.ing straight wall BAn of the nozzle at point An. For this point, as for the other points An_2' An_I' and An of intersection of the characteristic wit.h the straight lines emerging from source 0, we r.an calculate the relevant Mach numbers and the angles fL = -sin-l (11M). The flow region OAAn with a known velocity field confllled by the characteristic AAn and the straight walls of the nozzle is called a triangle of definiteness. The shape of this flow is preserved if we change the wall behind point An so that the radial flow on the Mach line gradually transforms into a plane parallel flow at the exit.
Ch, 5, Method of Characteristics
\'"ith a "iew to
thi~
'233
ronciilioll, we can
of the nr!'t family eml'l'ging from point ~traight lirH', If we now determine the the ehMueteristie!' AA" /Iud ADm' we lillc~. I t is exactly the .streamline pa.<;:-ing ('idl'~ with the .shaped ('ontol1r of titl' To rietN'minc tIlt'> wloeit\' field, \\.(' dimensional tlow being (,(;ll:"idl'r('ri the rhara('1l'ristic.'< of theftr.!'t family elll(,l'ging from point,;.: /t" -2 ... 1". l' ilS \\'('11 as the rhnraeteristir .. iD m art' slraight lillc.s Wig, :;,5,:~), Till' inclination of ead( charartN'islic 10 a radial line is II(>terlllin('d by thcenrri'5pOndIng ~!ar1t anglc jl( _. sin- J (J'Il!I). ~12 sin· 1 (t/M 2 ), C!t<'., while the vclocity Iklong a ('haJ'art(lri~ti(' is dl'terminod hy ils l'I'I(','allt ,'ahlent L1tl' initial pOints AI' A z, . Let liS ron."ill(>1' Ihe streamliu(' (lllll'l'giug from point ~In' Ttu~ initial part of thi:- tiTle ('oillel()c;': with the directioll of th(! "elodty at point An and is II strllight line that is all exten~ioll of contour /JAn lip 10 its iHtC'r,:cction poi)1t V 1 with the dlllrllC'lcri':Uc of the lirst family A"_lfI". Behind point IJ I , the streamline elplnent ('oillddcs with th(' dire(,tion of the \"elorily at point D1 equal to the ,'elodty at point An_I. Drawing t!trough point DI II straight line pllrallel to ray OA',_I lip 10 its iul('J';o'l'rtioll point D2 with the dllu'a('.leristic A,,_zD~, we obtain the next part of t.he stt'eamlill€" Behind poinl D 2 • part DzD m_? of the strenmlinc (poin\. /)",-2 i.~ on lhe ('hnrncll'ri~tic of the Ilrst family A 2D m_2 ) is parallel 10 straight linc 0 .. 1,,_2. The remaining parts of the streamline arc ('ollstrw:ted sitllil<"lrly. Bchilld pointD m, which ison the ('.hurac\.eristir AIJ m , the of thc ,:Ireallllille is panlUl:!l 10 tlte nxis of the uoule. The of the HOlz1ecoinciding with the streamline AnI)", I-I arrd rOllstru('\rri in thc form of a smooth cnrve en~ure,<; II parallel .~l(pcr.~oTlir flow with lhe presf>t Humber M"" lit tire nozzle exit,
6 Airfoil .nd Finite-Spin Wing in an Incompressible Flow
Let HS consider the problems associated with the application of the aerodynamic theory to calculating the flow past an airfoiL A fea~ ture of thb flow is the formation of a two-dimensional disturbed flow over the airfoil. We shall lise simpler equations of aerodynamies than for three-dimen:-;ional flows to investigate it. The flow over an airfoil trc
Let us consider the method of cal(,ulating the steady l10w of an in{'ompressible fluid past a thin slightlr bent airfoil at a small angle -of attack (Fig. G.1.1). The acrodynamic charaderistic.s of the airfoil obtained as a result of t.hese ('al('ulations can be used directly for flight at low subsonic spceds (M", < 0.3·0.4) when the air may he considered as an incompressible fluid. They can also be lIsed as
Ch. 6. Airfoil ~nd Fjnite-Sp~n Wing in Incompressible Flow
235
fI,.6.t.t Thln airfoil in an incompressible flow
the initial data when performing aerodynamic rulculations of airfoils having a given configuration in a !iuhsonic compre~~ible flow. Since the airfoil is thin. and the angle of atlack i~ not large. the velocity of the flow near it differs only slightly from that. of the llndi~turbed now. Such a now is called nearly uniForm. We can write the following condition for tilt' ,'('lority of a neal'ly !lniform flow: V, ,~ V ~,;- u (u~ V,), V, ~" (r~ \'.) (6,1.1) where II- and v are the components of lhe ,'cloeHy or small dis-turhanees. In accordance with this condition. V" ~"Vi T v~ = (V"" -i- u)~ -:- I'~ ~ v:;, .;...2V""u (6.1.2) Let us now determine the pressure in a nearly uniform now. From the 13ernOlllli equalion (3.4.13), in which we assume that the ronstant C 2 equals p"",'pcc " V~/2 anci (l ....: const, we obtain the excess pressure p - p. ~ p. (V~/2 - 1"/2) '~ -p.V.u (6,1.:1) We cleline u in terms of the "elodty potential, u .""' Ocr/iJx: p - Poe = -p",V""i}lrfo.I (G.1.4) The correspoJlding pressure coefficient i!<
Ii ~
(p - P.)·'q. ~ -2uIV. ~ -(211'.) iJqlax
(6,1.5)
By (G. 1.4), the excess pressure on the lJOttom surfact' of the airfoil is Pb - P ~ ~ -(a~bla,) V.p. and on it!; upper surface is p~ -
p ... = -(otpu/Ox) l' ~fI""
236
PI. I. Theory. Aerodynamics of an Airfoil and a Wing
where
,....
(Ph - Pu) dx "-' -V .. poo (OWb!aX - iJ'Pu1ox) dx
.. I
while the lift force for the entire airfoil with. the chord b is Y;I,1c = -VooP..
(a:: - 8izU )dx
Assuming the upper limit b' of the integral to be approximately equal 10 the chord b (because of the smallness of the angle of attack), we find t.hc rorresponding lift r.oeffiC'-ient: Y".1c
C,,;,.iC
=q:r-=--
2
l:,
i1qIh
0%
V~b ~ (/iX---a;-)dx
(6.1.6)
Let liS ('onsider tlle circulation of the velocity over a contour that is a rectangle with the dimensions dx and dy and encloses an element of the airfoil. In arcordallce with Fig. 6.1. t. the circulation is
dr:~ (Voo.i
ac:u ) dx- (v... -,- a::) dx+ (88~1 -
az
r ) dy
where 'PI and Ifr are the velocity polentials on the left and right surfaces. respectively. Let us introduce the concept of the intensity of circulation (of a vortex) determined by the derivative drldx .= y (x). The magnitude of this int.em:ity is
y(x)~ a:; - aa~b +.~( aa~1 -
0:;)
Since for a thin airfoil, the angular coefficient dyldx is small, the product of this coefficient and the difference of the vertical component of the velocities is a second-order infinitesimal and, consequently, y (x) ~ "'P.lax - .'fh/ax
. .i
(6.1.7)
Hence, (6.1.6) can be written in the form
cII".
j
y(x) dx o 'Ve calculate the moment coeHicient in a similar way: ic= V:b
m"t1' ic= -
V~bt
y(x)xdx
(6.1.8)
(6.1.9)
Ch. 6. Airfoil and Finite-Spen Wing in Incompressible Flow
237
In ac.cordancc with formulns (6.1.8) and (6.1.9), Lhe coefficients of the lift force and the moment pro(lnced by the pressllre depend on the distribution of the inlensity of the circillation a!on~ the airfoil. This signifies that the now over an airfoil cau iw(:alculatcd by replae.ing it with a sy~lcm of continlJollsly distrib\lted \"orLiees. According to the Biot-SaYart formula (~.7.12), an element of a distributed vortex with the rirr.ulaLion dl' 'Y (xl dx induces the vertical vclodty at a poillt with the uusrissa ; (Fig. (j.t. t) de ~ dl'!l2n (r -.)1 ~ ~ (x) dx/lh (. -
<)1
The velocity induced at this point b)' all the vortices i:;
,
v=-inJ
.,!~~~Z
(6.1.10)
o
ACcol'ding to the houndary condilion, vl(V .. -:- ()(fldx) ...", dyftk. Taking int,o account that the airfoil ig thin. wo can assnmo that V..,-i-!lCrlrJ.r :::::::: V"". and calculate the derivative dyfd.t from the equation of the mean c"mher line
Y
=-0
Y (x) """ 0.5 Wu -i- Yb)
(6.1-11)
where y" '-- y"V) anel Yb ; Yb(X) nre the equal-ions of the upper and bollom contours of the airfoil. FI'l:o;poctively. Hence_ the intensity of vortex. circulation'Y (.r) i" dC'termineu by the integral eqnation (0.1.12) Let us jnl-roduce instead of x a !le\\" inrlependent variable e determi.ned by the equation r ~ (b/2) (1 cos U) (0.1.13) We fmd the solution of Eq. (6.1.12) in the form of a trigonometric Fourier series:
1'(00) =2lT '"
[AD cot ~+ ~ A",Sin:(nOo)j ., ... t
in which the variable 6 0 in accordance wit.h (6.1.13) is related t.o the coordinat.e x --" ~ by the equation
£=
(bJ2)
(1 - cos 8 0 )
(6.1.15)
We chauge tile- yariables in (l),1.12). fir diRerential-ion of (6.1.13). we find (6.1.16) d.l' =-= (hIZ) sin 0 dO
238
Pt. I. Theory. Aerodynamics of an Airfoil artd a Wing
Substitution~
yield
where ~ (e,) ~ (dy/dx),. After introducing the values of the integral" ill (6.1.17) ealculated in [161. we find
(6.1.18) By integrating both sides of Eq. (6.1.18) from 0 to n, we obtain a relation for the coefficient A 0 of the Fourier series:
A,~
-+ I, ~(9)d9
(6.1.19)
By multiplying (o.1.Ul) in turn by cos B. cos 20, •.• , cos (nEl), we find expressions for the coefficients All A 2 • ••• , An. respectively. Hence, 2
An=-:l
rJ
~(O)eos(nO)dB
(6.1.20)
Let liS consider relation (6.1.8) for the lift coefficient. Going over to the variable e and introducing (13.1.16), we obtain n
eva,le :-.
v~ )
y(B) sin e de
(6.1.21)
o With a view to (6.1.14) Clla.le
= 2Ao
~ cot -} sin B dB + 2 ~
o
,,=1
An
I
sin (ne) sin e de
0
Integration yields (6.1.22)
Hence, the lilt cucfficient depends on the first two coefficient.; of the series. Introdll('.ing into (6.1.22) expression (6.1.19) and formllla (6.1.20) in which n = 1, we have Co,.
,,~
-21 ~ o
(e) (I-cos e)
de
(6.1.23)
Ch. D. Airfoil and Finite-Span Wing in Incompressible Flow
Sl39
Fig.6.U
~Iean
camber line of an airfoil
We change the variables ill (tU.9) for the moment coefficient:
~-~
m:".I.. =
Jy(9)(1-cos8)Rin9d8
(G.1.24}
With a view to expresRion (6.1.14) for y (9). we bave
m~a'
k
= -Ao
j (1 -cos 9,2 d8- ~ An J sin (n8) \1-cos8) sin 8d& II
n=1
0
IntegraUon yields mza_le = -
(;-t;2) (Ao
-T At -
A~,'2)
(6.1.25}
(1;4) cVale
(6.1.26}
With a view to (6.1.22), we have (n;4) (A I - A 2) -
Tn'a.le = -
By determining Al Rnd ...12 from (6.-1.20) and introdllciugtheir values into (6.1.26), we obtain 1
l!
1
mz", '" = -"2 ~ ~(9) (cos8-cos20) cla-Teu",
Ie
(6.1.27)-
Let us consider the body axis coordinates Xl lind Vt in which the equaUon of the mean camber lille is Yt = Yl (XI)' and the sJop~ of a tangent is determined hy the derivative dYl,'dx1. This angle (Fig. 6.1.2) is ~1 = ~ + ct, where p:::::J dy/dx. Hence w(' d('termine th£" angle ~ = ~l - a and introduce its value into (6.1.23) anll (G.t.27): e'a.le = 2n (a; ..!... eo) (G.1.28) mZa' ie'"'"
-2
(ffo-l1o)-fclla. Ie
(6.1.2H}
240
PI. I. Theory. Aerodynemics of an Airfoil and a Wing
where
eo=+!f31(1-COSe)d~; 1l0=-+If3~(1-COS2e)de
(6.1.30)
A glance at (6.1.28) reveals that when ex = -eo, the lift coefficient equals zero. The angle ex = -eo is called tlte angle of zero li.ft. The coe[Jicients eo and J.Lo are evaluated according to the given equation of the mean camber line of the airfoil provided that in the expression for the function f3I (tl) the variable Xl = xi/b is replaced .according to (6.1.13) with the relation Xl = (112) (1- cos 9). We can use the found values of mla and c Yd to approximately determine the coefficient of the centre of pressure cp = xp/b and the relative coordinate of the aerodynamic centre ZAC = xAclb: =
C
p
_~_J..+ c!l~ -
4
(:t/4)p.o-f-Io 1t (a;T~o)
I
x
AC
=
-~=...!.. iJc lIa
4
(6.1.31)
It follows from these relations that the coordinate of the centre -of pressure depends on the angle of attack and the shape of the airfoil. wl~ereas the aerodynamic centre is at a fixed point at a distance of one-fourth of the chord length from the leading edge. For a symmetric airfoil eo = J.Lo = 0 [which follows from formulas (6.1.30) in which f3I = dyIldx l = OJ. therefore the value of Cpt like that of the l"elative coordinate of the aerodynamic centre, is 114. 6.2. Transverse Flow over II Thin Plate
Let us consider the flow of an incompressible fluid over a very simple airfoil in the form of a thin plato arranged at rigllt angles to the direction of the free-stream velocity. We can solve this problem by means of the conformal transformation (or mapping) method. We shall use the results of the solution in real cases to determine the aerodynamic characteristics of finite-span ";vings. Let lLS arrange the plato in the plane of the complex variable o = x + iy along the real axis x (Fig. 6.2.tb). ~ow the free-stream velocity vector V will coincide with the direction of the imaginary axis y. In this case, the flow over such a plate can be obtained by conformal transformation of the flow over a Circular cylinder on the plane S = + iT'[ (Fig. 6.2.1a). If we know the function W = f (~) that is a complex potential for the flow over a circular cylinder and the analytical function S = F (a) of the complex varia hIe a = x -i+ iy (known as the conformal function) that transforms t.he contonr (lf~a ~circle on the plane ~ into a segment of a straight line placed across
s
Ch. 6. Airfoil .nd Finite.Span Wing in Incomprenible Flow
241
(a)
FiS!.6.l.t
Conformal transformation or the flow past a round cylinder (0) into a Dow past a flat plate arranged (b) at right angles to the direction of the free-stream velocity, or (r) along the flow
the now on the plane cr, then the fUllction W = f [F (0')1 is the complex potential for the flow over the plate. Let us sec how we can determine the complex potential for the [low over a circular cylinder. For this purpose, we shall again revert to the meLhod of conrormal transformation, using the known function of Lhe complex potential for the flow over a plate arranged along it. This function has the form W =
+ i1j;
= -iV (x .. - i.1ll
=--0
iVcr
(6.2.1)
We shall show that the conformal fUllclion
*"
(6.2.2)
transforms a segment of a straight line arranged along (he flow on the plane 0" into a circle on thf' plane ~ (Fig. f::j.~.1c.a).lndeed, examination of (6.2.2) reveals that since fOf points satisfying the conditions - 0 ~ y ~ a, x = 0, and a = ~R, ,ve ha\"(~ 0" = iy, then from the solutionoftheqlladraticeqlJation ';;2 _ O"~ _ R2 = 0, we obtain {~iyI2±VR'-Y'14~S+i~
(6.2.3)
Separating the rcal and imaginary parts, we obtain ~c'yI2,
,~±VIl'-Y'14
whence ;2 -i- 11~ R2. Consequently, points on the circle in the plane t correspond to points in the plane (J on the vertical segment. Substituting for (J in (6.2.1) its value from (G.2.2). we obtain the complex potential of the flow over a circular cylinder or radius R in a plane parallel now
242
pt.
I_ Theory. Aerodyn.mic5 of .n Airfoil and a WJng
at the velocity V: (6.2.4)
To obtain the complex potential for the flow over a plate arranged across the flow (Fig. 6.2.1b), let us substitute for S in (6.2.4) the following value obtained from the conformal formula 0" = S + + RZ1s transforming a circle of radius R (plane t) into a segment of a straight line across the flow (plane 0): ~=0/2±V02/4-R2
(6.2.5)
Therefore,
W=-iV
(f±l/ {_R2
Since for points on the plate we have and a = 2R, then
(6.2.6) 0"
= x, where -a ~ x ~ a
W=rp-i-ilP=-iV(+±q/RZ-f-
x/2.1:f~~t
:;:2/4)
whence the potential function is
(6.2.7)
where the plus sign corresponds to the upper surface, and the minus sign to the bottom olle. By evaluating the derivatives iJrpl8x, we can find the velocity on the plate and calculate the pressure. It follows from the data obtained that the pressure on the upper and bottom sides of the plate is the same. Consequently, ill the case being considered of the transverse free streamline flow of an ideal (in viscid) fluid over a plate, drag of the plate is absent. This interesting aerodynamic effect is considered below in Sec. 6.3 using the example of flow over a flat plate arranged at a certain finite angle of attack. A flow characterized by the potential function (6.2.7) is sho,vn in Fig. 6.2.1b. It is a non-circulatoryftow past the plate obtained upon the superposition onto an undisturbed flow with the potential W~~-iVS
(6.2.8)
of the flow from a doublet with the potential Wd = iVRzl1;
where S is determined by the conformal function (6.2.5).
(6.2.9)
ct..
6. Airfoil end Finite-Spen Wing in Incompressible Flow
243
6.3. Thin Plate at an Angle of Attack
Let us calculate the potential function for the disturbed flow of an incompressible fluid over a thin plate at the angle of attack ct using, as in the preceding problem, the method of conformal transformation. We arrange the plate in the plane of the complex variable 0' = X + iy along the real axis x. If we assume that t.he flow being considered is a non-circulatory one, the complex potential of such a flow can be written as the sum of the potentials of the longitudinal WI = V 000' and transverse W z flows at the velocity of the undisturbed flow V = aV"" (Fig. 6.3.1). The total complex potential is
W=W 1 + Wz=V",O'-ictV""
[-%-± V f - R z
-RZ(-%-±V (/41 -Rz)-I]=Voo('f±iaVooVcrZ-4RZ
(6.3.1)
,"Vith account taken of formula (G.2.7), the total velocit.y potcntial on the plate is q:=1'""x±ctV""Va2 -xZ (6.3.2) We use this \'alue of the potential to find the velocity component
V x =V""=FaV",,,xIVaZ-x2
(13.3.3)
The second veloci~y component on the thin plate Vy = O. A close look at (6.3.3) reveals that at the leading (x = -a) and trailing (x = a) edges, the velocity V:t is infinite. Physically, such a flow is impossible. The velocity at one of the edgcs, for example, the trailing one, can be restricted by superposition of a circulatory flow onto the flow being considered. In the pI aIle 1;, the potential of the circulatory flow is determined in accordance with (2.9.22) hy thc expression (6.3.4) w, ~ - (if/2n) In S Let us substitute for S its value from (6.2.5):
Ws=-
J~
In
(f± vf-
RZ)
(6.3.5)
Summation of (6.3.1) and (6.3.5) yields the complex potential of a circulatory-forward flow over an inclined plate: W= WI + W z + W3 = V""cr± iaV""Vcr 2 _ 4R2 (6.3.6)
244
Pt. I. Theory_ Aerodynamics of an Airfoil and a Wing
Flg.6.3.t
Flo\v over a flat plate at an angle or attack
We determine the circulation r on the basis of the ZhukovskyCbaplygin hypothesis, according to which the velocity at the trailing edge of the plate is finite. This value of the velocity can be obtained, as we already know from the theory of conformal transformation, in the form of the aerivative dWida of the complex potential W for a cylinder. This potential, in turn, makes it possible t.o nnd the complex velocity at the relevant point on the cylinder as the derivative dWidS;, which according to the rules of differentiation of a complex function is (6.3.7)
where Wand Ware the complex potentials for the cylinder and plate, respectively. According to the Zhukovsky-Chaplygin hypothesis, the quantity dWlda is limited in magnitude. Since the derivative dalds calculated by the formula daldS; = 1 _
R2/~'J.
equals zero on the cylinder at the point S; = R corresponding to a point on the trailing edge of the plate, the derivative dWidS; = O. The complex potential for a cylinder is
W=~+W;+W; The potential Wr characterizes the flow in an axial direction correspondin~ to the flow over the plate in the same direction at the velocity V .... It is obtained by substituting the value a = S; + + R2/S for (J in the formula WI = V""a:
W;
~ V~ (,
+ R'I,)
Ch. 6. Airfoil and Finite-Span Wing in Inccompr.ssible Flow
24&
The complex potentials Wt and W; are determined by Eqs. (6.2.4) and (6.3.4), respectively. We 511al1 substitute a.l' ~ for V in the rlrst of them and lind the total complex potential: V ~ (, + H'/,) - letV. (6 - H',,) - (ir,2n) In 6 (6,3.8)
w:.
We calculate the derivative:
"·r
dWid6 ~ V. (I - H'i 6') - let V • (I R', 6') - Ir, (2~ 6) (6.3.8') For the coordinate ~ = R corresponding to a point on the trailing edge of the plate, the derivative dWJd~ is zero. i.e. 2ia V "" - iri(2nR)
=0
Hence
r or. since 2R
= a,
~
-4naHV.
(6.3,9)
r=
-2naaV ....
«(;,3.9')
After inserting this result into (tt3.6) and differentiating with respect to a. we fmd the complex velocity:
~: =v":.;-iVy= il..,± Y~~~:R2
-r 2J\i;~V'" • a ± y~
(1- ±
y' a~~41l~ )
Simple transformations and the substitution 2R
=a
yield
dWldo~ Vz-tVv~ 1'.(1 'F iet.V (o-a)/(o+a)
16.3.10)
On the surface of the plate (0 = x) dWldo~Vz-IV,~V-t(I'FI"V(x
a)/(.: a)) (6.3.11)
Since I x I ~ a. we have dWldo~ Vz-iV,~ V.(I±etV(a
.i(a+.))
(6.3.11')
whence it follows that the total velocity on the plate is
v~
V. ~ V. (I ± "V(a
.)I(a -,. x))
(6.3.12)
The plus sign relates to the upper surface, and the minus sign to the bottom one. The velocity on the trailing edge (or = a = 2R) is V .... and on the leading edge (x = -a) it is found to be infinite. In real conditions, the tbickness of the leading edge is not zero; particularly, the nose may have a finite. although small. radius of curvature. Therefore the velocit.ies on such an edge have high. but tinite values.
~46
PI. I. Theory. Aerodynamics of an Airfoil and a Wing
y"",
Flg.6.1.1
Contour in the flow of an in-
""--cc'---'.L----'_ _ _-'''--
compressible fluid
To determine the force acting on the plate, we shall use the general expression for the principal vector of the hydrodynamic pressure forces applied to a stationary cylindrical body of an arbitrary shape in the steady flow of an incompressible fluid. By analogy with the complex: velocity, let us introdtlce the concept of the complex force Ra = X - iY, determining this force as the mirror reflection of the principal vector R;. of the pressure forces with respect to the renl axis. The vector R/J, being considered is detl3rmined by formulas (1.3.2) and (1.3.3) in which the friction coefficient el.:e is taken equal to zl3ro:
Ra=X -iY =
-~ p [cos ( 0 ) - i cos (.;.-u)] ds c
=
-~ p(sin9+icos9)ds= - i ~pe-i(jds
c
c
(6.3.13)
where n is tne direction of an outward normal to contour C of the body in the flow and 9 is the angle between the olement ds of the contour and tne x-axis (Fig. 6.3.2). Since d~ = dx + i dy = ds (cos 9 + i sin 9) = eie ds do ~ ax - i dy ~ as (cos 8 _ i sin 8) ~ .-"ds (6.3.14) I
I
and the pressure is determined by the Bernoulli equation p ~ p_ pV'_12 - pV'12 ~ C, -- pV'12 then
+
~~-~i~+f§V'~~ftV'~
(6.3.15)
c c c Taking into account that by (6.3.14) we have du = e- 2ie dCJ, and also that in accordance with the condition of flow without separa-
Ch. 6. Airfoil and Finite-Span Wing in Incompressible Flow
247
t.ion the complex velocity at the point of the contour being considered is
V=
V;I' - iV, = V cos
e-
tV sin
a=
Ve- iO
(6.3.16)
we find Ra=X-iY--t- ~ Vida c
(6.3.17)
Since for a potential now, the complex velocity is have
Ra=X-iY
V = dWlda. we
={£- ~ (~~)2 do
(6.3.18)
C
Expression (6.3.18) is called the Zhukovsky-Chaplygin formula. Integration in (6.3.18) can be performed over another contOllr enveloping the given contour C of the body in the flow, for example, over a circle K whose complex velocity in the plane 0" is written as follows: (6.3.19) dWldcr = V ~ - tf/(2ncr) + Alcr' where V cD = V;I''''' - tV,,,,,= V ... e- i9"", and A is a coe£licient determined with the aid of an equation similar to (6.3.8). The square of the complex velocity is
( ~~ )2 i'!.- I~~"" + "C"
2AV"'-:~'2/(4;1~)
+ ...
Introducing this value into (0.3.18), we find
Ra=X-iY=*[V!'~da_lr:", ~~ K
•
K
+(2AV_-L~,)t ~~ + ... ] Here the first and third integrals equal zero. The integral ~dO"/O" is evaluated with a view to the formula 6.3.2) and equals
~
0"
= X + iy
K
= re;qI
(Fig.
'!:' =lnO" IK=ln(rei~) IK=2sti
K
Hence,
Ro=X-iY=ipjl.. r
or
X-iY=ipV",re- to.. (6.3.20)
where V .. is the magnitude of the frec-stream velocity.
248
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
If its direction coincides with the horizontal axis (a .... = n), we have Ra = X - iY = -ipV ... (6.3.21)
r
whence it follows that (6.3.22) I R. I = Y = Y. = pv.r Expression (6.3.22) is known as the Zhukovsky formula. It follows from (6.3.21) that the force is a vector perpendicular to the free-stream velocity vector, and, therefore, is a lift force. Its direction coincides with that of the vector obtained by turning the velocity vector V ... through the angle nl2 against the circulation. A glance at (6.3.21) reveals that X = X a = O. Hence, when an in viscid incompressible fluid flows over a surface without separation, the resistance associated with the distribution of the pressure is zero. This aerodynamic effect is known as the Euler-D'Alcmbert paradox. This name corresponds to the fact that actually resistance is present because there always occurs a relevant redistribution of the pressure caused by separation of the flow and the action of skin friction (viscosity). Let us use formula (6.3.22) to evaluate the lift force of a flat plate. Taking into account the value of the circulation (6.3.9'), we obtain (Fig. 6.3.3) (6.3.23) Y .. = 2naapV!. Let us determine the normal Y and longitudinal X components of the force Y II: (6.3.24) Y = Y. cos a ~ Ya= 2:laapV:O (6.3.25) X = - Y. sin a ~ -Y.a = -2na2apV:c. The arising of the force X acting along the plate opposite to the flow seems to be paradoxical because all the elementary forces produced by the pressure are directed along a normal to the surface.
Ch. 6. Airfoil lind Finlte-Splln Wing in Incompressible Flow
'249
This component is called the suction force. The plty~ic(ll nature of its appearance consists in the following. A~snnlt' Lhllt the leading edge is slightly rounded. :\'ow the velociLies ne
T
.~
-X "" npc'
(0.3.27)
where (6.3.28)· We shall write the quanl.ily c!~-
c~
in the form of the limit
lim (r~2 {.r-Xl.e)j
(C>.3.28')
X-"'J.t
where ;fl.e is the abscissa of the lending edge (Xt.e - -a) . .( is the rUlining abscissa of points of the plate, afl{1 v.~ is the longitudinal component of the disturbed velocity on the upper surface of a wing. Formula (6.3,28') can be shown to be correct. For this purpose, we shall insert expression (6.3.2G) into (6.3.2t;'):
c!= }~~la [v!,a2
(
:~:
)
(:r·t
a)] =
2aa~r;,
Dy introducing this value into (6.3.27), we obtain formula (0.3.25). The conclusions on an incompressible Circulatory-forward flow over an airfoil in the form of a thill plflte (Ire used ill studying the aerodynamic characteristics of airfoils encountered in practice, and also of finite-span wings in flows of an incompressible fluid or of a compressible gas.
6.... Finite-Span Wing in an Incompressible Flow Up to now, we considered an airfoil in an incompressible flow. We· call presum~ that such an airfoil relates to an elementary part of a lifting SUI face belonging to an infinite-span wing in a plane parallel
ftO
pt. I. Theory. Aerodynamics of en Airfoil end eWing
'f11.U.t
:System of equivalent vortices for a rectangular wing
Dow. Hence. the theory of this 110w is also a foundation of the aerodynamics of an infinite-span wing. Er formula (6.3.22), the lift force of a unit-span wing part is Y!l) = p""v... r (Fig. GAia). Consequently, there is a circulation flow around the airfoil with the velocity circulation r. If the circulation is clockwise. the velocities on the upper surface of the airfoil are higher (a circulation flow having the same direction as the on·coming one is superposed on it), while on the bottom surface they are lower (the circulation flow does not coincide with the direction -of the oncoming flow). Therefore, in accordance with the Bernoulli .equation. the pressure from above is lower than from below, and the lift force is directed upward as shown in Fig. O.4.ib. Since by (2.7.8) and (2.7.8'), t·he circulation equals the vorticity (vortex strength) ")(, the part of the wing can be replaced with an 'equh'alent vortex of the indicated strength passing along its span. N. ZhukovskY used the term bound to designate this vortex. Hence, in the hydrodynamic sense, an inrmite-span wing is equivalent to a l)ound vortex. Let us now consider an approximate scheme oj the flow past a finite.$pan wing with a rectangular planform. As established by S. Chaplygin. a bound vortex near the side edges turns and is cast off the wing in the form of a pair of vortex cores approximately coinciding with the direction of the free-stream velocity. The distance e {Fig. 6.4.ic) from a vortex core to the relevant side edge depends on the geometry of the wing. Consequently, the hydrodynamic effect -of a finite-span wing can be obtained by replacing it with a bound vortex and a pair of free horseshoe vortices. This wing pattern is -called Cbaplygin's horseshoe one. A vortex system equivalent to a finite-span wing induces addHion.al velocities in the flow and thus causes downwash. which is a fea-
Ch. 6. Airfoil and Finite-Span Wing in Incompressible Flow
2fi1
f.lotl!!d·lI.p Z v.'rile!!.>
fI,.6.U Vortex ,neet and rolled-up vortices behlDd a WiDg
ture of now past a finite-span wing. The calculations of the induced velocities and tho down wash angle caused by the free \'ortices are based 011 the following theorems rormulated by II. IIl'lmllOltz: (1) the strength along a vortex docs not change in magnitude, and a~ n restJlt, a vortex cannot end somewhere ill a fluid. It must either be closed or reach the houndary of the nuid; (2) the slrength of a vortex does not depend Oil t he time; (:'\) a vortex is not disrupted in an ideal fiuid, In the considered scheme of a rectangular wing. the circulation along the span was presumed to be constant in accordance wilh the assumption that the lift forct! of each elementary part of the wing is identical. Actnally, the lift force along tile spall of a \\'ing having such II rectangular phl.nform varies. This change is not great ill the middle part of the wing and is more 1I0ticeahl~ at the side edges. For n wing with an arbitn'lrY planform, the change in the circulation is cle
2152
Pt. I. Theory. Aerodynamics of an AirFoil and" Wing
Flg.6.U
r~~Wapdp~~;a:~~hof tinad:::l dar~: called the setting angle). The appearance of downwash of the flow behind the wing at the angle € leads to the fact that the flow over the wing in the section being considered is characterized by values of the velocity V~ and the angle of attack at differing from the corresponding values of V"" and a that determine the flow over an infinite-span wing. The true angle of attack Ctt of a section of a finitespan wing is smaller than the setting angle by the magnitude of the downwash angle e (Fig. 6.4.3), i.e. Ctt = a-E. The downwash angle E = -wiV varies along the span of the wing, increasing toward its tips. For convenience, the concept of the span-averaged down wash 00
1/2
angle is introduced, determined by the formula em Accordingly, the true angle of attack of the wing is tXt=~-em
={ J e dz. -1/2
(6.4.1)
The existence of a downwash angle leads to a change in the forces acting on a body in the flow. If flow downwash is absent, the vector of the aerodynamic force by (6.3.21) is normal to the direction of the velocity of the undisturbed ftow V"". If downwash is present, the vector of the resultant aerodynamic force is oriented along a normal to the direction of the true velocity V~ (Fig. 6.4.3). Such deviation of the resultant force through the angle till causes the component XI to appear in the direction of the undisturbed flow. This additional force X i appearing as a result of flow downwash is called the induced r('sistance or induced drag. From the physical viewpoint, an induced drag is due to losses of a portion of the kinetic energy of a moving wing spent on the formation of vortices cast off its trailing edge. The magnitude of this drag is determined in accordance with Fig. 6.4.3 from the expression XI = YaEm
(6.4.2)
The lift force Y a , owing to the smallness of the downwash, is determined in the same way as for an infinite-spall wing. 1f we divide XI by the quantity (PooV!.l2) Sw, we obtain the induced drag c0efficient
Ch. 6. Airloil .nd Finite_Span Wing in Incompressible Flow
253
'190 6.'" Replacement uf a finite-span wing with a loaded linc: J-dlstribuUOll ot c:in:ulauOll; 1'-loadtd line; 3-d1s1ribuUOll or inducrd velOCities wash angles
I.If
down-
We sllaU determine thls coefficicut on the basis or the theory of a "loaded line". According to this theory, a finite-span wing is replaced with a single hound vortex (a loaded line). Tilt' circulation r (z) for tlte loaded line is the same as for the corresponding sections of the wing itself (Fig. 6.4.4). Upon such a replacement, a plane vortex sheet begins directly on the loaded line and has a strength dr (z)/th that varies along the span. The flow down wash in a gil'en section is determined for a semi-infinite vortex core having the strength (dr (z)fthJ th_ Accordingly, the tot.al downwash angle for a section, as can be seen from Fig. 6.4.4. is e= -
v:
= -
,,2
J dizf). /:"'1.
4!t~...
(6.4.4)
-1/2
Here the improper integral must bc considered in the sense of its principal value. By (6.4.4), the mean dowilwash angle over a span is
'm=--', \'2 [I?J dr(.t·).~." ]dz 4:tV""l. d% % _%
-1/1
(6.4.5)
-1/2
The lift force coetticient cY " can be determined according t.o the known law of circulation distribution along the span. With such a determination, we caft proceed from the hypothesis of plane sections according to which the flow ovcr a wing element being considered is
254
Pt. I. Theory. Aerodynllmic5 of an Airfoil and a Wing
the same as over the corresponding airfoil belonging to a cylindrical inlinHe-span wing. Calculations on the basis of this hypothesis give a satisfactory accuracy for wings with a low sweep and an aspect ratio of Aw> 3-4. According to the Zhukovsky formula, the lift force of an airfoil is dY, _ p_V_f (,) d, (6.4.6) its total value for a wing is '/2
Y,-p_V_
j
f(,)d,
(6.4.6')
-[/2
and the corresponding aerodynamic coefficient is Cila =
q:Sw
In = V...2S W
)
r
(z) dz
(6.4.6")
-1/2
To find the law of circulation distribution r (z), let us consider an airfoil in an arbitrary section of a wing for which the lift force can be wrilten in the form dY a = Cil (z) q ... b(z) dz (6.4.7) where b(z) is the chord of the airfoil in the section being considered. With a view to the Zhukovsky formula (6.4.6), we find a coupling equation determining the relation for the velocity circulation in a given section: (6.4.8) f(,) - 0.5", (,) b(,) V_ According to the hypothesis of plane sections, the lift coefficient clIa(z) of a section being considered is the same as for the corresponding infinite-span cylindrical wing. Its magnitude can be determined with account taken of the downwash angle by the formula cila (z) = c~a (z) (0: -
e)
(6.4.9)
where the derivative c~a(z) = (}clIa(z)/oo: is determined for an infinite-span wing and a range of angles of attack corresponding to the linear section of the curve c1Io (0:). After inserting into (6.4.8) the values from (6.4.4) and (6.4.9), we obtain [(z)=(1I2)
~a(z)b(z)V...
(a+
41t~...
1/2
)
d~~;') ./~1.) (6.4.10)
-1/2
This equation is called the fundamental integro-differential equation of a finite-span wing. It allows one to find the law of circulation distribution r (z) for a given wing shape according to the known conditions of its flight. The angle of attack 0: in (6.4.10) may be fixed
Ch. 6, Airfoil .nd Finite-Span Wing in Incompressible Flow
2&5,
(i.e. the same for all sections) or may vary along the span if geometric warp of the wing is present. One of the most favoured ways of solving Eq. (6.4.10) is based on expanding the required function r (z) into a trigonometric seris (the Glauert-Trefftz method)
r (z) = 2lV ""=~t An sin (nO)
(o.4.11}
where the new variable e is related to the variable z by the expression z = - (U2) cos e (see Io'ig. 6.S.1), aud An are cOllstant coefficients determined with the aid of Eq. (6.4.10). Since the series (6.4.11) is a rapidly descending one, a small number m of terms is usually sufficient. To det.ermine the unknowncoefficients Am, we compile m algebraic equations (according to the Ilumber of selected sections). Each of these equations is obtained by introducing into (6.4.10) the value of the circulation r (z) = = 2lV"" :LAn '/1,=1
SiD
(118) for the corresponding sectiOll.
For conventional wings that are symmetric about the central (root} chord, the distribution of the circulation over the span is also symmetric, i.e. the equality r (e) = r (:r - 6) holds (see Fig. 6.5.1). Accordingly, the terms of the series with even indices equal zero. and the circulation can be written in the approximate form r = 2lV "" (AI sin 6
+ A, sin (3e) + ... + Am sin (me))
Consequently, the number of algebraic equations needed to determine the coefficients Am diminishes. The procedure followed to determine these coefficients is set out in detail in 1161. Using the found law of distribution of the circulalion in the form of a series, we can determine the flow dov,,-nwash and the corresponding aerodynamic coeJJicients (see 1131). By goillg over in (6.4.6") from the variable z to the variable 0 in accordance with the expression dz = (1I2) sin 8 ae, introducing the formula for the circulation in the form of a series, and taking into account that l'l./S ..... = Aw, we find a relation for the lift coefficient: e'a = 1tA".A 1 (f1.4.12) Using (6.4.5), we determine the me-an down wash angle: 8 m = (c"lnl.w) (I + .) (6.4.13) where ratio:
't"
is a coefficient taking into account the inrJuence of the aspect (6.4.14)
2~6
Pt. I. Theory. Aerodynamics 01 an Airfoil and a Wing
JI'rom (6.4.3), we obtain the relevant induced drag loefiicient. Its value can be determined more precisely, however, by going over from the mean downwash angle to its local value in accordance with the expression dX, = e dY a • By introducing into this expression instead of t and dY. their values from (6.4.4) and (6.4.6), integratiog, and determining the induced drag coefficient. we obtain -t
ex.
I
?,I (z) ['1,2j ~'"'i""=Z dr(,') d..!:" Jdz
= 21I:''-:''S"" J
-1/2
(6.4.15)
-l!2
Introducing the value of r(z) and substituting have
~.
for l'l/Sw, we (6.4.16)
where the coellicient 6 taking into account t.he influence of the aspect ratio on the drag depending on the lift is
6=.f .-, nA~.IA:
(6.4.17)
The coefficient.s T and 6 for wings of various planforms can be determined according to the dat.a given iu (13, 161. The results of the general theory of a loaded line obtained can be seen to be charact.erized by a comparative simplicity of the aerodynamic relations, proYide a clear notion of the physical phenomena attending flow past finite-span wings, and allow one t.o reveal the mechanism of formation or the lift force and induced drag. The applicat.ion of this theory, however. is limit.ed to wings ' .... ith a sufficiently small sweep and a relatively large aspect ratio. In modern aerodynamics, more accurate and more general solut.ions are worked out. They arc described in special literature. At the same time. the development of ways of evaluating the aero'dynamic properties of wings by constructing approximat.e models ·of the flow over fmite-span wings is of practical signilicance. Let us consider one of them based on the representation of the aerodynamic scheme of a wing in the form of a bound and a pair of free vortex .cores. This representation is based on experimental data according to which a vortex sheet is not stable and at a comparatively short distance from the wing rolls up into two parallel vortex cores (see Fig. 6.4.2). The basic element of this problem is the fmding of the distance lo between the free (rolled up) vortices. We proceed here from the fact that for a wing with a span of I, the vortex pattern of the wing may be replaced with a single horseshoe vortex with the constant circulation ro corresponding to the root section. We also assume that the bound vortex (the loaded line) passes through the aerodynamic cen-
Ch. 6. Airfoil and Finite-Span Wing in Incompressible Flow
~7
tre of the wing with the coordinate XFa' The magnitude of this circulation can be determined by a coupling equation according to which ro = O·5c v•oboV 00
(6.4.18)
where ev• o and ho are the lift coefficient and t.he section chord, respectively. A similar expression can be compiled for the mean circl1ialion over the span. the sarue as in the section with t.he chol'd bIn: (GA. in)
where eVa is the lift coefficient of the wing. From Eqs. (6.4,18) and (6.4.19), we find the relation between the circulations: (6 ... 20) As stipulated, the adopted vortex systems rill and r 0 correspond to the same lift force (Y II = pooroorml = p ... T',.Tolo). hence rml = rolo. Therefore. by Eq. (6.4.20). we have
(6.4.21)
now. according to the known arrangement of the horsesho(' \'ortex system, we cau use formula (2.7.13) to determine the do\vllwash angle at each point behind a wing, also taking into l\cconnt the induction of the bound \·orlex. To find the induced drag coefficient by formula (6.4.3). we must find the mean down wash IlIlgle
£..,
'"
~ (ill) ~ e dz all the IOl\ded linc, -~"2
t.aking account onll' of 1\ pltir of free vortices with the aid of formu· la (2.7.13). According to raiculatiolls (sel' (131), this angle is (6 t,,22)
wbere ro is determined by (6.4.18). Having in view that lIb o = A.w (bmlb o). we finally obtain
=
(6.4.23)
We can use this downwash angle to estimate the \'aille of the indu('ed drag cocffident with the aid of (6.4.3). In a particular case for an infmite-span wing (Aw- 00). the downwash angle is absent and, consequently, the induced drag vanishes. 11-onu
2~8
Pt. I. Theory. Aerodyn"mics of an Airfoil and
/I
Wing
6.5. Wing with Optimal Planform COnYW,lon of CoefIkIeIdI and (:c. 1 frOID One Wing A,pen Ratio
cv
to Another
A finite-span lifting plano having the minimum induced drag is called a wing with the optimal planform. A glance at formula (6.4.16) reveals that "t the given values of c Va and Aw , the vallie of C:/:.! wi1l be minimum when the coefficicnt 6 = 0, or
i; nA:IA: = O. It follows -,
from this equality that all the weflidcnts of the expansion of the circ.111atioll r into a series (6.4.11), except for AI' equal zero (As = = A5 = . . . .- 0). Hence, the circulatioll in an arbitrary section is
r (e)
~
21V ~A, sin
e
(6.5.1)
For the celltral ::;ection (0 = 1[.'2), the dr{:lIlation is maximum and equals 1'0 = 2lV ",AI' lIenee, r (0) = rosin S. Inserting into this formnla sill 0 Vt-{2zll)2, we obtain 1" (,)/f:
+
z'I(0.51)'
=1
(6.5.2)
This equation determines the ~lliptic law of distrib\ltion of the circulation oyer the span of a wing IHlving the optimal planform (Fig. 6.5.1). For such a wing, the illdnced rlrag coefficient c",,! = "..., c;/{nAw), while the mean now downwasl! angle Em = cy/(nA w ); by analysing (6..1.4), we can convince o\lrselves that the quantity 10 m C'xactly equals the downwash angle in an arbitrary section. In other words. the downwash angle does not change along the span in a wing with the minimum induced drag. In accordance with the coupling equ
Let liS consider a wing at tlte angle of attack" having section profiles that
Ch. 6. Airfoil and Finite~Span Wing in Incompressible Flow
~9
'f(;;;- --~-:)!'\\ ~\'i:it~!l
circulation
distribulion of the and a geometric of the constants
~n!~jr8tation
::;:'
r
"a
i.lf?
are the same: at = CJ. - e, and, therefore, the lift eoefficienl~ of the profiles are the same. i.e. the ratio c!/a/cYaO = 1. Hence, the local cliOI'd of the wing is (letermined by the relation
b (,) ~ b. V 1- (,10.51)' (6.3.4) III accordance with the result obtained, a wing with the minimum induced drag has all elliplical planform. In the case heing considered, a cons\.ant coefficient c II c~. at is ensured a~ a result of the constancy along the span ~f the "quantity at an!l the derivatiq" iuentieal for the same proliles. For a wing with an elliptical planform. the same effert cau he achieved by varying the parameter~ c~a and at, i.e. by selecting the corresponding prol'tles ill comhination with gcom(,tric warp (UpOIl which the seetions arc set at different geometric angles of attack). I t must be noted that j(]~t. .<;;urh 11 method of varying lhe geometric /lnd aerodYllamic parameters is used to achieve an elliptical (or dose to it) distribution of the circulation on modem wing!>. IIere three factors c.hange in the corresponding way simultaneously. These factors, namely, c~a (z), at (z), and b (z) characteri?e the circulation 1'(:) --: O.5c~.atbF ox- thal is di~tl"ihllted according to the elliptical law (u.5.2) .., In the simplesl case of a wing wilh a rectangular planform. b '-'" = const, t.his law of circ.ll1atioll di~trih\ltion il' cnsured by only varying tlte va\lIes of c~ and at (by selecting the profiles and warping, rl'sp(!clivcly). It mllst"be horne i~ mind that for wings of a nOll·cllip· tical planform, including rectallgular ones, an elliptk.al distribution of the eirculatiofl can be achieved only at a definite angle of attack, and the distribution will be different if the value of the angle is changed. It is intercsting to appraise by how much wings with a difrcrent planform differ from their elliptical counterparts in their aerodynamic properties. Table 6.5.1 gives the results of e\'a!uating the coefti-
c;a
260
Pt. I. Theory. A.rodynemlcs of
.In
Airfoil end eWing
Table 6.S.1 Piaarorm ot willi
Elliptical Trapezoidal ('1w = 2-3) Rectangular (A. w = 5-8) RectaDgular with rounded tips
o o
o o
o
0.179 0.14i
0,053
eients 15 and T, called corrections for wings of a non-elliptical planform. These coefficients depend mainly on the planform of a wing and its aspect ratio. A glance at the table reveals that trapezoidal wings do not virtually differ from elliptical ones. A slight deviation in the values of 15 and 't is observed in rectangular wings with straight tips. Rounding of those tips leads to complete analogy with an elliptical wing as regards the value of 15 and to a smaller difference in the coefficients 'to The relations obtained for determining the aerodynamic coefficients of fmite-span wings make it possible to solve an important problem associated with the fmding of these coefficients when going over from one wing aspect ratio AWl to another one Aw2' This solution appreciably facilitates aerodynamic calculations because it makes it possible to use the data obtained when testing serial models of wings with the adopted (standard) value of the aspect ratio in wind tunnels. Let us aSS\lme that for a wing with the aspect ratio AWl we know the coefficients c lla and cx.1 and we have to convert these coefficients to another aspect ratio A 912 of a wing having the same set of sections, but differing in its planform. If both of these wings have the same lift coefficient c Ua ' the mean downwash angles behind them are gmt
= (c"lrrJ··W1) (1
+ 't1)'
em2 = (c,/nlw2) (1
+ '(2)
Identical values of cr. for wings are determined hy identical true angles of attack, i.e. au = au, where au = al - eDll' au :-= a2 - em2 Consequently, for the setting angle of the second wing, we obtain the expression
Ch. 6. Airfoil and Fi"ile-SplI" Wing in Incompressible Flow
Flg. Mol COD version
261
of aerodynamic coef-
ficients of II. wing from aspect ratio to another
one
wash angle (Ew2 > "'WI) must haye an increased setting nngle ('Z:z > al) to obtain the same coefficjent c,Ia The induced drag coefficients for two wings wilh the aspect ratios AWl and A.w2 and with identical coefficients cYa are determined by the following formulas, respectively:
>
cx .11 = (c,/",-AwJ (1
+ 61),
Cx.12
= (Cy/J1f. ~d (1
+ 62)
Accordingly. for a wing with the aspect ratio AwZ. the indllced drag coefficient is C;I;,U.
=
Cx.1l -
(c,/
+ 61)/Awl
-
(1
+ 6z)/A.w 1 2
(6.5.6)
This formula is used to convert the value of c.", 1 for a wing with the aspect ratio AWl to its value for the aspect ratio Avo'2' If this new ratio is smaller than the given one C)"w2 < ~'''l)' a large dowllwash appears (em:z> eml)' and, consequently, the mduced drag Coofflclent grows (C x,12 > cx.Ii)' Figure 6.5.2 shows graphically how the coellicients c,III ant! Cx,1 are converted from the aspect ratio AWl to Aw2 in accordance with formulas (6.5.5) and (6.5.6). First the curves c'a "" II (al) and CUll = = Cfl (cx,n) are plotted for a wing with the given aspect ratio AWl' Next. setting ~ number of values of c,Ia' we lise the plot to determine the correspondmg values of a 1 and c;.;.Il' and then calculate l.he angles of attack a2 and the coefficients c.~ 12 for the second WHlg by formulas (6.5.5) and (6.5.6). The corresponding points are laid off, and the curves C,I. = h. (0:.2) arId c,In -= (f2 (c x.l2) are plotted, For wings with smaller aspect ratios ().w:z < AWl)' these l'Uf\'eS will be to the right of those of tile ratio Awl because with the same true angle of attack, tIle incrcage in the downwash angle €w2 [Ol' the second wing with a smaller A,,2 is compensated by the growth in the setting angle of attack (a2 > a 1); ill turn, all illcreased coefiici('nt C.~.12 corresponds to tho greater downwash angle 1:'''''2, and this i:5 exactly what is shown in Fig. 6.5.2. The plots in Fig. 6.5.2 can be llsed to convert clla and Cx .! to a wing with an infini tely large aspect ratio (airfoil). For this pt1rpose,
262
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
the second terms in the brackets in formulas (6.5.5) and (6.5.6) must be taken equal to zero. The converted curves Clla ~-..: 12 (a2) and clla = = 1"f2 (C~.I2) will occupy a position at the extreme left of the corresponding curves clI , = h (al) and cu, = 'PI (c~.Il). Mean Aerodynamic ChoM. When performing aerodynamic calculations of finite-span wings. the mean aerodynamic clIord is selected as the characteristic geometric dimension of the span. Such a chord belongs to a conditional wing having a rectangular planform for which the planform area. the aerodynamic force, and the pitching moment are the same as for a wing of the given planform. The mean aerodynamic chord allows us to compare the moment characteristics of various wings with a varying chord along their span. One of such characteristics is the pitching moment coefticient determined as mlA = Mz/(bAqSw). where bA is the mean aerodynamic chord. The quantity mzA is sufficiently stable when the planform of a wing and its dimensions change. The value of the chord h. and those of the coordinates of its leading point XA. YA (relative to coordinate axes passing through the apex of the given wing), are usually determined approximately assuming equality of the aerodynamic coefficients (of the moment, drag, and lift) of a wing as a whole and its individual section (profile). Accordingly. we have
bA=S~
'/'
1/'
) b2 (z)dz,
o
1
z·--i;-I%b(Z)d'l
(6.5.7)
'/2
YA-
L Jo yb('Jd,
J
where x, yare the longitudinal and vertical coordinates of the leading point of a section with the varying chord b (z). Evidently, knowing and the equation of the leading edge of a given wing, we can determine the lateral coordinate ZA of the mean aerodynamic chord. Formulas (6.5.7) make it possible to calculate the magnitude of the chord bA, by numerical or graphical integration and determine its position. The middle of bA coincides with the centre of gravity of the wing area. For a broad range of wings having a trapezoidal planform, we can find analytical relations for bAr xA. NA' and ZA' For such wings
XI.
b(.)-bo[I-(~w-IJ'/~wJ } Sw-b,l(~w+I)/(2'1w)
z=ztan"o;
y=ztan1p
(6.5.8)
Ch. 6. Airloil and Finite-Span Wing in Incompressible Flow
263
where f). = bJb A is the taper ratio of the wing, XI' is the sweep angle of the leading edge. ~: is the dihedral angle. and i""" 2-;11. Introduction of (6.5.8) inlo «(i.5.7) yields
bA =4Sw [1-~"/(tl"+ 1)'[/(31) } ZA,= 1(f)w+ 2) tan xo/rO (11,,·+ 1)1 YA=I(~w" 2)1.n~·/[6(~ .. -< 1)[
(U.5.U)
'fhe lateral coordinate of the (.hord bit, is found from the formula ZA ..." XIt, !tan Xo_ [f 8 wing tla:o; no dihedral, the coordinate YA, is evidently zero.
An Airfoil in a Compressible Flow
7
7.1. Subsonic Flow over
iI
Thin Airfoil
LInearization of the Equation lor the Velocity Potential
The calculation of a subsonic flow over an airfoil i.e; associated with solution of the equation for the velocity potential of a plane two~ dimensional flow that is obtained from (5.1.8) provided that e = 0 and has the form (Vi-a 2)
;;~
+2V:rVII a!2:y
+(V~-a2) :~ s::::0.
(7.1.1)
This partial differential equation of the second order is non-linear in the unknowll function
a'
~
a?,
+ I(k -
1)/21 (V? -
V')
(7.1.2)
Substituting for 1'Z its value from (6.1.2), we obtain a Z = a!" -
(k -
1) 1'""u
(7.1.2')
Inserting into (7.1.1) the value of a2 from (7.1.2'), aud also V" = = 1'.., -i- u, Vv = v, Vi=J12"" 21',.,u, V; ~ v 2 , and taking into account that the total velocity potential of a linearized flow can be writlen ill the form q> = ((I"" cp', where {foo is tile velocity potential of the oncoming flow, while the additional potential according to
+
+
eh. 7. An Airfoil In e
Compre~~ible Flow
26a
comlition (6.1.1) is
[V~-a!.+(k+1)V",ul ~:~' +2(V",+u)v ~~:
+ ru2 -a'!.+ (k-1) V",u]
~~' =0
(7.1.3)
The second partial derivatives of the potential (p' with respect to the ('.oordinate~ x and Y Ilre f1rst·order inrmitesimals: o2cp'
OIL
""Tz2 = Tz '
oZ(p'
ilu
ill)
a;ay = Ty = 8Z t
iI~'
ifr;
-ays = 8Y
With this in view, ill (i.1.3), \ve ('an determine the gronp of terms that are second· and third-order infmitesimals; disregarding them, we obtain a linearized equation in the following form: (a!o-V!o)
~:~' +a!o~-O
(7.!.4)
or (1-M!,) ~~' ~~. ==0 (7.1.4') where M", = V ",/a..,. Let us consider the expression for the pressure in a linearized Dow. To do this. we Sh811 use formula (3.6.26), which we shall write in the form
+
pip.., = (1 -
y2/V:nh)
"/(11_1)
(1 -
v:../l'~a"tll/(~-l)
Inserting the value of V' from (6.1.2), we have pip"", = [1 - 2V..,u/(Vlssa" - P!..,)lk/(k-lj Taking into a(',count that by formula (3,G.22) l"max- V ;,=
k':"t a;, = k~t'~
we find
:. :-..:(1_k;1.
P"'pv~"'lIr/(lI-l)
(7.1.5)
Let us expand the expression Oil the right-hand side into a binomial series and retaiu the second term in the expansion: p/p~ ~ I - p~V~u/p~ (i.1.5·) Hence we find the excess pressure p - p ... = -!}",V ...rL and the pressure coe[(icient p- = -2u1V i.e. we obtain the same relations (6.1.3) and (6.1.5) as for an incompressible fluid. But when using these relations for Iligh ~ub~onic velocities. one lU\lst take into account that the disturbance velocity u = iJq//ox is detel'roined with a view to compressibility. 00.
·266
Pt. I. Theory. Aerodynamics 01 an Airfoil and
I
Wing
hlllllon letween the Parameters of • Clllllpreuible and InCOlllpresslbi. FIIlId Flow DVer • Thin Airfoil
Tbe flow over a thin airfoil at a small angle of attack in a compressible subsonic Dow is investigated with the aid of Eq. (7.1.4') in which M. < 1. Let us cb.ange the variables in this equation in .accordance with the relations
%,=%, y,=yVI-M:', ~;=~'vV...IV_
(7.1.6)
where,.. is an arbitrary parameter. and V.o is the velocity of a conditional Dow (fictitious velocity) that in the general case differs from the veloci ty V. of a given flow. By introducing fl.1.6) into (7.1.4'). we obtain the Laplace equation for determining the veloeHy potential rp~ of an in('".ompressible flow in the plane Zoo Yo: 8'~;.'az: + a'~;.'art, = 0 (1.1.4") Hence, the problem of a compressible flow over a given airfoil can be solved by using the results of solving the problem on an incompressible flow at the fictitious velocity V. o over 8 modified airfoil. Let us find the relation between the ('.orresponding parameters of flow over the airfoils and between thoir geomotric characteristics. The relation between the disturbance velocities U o in the incompressible and u in the compressible flows is established in accordance with (7.1.6) as follows:
~= :~ = ~~'
V ~:o
=uy ~.:o
(7.1.7)
Using (6.1.5), we find the pressure coefficients in an incompressible fluid Pic = -2uolV.o and in a compressible gas p = -2uJV"". Consequently, with a view to (7.1.7), we obtain (7.1.8)
From the formulas cg,JC = ~ Pic dio, mz.l c = ~ FICXO eGo, in which ;;, = x/b, and expression (7.1.8). we lind the relation between the corresponding lift and moment coefficients:
C".Jc=.,~pdi='\'C"a;
mZatc=y~pzii-ym'a
(7.1.9)
where ; = :db. Let us establish the relation between the configurations of airfoils and the angles of attack. For this purpose, we shall first determine
Ch. 7. An Airfoil in .. Compressible Flow
261
'·'~'~I_..
""~
-,---
,---*1
/)'
:'
~
Pig. 7.1.1 Airfoils in a nearly unifonn incompressible (a) and compressible (b) flows: A lind A.-st&lnltlon poln\l
the relation between the vertical velocity components. In accordance with (7.1.6), we have
vo= aq:o = alp' .
ayo
ay
• V_o =tI
t
V1-Jl'!.,
v_
V
V 1-
• ,,·...G
M:"
v_
(i.1.10)
In an incompressible fluid by Eq. (3.3.17'), in which the function
~ni!i~~~i~nll~~~~~/(~!: -=!_ f~JxQ) ~y~7ck: ~~,(~:ti::~:;; a~~~:~~~i!~~
Uo <: V"'o, assume that "Ill V ""0 ...: dYo/d:t o. Similarly, we rind for an airfoil in a compres.'1ible gas that vlV,", '= dyldx. Consequently, (1,.''/1') (l'""IVooll) = (dYoldy) dl'fdx o• Taking (7.1.6) and (7.1.10) into account, we find dYoldy = I'IV 1 - JP"". Integration ror the condition that for y = 0 the quantity Yo ...,., 0 yields an equation relating the vertical coordinates of the fictitious and gh'en airfoils: Yo ~
yytVl - M:.
(7,1.11)
At the same time. as follows from (7.1.6). the horizontal coordinates of the airfoils do not change. With this in view, the angles of attack can be written in the forlll 0;10 = yJ(b' x) and 0; = y/(b' - x) where b' is the distance to the trailing edge of the airfoil, and x is the horizontal coordinate (Fig. 7.1.1.). Hencp., b)" (7.1.11), we ha,"e
"" ~"ytVl-M:. (7.1.12) Let us assume that the arbitrary parameter I' = 1.. Therefore.
p~p,,, Y=YoVI-M~, "~""VI-M:'} era = ell_let
m,_ = m,.1c
(7,1.13)
268
Pt. I. Theory. Aerodyn.mic:s of tin Airfoil tlnd
tI
Wing
Consequently. if the pre!!sure coefficients arc the same at corresponding points of thin airfoils in compressible and incompressible flows. then in the compressible flow the airfoil is thinner than in the incompressible one V 1 - M!, time!!. The angle of attack decreases to the same extent. Let us consider the case when y = V 1 - M!, and. therefore.
p""'Plc/V1-M!"
Y=Yo,
cr;=al e
c" ~ c",,/V 1- M:'. m" ~ m",JV 1- M!.
}
(7.1.14)
According to the results obtained, for two identical airfoils at the same angle of attack, the pressure coefficients at corresponding points of the airfoils. and also their overall lift and moment coefficients in a compressible flow are larger than those in an incompressible one 1N 1 - M!.. times. Hence follows the conclusion that compressibility leads to an increase in the pressure and lift force. The coefficient 1tV 1 - M!, is known as the PrandtJ-Glauert correction (for the compressibility eUect). The corresponding relation (7.1.14) for ii is known as the Prandtl-Glauert formula. It can be considered as a first approximation when calculating the pressure coefficient ill a compressible flow according to the relevant value of Pie' More accurate results relating to thickened airfoils and increased angles of attack are obtained by the Karman-Talen formula
-P=PIO - (V-M' PI )-' 1-M!.+ 1+Y1-=-M!. +
(7.1.15)
The lise of formulas (7.1.14) and (7.1.15) for p leads to quite a large error when determining the pressure coefficient at the stagnation point where the velocities and, consequently, the local Mach number, equal zero. For example, for this point. at which POle = 1, formula (7.1.14) when M .. = 0.8 yields a coefficient Po = 1.67, and formula (7.1.15), Po = 1.26 instead of the actual value of 1.17. At the stagnation point, the pressure coefficient Po = 2 (Po - pcc)/(kp...M:O) for arbitrary numbers Moo < 1 is evaluated with the aid of an expression obtained from (3.6.30) provided that M = 1:
Po =
~[( 1+
";1 h/';;.. Y""-I) -1J
(7.1.16)
When M"" < 1, the quantity in parentheses can be written in the form of a series in which the first three terms are retained:
Po=1+ ~:.. +2;"JU!,
(7.1.17)
This relation is suitable for quite a broad runge of values of 0 ";;M~";;1.
~
Ch, 7. An Airfoil in e CompressIble Flow
269
7.2. Khrlsllanovlch MoIhotI content of 1M Method If an airfoil or another body in a flow introduces finite disturbances into it, the linearized equations are not suitable. Non-linear equations of gas dynamics must be used when studying such a flow. Many problems on supersonic. flow can be solved by employing, particularly, the method of characteristics. The number of solvable problems of supcrsonic. aerodynamics increases still more owing to the numerical methods of integrating the equations of motion. It is mll('h more difficult to analyse subsonic flow. This is explained mathematically by the different nature of the equations: for snpersonic flows they are hyperbolic, and for subsonic ones, elliptic. The lise of imaginary characteristics in the elliptic equations does not yield appreciable simpli{i('.ations. The more ('.omplicnted nature of studying sub50nic now~ i~ explained physic,ally by the fact that disturbances in them propagate into all the regions of motion, whereas the disturbances in ~npersonic flows are conlined within conkal ~urfaces issuing from the point of disturbance and propagating only downstream. Owing to the linearized nature of the equations, the im'esLigation of nearly uniform subsonic flows i~ !!omewhat simpler than of subsonic. flows with finite disturbances o\"er, for c:'\ample, thick airfoils. A meLhod of studying such flows is de5Cribed by Academician S. Chaplygin [t71. He gives equations forming thc mathematical basis of the theory- of potential subsonic flows. They are known in gas dynamics as the Chaplygin equations, Their feature i!'l that they describe the flow of a gas Ilot in the plane y. x, but in that of the special ('·oordinates 't" and ~ (or '""" V~ is the square of the lotal \"elocitr at a given point in the flo\\,. and ~ is the polar angle determined from the condition V", ,.,,, V ('os ~). Unlike the (',ollventional e(lua· tions, they are Unear because the coeflicient.<; of the equations are functions of the independent "ariables"t and ~. Tbese equations were so!\'ed by Chaplygin lor a nllmber of cases of flow at high subsonic \"elocities. The Chaplygin equations underlie many other methods in the field of high-speed aerodynamics. Academician S. Khristiano"ich, using these equation!'!, dcveloped a method making it possible to take into account the influence of compressibiJity Oil the subsonic Dow over aidons of an arbitrary configuration. The theoretical tenets of this method are desc.ribed in detail in [:iJ. We shall consider the basic content of the method and its application to the !'!ollition of various problems of a compressible subsonic flow over airfoils. When considering a compressible flow o\'er an airfoil of a gh'en configuration. Khristianovich showed that the flow equations c_an be
270
(a)
Pt. I. Theory. Aerodynamic:s of lin Airfoil lind
y
II
Wing
(b)
p
" FIg.7.1.t
Calculation of the pressure on an airfoil in D. compressible now: a-VUe no_I b_flcUltoll!l flOw; I-given llrfoll; R-i\clltiOUI alrfotl
reduced to e(luatiOn.s of an incompressible flow over an air/oil u:ith a modified configuration (Fig. 7.2.1). Hcn('e, according to Khristianovich's approach, lirst the relatively simple problem of a conditional (flCtitious) incompressible flow oeer a fictitious airfoil is solved, and then the parameters obtained are converted to the conditions of a compressible flow over the given airfoil. Table 7.2.1
A 1~ 1~::::oo 1::~981 ~:::93 [:;:83 1:~67 [:~;43 [ ~~'"
[~:i~621 ::~71 ::~" :;~51 ~:;;21 ::~, 1:~:o , 1::;~;1 ::;!, 1~:;;O681 :;;;71 ::~8~71 :::~ 1:::~:Il 1:::;;3 'I ::24 1:~;, I ::;~ I::;;·1 :.7517 1 :;;" 1
A
1
This conversion is hased on the usc of a funetional relation between the true speed ratio A = Vla* of a compressible Dow and the lictiHOllS speed ratio A = Vola* at the corresponding points of the gi. ven and flCtitiuus airfoils (Table 7.2.1). The method being considered makes it possible to reconstrud the given airfoil to the fictitious
Ch. 7. An Airfoil in " Compressible Flow
271
As shown by Khristianovich, for prolate airfoils, the difference between the configurations of the given and fictitious airfoils may be disregarded. In this rase, the Khristianovieh method allows olle to ronvert the parameters of the now over an airfoil (pressure. veloe· ity) to any number M"", > 0, Le. to lake aceount of compressibilitYaif the distribution o[ these parameters over the same airfoil is known for a low-speed flow wh('1I the innnence of compressibility is absent (M ... ~ 0). In addition, this method allows us to convert the flow parameters from OIU) number M ~1 > 0 to ,\llother one M <"~ > o. The Khristianovich method is sllitable provided that- the velocity is suhsunic o\"er the enlil'tl airfoil. This condition is satisfied if the Mach number of the oncoming now is les.." thnll the critical value M cr' Conseqllently, hefore p(>rfol'ming calculations, one mllst find tlds criticnl value nnd determine the numbf'r M"" < M "-'. er for whirh calculations are possible. The critical 1lI1l11bN Moo .... , can also be found by the Khristianovich method. 0<""
Conversion of the Pressure CoeHldeni for an IncompreSJlble FlUid to the Humber Moo>O
Assume that we know thr {listrilmtiOIl of the pressure coefficient over an airfoil in Iln incompressible flow, i.e. we know the form of thC' function Pic c- Pie (x) (Fig. 7.2.2). This fnnction is converted to the number M"",.er > Moo > 0 n~ follows. We determine the speed ratio accorciillg to the given number M~>O:
A"" _=
Ikt I M;, (1-1- k-;t .ill;' r!r 2
(7.2.J)
From Tahle 7.2.1, we (ind the i'lflitioH$ speed ralio .\"" of an ill_ eOlllprf'ssible flmv corresponding to the value ;."'" and for tl\{> choSE'Q value of frolll th(> Bernoulli e(lllation
Pic
(7.2.2) we determine the local fictitious speed ratio
(7.2.2') Kllowing A, we usc Table 7.2.1 to find t.he loeal trlle !
:272
pt. I. Thtiory. Aerodynamics of an Airfoil and
fII
Wing
'FIg.7.1.J:
Nature of &he pressure distrl· button 011 one side 01 aD airfoil at diD'erent valun of M ...
ii
p
formula = 2 (pIp., - 1)/(kM:'). The curve = p(x) converted to the given number M., is shown by a dashed line in Fig. 7.2.2. ConWlnlon of the Pruaur. Coefficient Irom M ... 1 >0 to lJI...a>'>
Assume that we know the distribution of the pressure coefficient a compressible fluid PI = P; (x) nt a certain number M""1 (where M -,Cr > M _1 > 0). To COil vert this distribution to the number M"' 2 > 0, it is necessary to flrst ca!c.ulate the speed ratios J.. ... I and A"'2 corresponding to the numbers M ""I and M"2 by formula (7.2.1) and use Table 7.2.1 to lind the speed ratios A ...t and A"' 2 of the ficti· 1.i01ls incompressible flow. After l'I$Signing a value t.o the pressure coefficient PI' from the formula PI = 2 (PI - poo)/(kM ""IP""I) we rand t.he absolute pressure:
·~f
p, ~ p~, I;' (kMl.,t2)+ 11 and evaluate the local speed ratio of the given flow:
Af c:
{::::! [t - (To r"-I>I"]} 1/2
(7.2.3) (7.2.4)
From Table 7.2.1 according to the value of A.h we fand the local speed ratio of the fictitious incompressible flow, and from the Bernoulli equation (i.2.2), we c.a1culate the corresponding pressure coeffi·dent:
Pic = 1 -
(A/A""I)2
The pressure ('.oefiicient P2 for the number M ",,2 is determined according to this value of Pie in the same way as the pressure coeffi· cient of an incompressible now is converted to the number M"" > 0 .according to thE' procedure set nut. previonsly.
Ch. 7. An Airfoil in .. Compressible Flow
273
Determination 01 the Critical Number .V
According to Khristianovich's hypothesis, a local sonic t'elocity, which the crUical JJach number.u "',er oj lhe ollcomin/f flow corresponds to, appea.rs near all air/oil where the ma:J;imum rar~/action. is obsern:d tn all incompressible flou:. Khri.~t.iallo\"ich establisheu the relation betweell the minimum coefficiellt PIc, nlln corresponding 10 this maximum rarefilction and the number Moo. cr' lIence, to lind the critical \Iach Illimber, it is Ileccssary in some war or other, for example, by hlowillg air over a model ill ala\\"speed willu tnnnel, to determine lhe magnitude of tliC maximum rarefacLlon PIc, mIll' If us aresult we linclthe distribution of the pressure with account taken of the compressibility for Moo, rr > M"" > 0, we Cllll determine the value of Pte, mIll by the COH\'ersion of Pic. min to the number Moo = O. Assume tha~ we know the value of lhe maximullI rarefaction P1C. IHln' Since}, -""", 1 \....here a sonic velocity appears, from Table i .2.1 we can find the corresponding value of the local spe(>.d ratio of the fictitiouS incompressible flow, i.e. A ~ 0.7577. Gsing the Bernoulli equation mIn = 1 - (A/.\",,)2 we can find the speed ratio for the fktitiollS oncorning flow:
Pic.
1\00
=
A/I ' 1- Pic, mill =. 0.7577 / I·
'1-
Pic,
mIll
(i .2.5)
while H~illg Table 7.2.1 and the value of .\"" we can deterrnine the critical speed ratio A",_ cr of the compre~sihle flow. The corresponding critical Mar-h number is
M",. cr =
"00, cr (k-~-1_ 1.:-;-1 A~, er) -1.'2
(i .2.G)
A plot of the critical Mach number "erslls "I~' 111111 con.slructed ae.cording to the results of the above calculation isshown in Fig. 7.2.3. An increase in the airfoil thickness is attended hy a decrease in the IIl1mber M Cr' The explanation i~ that such all inr-rease leads to contraction of tlte stream lilament and to an increase in the local flo\\' velocily. Conseqnently, sonie "elocity on a thickened airfoil is achie"ed at a lower fr('e-~tl"eam velocity. i.e. at a lower value of M 1'1"". cr- This conclusion follows (liree.tly frolll Fig. 7.2.3 in acconlnnce with which at all illcreased local yelodty lower Yulues of M",.cr correspond to a lower value of the coeffirient Pte. ml,,' Upon an inet'ease in the angle of attack, the /lumber M "". cr diminishes, which is also explained by tlw greater contraction of the streO'l.rn HIaments and by the 'ls.sociated increase in the loc.al subsonic velocity. QQ.
oX
-:
274
'·m
PI. I. Theory. Aerodynemics of en Airfoil end eWing
M_.,r
'-'
I .
a., I.a
a.8
I:---r-
0.7
0.6 Flg.7.:U A Khristianovich curve for determining .the crHical ~18ch number
a.5 M
-0.5 -'.'0 -,.~ Jii(.nri~
....rodynamlc Coefficients
Khristianovich's investigations made it possible to obtain more accurate relations for the lift and moment r.oefticients than relations (7.1.1'1) found on tlle basis of the Prandtl-Glauert formula for the pressure coefficient. These relations are as follows:
clIa=cy.lcL/V1-M!o;
ml:a=m%alcL2/V1-M!.,
(7.2.7)
where
L = 1-f- O.05M;'1 M!."
cr
Compressibility changes the position of the centre of pressure of an airfoil (the coordinate xp of this centre is measnred from the leading edge along the chord). It follows from (7.2.7) that in a compressible flow, the coefficient of the centre of pressure is (7.2.8) where Cp = xplb = -m 2/c!la' c".ie = -m'alc/cYalC
Examination of (7.2.8) reveals that the centre of pressure in a compreS!>ible flow ill c.omparisoll with an incompressible one is displaced toward the trailing edge. This is explained by the increase in the aerodynamic load on the tail sections of an airfoil at increased flow speech; and. as a consequenc.e, by the appearance of an additional stabilizing effect.
7.3. Flow at Supercrifical Velocity oYer an Airfoil (Moo>,ll"",cr)
Sn bsonic flow over an airfoil can be characterized by two cases. In the flrsl one, the local velocity of the flow on a surface does not exceed the speed of sound anywhere. This case is purely subsonic
Ch. 7. An Airfoil in • Compressible Flow
271!.
~~\~'~i~ flow over an airfoil at a supercritical ve1ocill': ,,-diagram or H," Row U5('d ror ealculllli("'~ ,,'hN' • local norlllal shuck Is p",~nl; bdi~lrfb"li"n 01 Ilw Jlrcssuro' ov .. r Ihr llirl"iI wb"n II local }.'$hap<'d shock rO""5
flow. It i~ gelle!ral kuowll'dgl' thalH! sur.h II "('Iodt)', 110 shocks ran form nt alLY point on nn airfoil. while! the lift forre lHul drug ;He determined with a dcw to tIll' {'ompressibility ami in the gellernl ca~e depend on Ihe lIormal pressllf(~ > M '-'. ("r' the lornl \"Clodl}" a! somE' poilllS ill the vicinily of an airfoil be("ornes higher tllall Ill(' speed of SOIIll/1, (111(\ a 1. Olle of local supersonic "elocities Ilppears. IT('re the flow o\"er all airfoil is chm· arleri1.ed hy II\(' flld thai hoth hphilHl it and ahc(ld of t.he leading edge t.he local \T.]odly is lowm' Ihall I.he speNI of ~ollnd. 11(>lIc(>, local compression shocks form in the trallsilioll from s upersonir. velocities 10 !'lubsonir oncs in the \'ieini ly of the wing (Fig. 7.3. t) . Such I.-shaped shorks {'ollsis!.. as it wcre , or two shods: a fronl cllrvp.d (ohlique) onC! DH and 11 r('ar alnlOsl !'Lraight. allc (Fig. 7.3 .1h) . Local shocks Ihol. may form on both tile uppE'r ,IIIIJ hollol1l si of au ;Iddiliollul wllve drng Xw' IIcnce . I./ic tolal drag of an "'irfoil X it consi~ts of the profile X pr ami t.he WIH"C X", drag!' , i.('. Xl. Xpr -i- X w ' or c'~a=-,,:c.'·.l'r + c .~.w . where c"" iJ; lilc tot:ll dl·:lg ('odTiricllt of Ihe i,lirroil. c",, lor allil c"'.w lire the profile and wave drag {'oefficiC'llls, I·cspcc.tin-ly. Let llS ('onsider a !'limple method of calcllla1.ing wave drng proposed hy Prof. G. Bllrago [161. Let liS assume tlHlt a local supersonic zone closed hy Hearly uormaJ slio('k IJC is formed on the llpper side of all airfoil (Fig. 7.:~.1a). Let usscpilrulc a stream IllaJllcnl passing throllgh this shock in the flo\\'. The paramctC'r.<: of the ga~ in the fllamellt di1"('ctly alwad of the sllock fll'() i"" Ill' PI' ami ,til> and hehind it are
en
276
Pt. J. Theory. Aerodyn.!lmics of .!In AirFoil .!Ind .!I Wing
V z, P2, Pz. and M z· Let us introduce two control surfaces I-I and II-I! to the left and right of the airfoil at a sufficiently large distance from it; let the parameters of tho gas along the left plane be V,oo. p,,,,, and PI'"'' and along the right plane be Va 00, P2o'" and pzoo. Using the theorem of the change in the momentum of the mass of a gas when flowing through the control surfaces, we ohtain
1v,~dm'~-1 v,~dm'·-1 (p,~-p,.)dy-Xw
(7.3.1)
where PI and Pzcc are the forces acting on the left and "right surfaces, X w is the force of the wave drag with which the airfoil acts on the flow, and dm p " . - p, ..... 1','" dYl""" dm2"" = P2""V2oo dY2"" are the rates of flow of the gas along the stream filaments that according to the condition of the constancy of the flow rale are equal, i.e. p,,,,,V1 dYloo = P2ooV2°O dyzoo (7.3.2) 00
°o
Let u::: assume (see [1(1) that levelling out of the velocities occurs hehind the wing at a l
Consequently,
Xw
-1 (p,~
-
p,~) dy
(7.3.3)
We can assume that for stream filaments not intersecting the shock. P2<>< = p,oo i.e. the pressure at a large distance from the airfoil behind it is restored to the value of the free-stream pressure. For the fllaments that pass through the shock, P200 < PlOO' Indeed, since PI"" = Pu ("1 - VIooIV:nad~J{h-t)
P2""= p~ ("1 _
where
p~
<
V~ooIV~ax)hJ(h-t).= p~ (1- VL"IV~ax)h/(h-l)
}
(7.3.4)
Po, we ha\'e p2~/p,oo
= p;lpo"-:
Vo
<
1
(7.3.5)
According to (7.3.5), formula (7.3,3) can be trans[ormed by taking inlo account lhe flow ratc equation p,,,,,V,,,, dylO" = Pll'l ds, in accordance with which dyl'c ~ dy -= (p,l't/pP,vloo) ds
where s is lhe length of the shock.
Ch. 7. An Airfoil in
(I
Compressible Flow
As a result of transformations, the wave drag force is s Xw=Pt<S> ~ PI~l (1-\'o)d.<; '0 PIOHI<S>
277
(7.3.G)
and the coefficient of this force is 2X. 2 ex. w = kp..,M!..b = k,V:'b
Let us expand the function n =M I -1: '\'0
'\'0
(n) ~.. "0(0) + ( ~:~
I' Pl""l',,,, p,r, (1 -
)d
Vo
I)
in (1.3.20) inLo
i.I
s
(7.3.6')
series in powers of
)"=0 n+ {( ~:~~~ )"=0 n Z (7.3.7)
In exprel'!sioll (i.a.i). the quantily Vo (0) '""" 1 because at .t11 = 1 (n = 0), thesJlOck lran"forllls inlo a wave of nn illfllllLe!'!iroal stl'ength. and the preS!;llrC p~ '.; PB' using (4.;U:J), (·1,;U5). IlIld (4.3.20), we (··au 111":'0 ~how thilt
(~:o )'I~O = ( g'~1 ).11,,. I=
0 and (
~:\:n }n._n ~ ( ;~i )_11.=1 =
0
Having in view that for thin airfoil.s al smAll angles of aUlU'k. the differcllce Ml - 1 is small, we- can limil oHrscives to the [ourln term in expansion (7.3.7): yo(n) = 1-:-,*" (
r;; L,.:.I(Mt-l)Z
(7.3.7')
Investigations have shown that. for a given airfoil. the quantity Ml - 1 call be considered approximately proporLional to the diff{'r~ ence M"" - Moo. cr' Designating tho relevant proportionality fact.or by Al and including it in the overall coefficient A determiocfl in the form
A---"-'-(""·') ak,":'b dMi
.II,~I
i~dS
0 PI_v1_
(7.3.8)
we obtain an expres.'1ion from (7.3.6') and (7.3.i') for the wave- drag coefficient: (7.3.\) ':r.w = A (Mo<> - M"".cr)3 The coefficit'Jlt A in the general case depends on the configuration of the aidoil, the aJlgle of attack, and the number Ifl_. It may be considered as approximately constant, however. Tests of modern airfoils installed at small angles of attack in wiud tunnels show that
278
Pt. I. Theory. Aerodyn.mic5 of an AirFoil and a Wing
.
C,
0.6
0.5
V_(M_) ~
rI-
D•
.,
,
'1,.7.:1.1
I
Drag o( an airroil in a nearl)"
sonic no",'
o.
..
H=
i-' D.'
0.'
the coefficient A ~ 11. At t.his nIue, satisfactory results of calcll~ lations by formula (7.3.9) arc obtained if the differenco M co - M .... et does not exceed 0.15. It. follows from formula (7.a.B) that the wave drag coellicient increases with increasing Moo. This is due to the fact that npon a growth in t.he airspeed, the shocks formed on an airfol I become more and more intensive and extended. To lower cx. w at a given M .... one must ollsure an increase in the number M _. cr' whit-.h is achieved mainly by reducing the thickness or the ail'foil. A similar result can be obtained when the angle of altaek is made smaller. }o'igure 7.3.2 shows an experimental (".uT\'e charaeterizing the change in the drag coefficient c'\"a -,:, c.~, Pr -I- c.~. w in nearly sonic flow. For values of M Of> < 0.45 -0.5, only profile drag is observed. while at M Of> > 0.5-0.55 (supercritical flow velo('.ities) a wave drag also ap~ pears that. is due to local shocks. 7.4. Supersonic Flow of a Gas with Constant Specific Heats over a Thin Plate Let us ('ollsidcr a very simple airfoil in the form of an inflllitely thin plate placed iu a suporsonic flow at the angle of attack (x. The flow over such a plate is shaWl! scllematically in Fig. 7.4.1. At its leading edge. the supersonic flow divides into two parts-an upper one (above the plate) and a bottom one (under it) that do not influence each ot.hel'. This is wh\' the supersonic flow over each side can be investigated independently. Let us consider the upper side of the plate. The flow here is a plane supersonic flow over a surface forming an angle larger than 18(J<' with the direction of the undisturbed flow. Such a flow is shown sc.hematically in }o'ig. 7.4.2, where plane OC corresponds to the upper
Ch. 7. An Airfoil in a Compressible Fiow
Jl79
,I
Fig.U.'
Supersonic flow over a thin platc: J-uplIJ\$ion ran:
~-8 t1ock
(') 'I ,),-"
":~1j~. ~
,~ Fig. 7.4.1
Prandtl-Meycr no\\': ,,_ph ysieal planl"; b-hmlograph pial"': c-d;agram of ;'\ nl"arl), uniform no\\'; ~ Ion ran : t-CI>ic),clo;,1
1- ~ x,,;,\I\
sillc of thc phill'. II wus rllost illwsligatcd hy L I'nUld11 and T. :\Icyer aud i!; <"-JlIINI a Prandtl-M(,ycr now. In aCl'Ql'tlance wilh til(' flow di
280
Pt. I. Theory. AerodynemiC5 of en Airfoil and a Wing
the disturbed velocity vector Voc at the angle ~LOC -...:: sin-I (1/Mod determined from the Madl number 0 the disturbed now along plane OC. The change in thc direcLiou of the now betwecn Mach lines OD and OE can be repre~ented as a consecutive set of deflections of the streamlines through the small angles A~. A straight Madl line isslling from point 0 corresponds to each of these defloctions iudie-aUng the formation of an additional disturhance. Hence, thc turning flow is filled with an infinite multitude of Mach line.~ forming a "fan" of disturbance lines that ("horacterizes a centered expansion wave. This r,entered wave, sometimes c.aUed a Prandtl-Mf"ycr fan, is defmed by straight Mach lincs along each of whir.h the now parameters arc COllstant, and this is why it belongs to the closs of simple expan:;ion waves. The problem on the disturbed molion of a gas ncar an obtuse angle, which is associated with the formation of a centered expansion wave, can be solved according to the method of characteristics. Point F' on an epicycloid-a charar.teristic in tI hodograph of the same family-corresponds to point F of intersection of a streamline belonging to the plarc parallel onrQming flow (the inclination of a streamline at this pomt is ~ _..: 0) with characlcristic OD in a J.hys-. ical plane. To be specirlc, we ran relate each of these characteristics to those of the firsl family. The equation ~ ." 0) -:- ~I is used for a characteristic of this family in Lile hodograph. Sinc.e we have assumed that ~:-: 0, therollstant ~I :::-: - ( I ) " , (Moo), where the angle (I)"" is fonnd from (5.3.30) according to the known number Moo. Consequently, the equation for the characteristic has the form ~ = (I) - 0)"", whence (7.1.1) By setting the inclination of a streamline on the small angle ~ = A~, we can calculate the corresponding angle co ,..", A~ + w"" and flDd the number M on the ncigIlbouring Mach line inclined to the new direction of a streamline at the angle j.l. "" sin- I (11M). The Mach number on plane OC with an inclination of ~ = ~oc "Ct, i.e. on the upper side of the plate, is determined according to the angle _ floc + .. (7.4.2) ~
~
The found vnlue of the local number Moe makes it pos!;ible to determine the Mach angle ~c sin _I (tlMod. A graphical solution of the problem on the Prandtl-Meyer flow is shown in Fig. 7.4.2h. The coordinate of point G' of intersection of the epicycloid with straight line O'G' parallel to plane OC determines the speed ratio hoc of the disturbed flow near plane OC. The point G of intersection of a streamline with characteristic OE corresponds to point G' in a physical plane.
Ch, 7, An Airfoil in
II
Compressible Flow
'281
AC'cordillg to the kl\own numbel' Moe (J.oe), hy using formula (3.6.30), we ("lln d('t('rmin(' till' pl'es!
Pu= Poe, u=: Poe [( 1+ k;-l M:")! (1 T k-;-1 Mf)C) ]"/(11- I) (7A.3)· and t.he cOfl'esponciiug vtllue of the pressure cocftident.! 1"1
POC,n -, 2 ({Joc, II
-
poo)/(k.lli.,p",)
At. hypersonic vclocilie~ (Moo:» t), thc cllicuhltiou of the Prilluitl\Ieyer flow is simplified, bec8u:-tc to detflrmine 01(' funrliOIl W wc call lise formula (5.3,/11). This allows mILo deL(,I'minf! the lot'lllliumher M directly. Suhstituting for the Rugles Woe ami 0,1.,.., ill (IA2) their \'1'11u('s ill ac('ordan("e wilh (5.3.!11), we oblilin
M,,=Moc=( ,u~
_k-;IPoc)-1
(7A.4)
The t'orresponding preS,'illre cal) be determined by formula (3.6.30). Lct us (".ollsidel' the Prandtl-\l€!'rer now appearing III hypel':-tonic \'clocitie!> with !.aunli now dellc("lions Poe -" -:t. At \'er~' hu-gc numbers M "", the hmun or 'Illch lilies i~"'lIillg fl'om point 0 will be very nan-ow. We Clln ("Ollsider [0 11 suftidently good appro:'(imatioll that the beam is ("olllpre:":"Cld into a single line on which [he £low imrnediat('ly turn:;, wilh cxpansion. This line ('an tlwt'cfore be ("ollliitiollally considered 11:-; Illl "expansion sho('.k" behilld whirh the velodties (;'\lacll Ilumbers) grow and the pre~s\lt'es lower. The angle ~l;)C of inclination of this line to the ve("tor V"" is ohtailled if we take "d\'an~ tage of the analogy with II. ("omprcssioll shot'k illld ("alculntc this· angle by Iormuln (4.6.1J) provided that. the angle ~~ ~.:. ~ -" ex in this formula is negflth'e, Assuming that O. . ~IOC' we lind
I:~i =2(t~6)+V "(1~W
+rb-
(7,4.5}
where I Poe I ::-. I ex I is the magnitude of the tllming angle of theDow (the angle of attack), ~lIld K~ -:: (M ""~od:.!. By llsing formula (4.6,11') to ("valnate t.he pressui'e coefficient with a view to the sign of the lingle ~oc :- a. we obtain
(U.S} At low numbers Moo aud small angles ~oc '--: -:t, a nearly uniform Ptandtl~:,\'leyer flow appears lIear the deflected surbce. Fill' such a f10\V, relation (1.1.2') fOl' the !:\pced of sOllnd holds. If we III)W find a: formula for the l'.1ach nllmber from (7.1.2), namely,
'H~=f.-=k~1(~-1)+ ~~
..
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
and introduce the value of a~ fl'om (7.1.2') into t,his formula, we obtain
;.=M2 . . . M;,(1-L
e:)r1-(k-1)
u~"'rl
(7.4.7)
In acC'ordance with this expressioll. we can assume in a first approximation lhat ill a nearly uniform flow M =::: M"" and, C'onsequently, the equation dy1dx tall (P ± J.l",,) is used for the cllaracteristics in a ph~'siC'1l1 pi nne. Since the flow den()('.tion angle P is small and ~ <:: 11.". then dy/dx =::: ± tan J.l "". COllsequently, the characteristics are Marh lines inclined to the x-axis at angles of ±J.l ..... For a PrandtlMeyer flow, we hHe a family of characteristics in the form of para[lei lilies inclined to the horizontal axis at the angle J.l .... Wig. 7.4.2c). We obtain dirfl!fenrp. p.quations fOf tile characteristics in the hodo:grapll of t\ supersonic Row from (.'}.3.21) and (5.3.22): dV!V =F tan p..dP -- 0
(704.8)
For a nearly uniform flow, we have AV - u, V -, V coo. dP
,p,
tan J.l:""": tan J.loo -: (iJP... _1)_1/2
Consequently.
ulV ~
±~!V M:, - 1
(7.4.8')
Int<erting the \'aille of u fl'om (7.4.8') into «(U.5), we ohtain (he pres.~urc ('O('ffiricnl (7.4.9)
p
dince we are considering an expanding flow for which < 0 and are ha\'ing in "iew that tile magnitude of the angle Ii is being found, we mll~t take the minus sign in formula (7.4.9). Acc.ordingly, on the upper side of the plate inclined at a small angle of attack Ii ." a, the pr~~ure roerriC'ient is
Pu -.':
POC.II = -2a.IY M:' - 1
(7.4.10)
Let liS rOllsicier tile botlOin side of the plate. The flow o\'er this side (see "'ig. 7.4.1) is attended by the formation of shork OF. issuing from n point on the leading edge and, consequently, by compression of the now. To ti(!termine the angle e~.OI~ of inclination of the shock, we shoulll use formula (4.3.25) in which we mll~t a~lIme that MI -~ ~.., M ... and ~$ _ a. At"t'_ording to the fOlilld value of 6 s'0l::. we rllld the ~ta('h number M2 -= MOC • b on the bottom side by (4.3.19) or (4.3.19'). When determining the nature of the flow in the region behind point C on Ihe trailing edge. we can proceed from the following -consiilel·atiOlls. On the upper side of the plate, the number Moe.u ahead of shock CD is larger than the number Moo ahead of shock OE
Ch. 7. An Airfoil in
iI
Compre~~ibfe
Flow
'283
formed on the loading o(lgo from below. If we ai'Sll[l)(' that behind point C the flow does not deflect from the direction of the undisturbed flow (streamline CF is parallel to the H'ctor V",). the losses in the upper shock will evidently be greater, and tllcrefDrc Mo·,\> < < iVCF• lI • The pr(':-;surc in the region abovc line CF will he greater than in that helow it. A pressure jump rannol be rehlill('(1 on a houndary i'lll'fac(' in
11
gas flow. althouglt the velodties may remain different. Tllerefore, itl real conditioll~. the diredion of :o.trPHmlinc CF differ.~ from that of the free-stream velocity. i.t'. a downwash of thc flow forms behind the pl
'H·W~·t~~~gli~~ ~IFe ~'~ll~;~; r~~if~~l l~el?tn(ttl~ltl'Ail~~re~igea.n~1 ~;al~d~t Meyer flow appears with the ~1acb 1\lllTlher MU.b d£'tt'rmi1l(,d with the aid of the formula WCF.h (')OLl> _,_ a. The pressure PI' ....., lhX:.\l on the llppe'r side of the phlte is dl'll'rmilled by forml11a (7.1•. :1). white the ("orresponding prpsslIrf> Ph .=-: /JOC.b on the bottom side- is ("
." ~ill
lift and drag ('oefficient~
'H'e
'I. CUa
The ('Ol'l'(>.~pouding \',ducs of !he }-,'(q",,/,) :1ud cx" . Xa'(q",).)·
r" .-
Intro
:-=
(j)h - POI) ros a,
c:r" '-'
(Pb -
P\I) .!'ill
a
(i.4.11)
The force X 3 Hppearillg !lpon supersonic fiow over the plate and <.:a\l~e(1 by the formation of ~hock waves and ordinary disturban('e wav!,." is I"all('d the waH' drag. lind the rOfl"esponding quantity eXa . ex'''' is ('ailed the wave drag coefficient. This drag dot's not equal zero even in an ideal (in\'iscid) nuid.
284
Pt. I. Theory. Aerodynamics 01 an Airfoil and a Wing
The Jinencss of the plate K = cy/cxa ,..,., cot Ct can be seen to be a function of only the anglc of aLt.Ack. Owing to Ihe uniform distribution of the pres:o;nrc ovcr the snrfate of the plate. the centre of pressure is at ils mhldle. Conseqllenlly, the moment of the pressurE" forces about the l(lading edge is M: :.." -YU2. and the corresponding moment coefficient is :...~ m:8 = Mz/(q""L'l) = -(.vb - Pu)/2
In:
(7.4.12)
At hypersonic velocities, the pressure c.oefJirient for the upper side oC the plate is determined approximately by (7.1i.6), and Cor its lower side, by (4..6.12). With this in view and assuming in (1.6.12) that ~8 - Ct, we obtain for the differenc.e of the pressure coefficienL<; Ph also known ns the pressure-drop coeffleient, the expression
Pn.
(7.4.13) Consequently, c,/a.'~4VVI4(1
6J'J+lIK'
17.4.14)
cJ.)a3=4Vlll4.(1
6}2j·I·1/Kz
(7.4.15)
nl z
/a.2.=
-2Vf/14(1
6)2J+ 11Kz
(7.4.16)
Formulas (i.1.1:J)-(7A.16)expre!'s the Jaw of hypersonic simitarity as applied to the flow oyer a thin plate. This law consists in that regardle!'s of the values of Moo and ex, but nt identical values of K = M""ex, the corresponding quantities pla.'I.. cyia. 2 • cxa.'tz,:i, and m:a/a.'I. for plates are identical. The pnrameter K = M ""ex is called the hypersonic iimilarity criterion. Examination of formulas (7.4.13)-(7.4.16) reveals that the relations for the pres.~lIre. lift, and moment coefficients are quadratic, and for the drag roeffir.ient~-cl1 bie. funclions of the angle of attack ex. At the limit., when K __ 00, we have (Pb -
pu)/a'!. = cila/a'! = cz/as = 2/(1 - 6)
m,,!a.'
.~
-li(1 - 6)
(i.4.17)
(7.4.18)
"'or a plate in a nearly uniform (linearized) flow, the pressure coefficicuts are calculated by formula (7.4.9) in which we should assume that ~ = Ct. The minus sign in this formuJa determines the pre!r.'!ure coefficient for the upper side, and the pIns sign-for tlle bottom side. Accordingly, the difference between the pressure coeDicients related to the angle of aitllck is
(P. -
p,)Ia. ~
"IV M!. - I
(7.4.19)
Ch. 7. An Airfoil in a Compressible Flow
Introducing (i.!L.19) into
formllla~
28~
('iAB) and (7.4.12). we obtain
cy/a =
4/V bl;, -1
(7.1.20)
cx/a'Z ==
4/V M!o -1
(;.4.21)
rnz/a= -2/J!M!o-1
(;.4.22)
In the given cuse. the number M", is the similarity criterion. 'Vhcn its valuc is retaincu, and regUl'dless of tllC yullic of the angle Q[ attack, the rOl·re~ponding. values of pia, cy/a, c,/a 2, alld m:/a are identical. If \\'e ronsider a nearly uniform flow at very large numbers ~f '" :» 1. rormulas similar to (7.1.1n)-(7.1.~2) cnn be wrilten in the [orlll {Ph - pu)/a2 '-" cy/a~ :-=; c:r./a:1 '""" 4'K (7.4.2:1) (;.4.24) m:/a2 = -';!JK Hence, the h},persouic similarity cl'itcriOIl K M ooa is also valid for a nearly uniform (linearized) now with large ~Iuch nHllIher~. It is evident that SUell a now can appear only at very small angles of attflrk.
7.5. Parameters of a Supersonic Flow over an Airfoil wHh an ArbHrary ConftguraHon Use of the Method of Characteristics
Let us consider a supersonic. flow over a sharp-nosed airfoil with an arbitrary configuration (Fig. 7.5.1). The upper contour of the airfoil is given by the equation Yu = fu (x), and the hottom one by the equation Yb = fll (x). Assume that the angle of attack is larger than the augle ~o.u formed by a tangent to the airfoil ('~ont·ollf on its upper side at point 0 on the leading edge. Consequently, a PrandtlMeyer now develops at this point. The flow pa.s!';cs through an expan-"'ion fan issuing from point 0 as from a sonrce of disturbances, and acquires a direction tangent to the contour at thi:-: point. Further expansion of the flow occurs behind point 0 along the cont.our. Con!'equently, tIle flow near the airfoil c·an be considered as a consecutit'e set oj Prandtl-J[eyer flows. Since the turning angle at the points on the coutour is inlinitely small, indivillnal Mach ljnes issue from them instead of a fan of expansion lines (Fig. 7.5.1). We lind the velocit:,>' at point 0 by formula (7.4.2). which we shall write in the form (7.;).1) w~ '-= Woo -7- CL - Bo... where
~o.
u
-:=
tan _l (dy /d.r)o.
286
Pt. I. Theory. Aerodynllmics of lin Airfoil lind
II
Wing
flg.7.S.t
Supersonic Dow over a sharp·nosed airfoil: J-~XpanBi ... n
fan; 2_Mach
lin~s
Vsing the value of 000, we determine the corresponding number Mf) from Table 5.3.1. At point C, which is at a small distance from the nose, the velority of the gas is calculated as for a Prandtl·Meyer flow by the forOlIlJa (7.5.2) We -'. (j)o -+- ~o.u - ~c where ~c tan· l (dy u1dx)c. IlIFerting Eq. (7.5.1) into (7.5.2), we obtain (7.5.3) We''='' woc+a-~e Hence, by (7.5.3) for any arbitrary point N on the contonr, we have (7.5.1) W.'/ Woo -+. a - ~N where~:s:
=. tall~l (dYu1dx)N is the angle calculat.ed wit.h a view t.() the sign (for the leading part of the cont.our the signs of the angles are positive, and for the trailing part, negative). Let. lIS assume, as for flow over a flat plate, t.hat the flow behind point H approximately retains the direction of the free stream. Therefore, at point E, t.he now moving ncar the contour at a velocity corresponding to the number Mn.u turns, and a shock BE appears issuing from point H. The angle Os.BF. of inclination of the shock and the parameters behind it are calculated wit.h the aid of the relevant for· Imllas of the shock t.heory according t.o the known valnes of the angle of attack Ct, the lIl1mber M II• u , and the contonr nose angle at point H on the upper side. The parameters on the upper side of the airfoil (pressure, velocity, etc.) are determined from the known value of the local :Mach numlJerwith the aid of the relations for an isentropic now 01 a gas. If the angle of attack equals the angle ~o. ", we have t.he limiting ca.~e of a Prandtl-Meyer flow at point 0 where the Mach number i~ AlO.b = M "". Formula (7.5.4) can be written as (7.5.4')
Ch. 7. An Airfoil in II Compressible Flow
287
Fig. 7.5..2
Supersonic flow over the bottom side of an airfoil with the fonnation of s shock: I-s1raight part of UtI' rolllollr of the Rlrfol] in tha flow; ll-cllrvr.ll'ilTt or lhp air/(Illronlour; III-curv .. <1 part 01 the shock; IV-slralf:ht part 01 tlw ~hntk
Calculation of thellow OVCl" Ihe bottom ~ide of the airfoil (Fig. 7.S.2) h('gins with detrrminatioll of the gas parameters at point V-directly behind the siwek. For this purpose, llsing formula (-'I.:i.25) anti the \'allies M. . M"" and ~{; - rx -f· ~O,b, we calcillate the :sponils to Of). lis length is determined as the distallc(, hom poinl () 10 pointJ thatis on the intersection of the shock with'l first fanlily characteristie issuing from point D. The now hehilld a straight shock is uorteJ·-free. cOJlseqlleHtly there is all isentropir flow over part of the contollr brhilld poinl D. To determine the velocity of sueh u flow at point F, we shall lise Eq. (i.4..1), from which we lind WI" .• Wn - (~D - ~~,), where (OJ) is the ,'alue of the angle CD calclilated hy formula (.").3.30) for the numher lJf on port OD of the contollr. 'fhe "allies of the angles ~[) and ~t' arc determined with a "ie\\" to the sign (ill the given case the ("\ioll of the shock and the characteristic. Downstream. cHrving of the shock is due to its intt'l"
.288
Pt. J. Theory. Aerodynamics of en Airfoil and eWing
second family characteristic J l/ is the boundary of this vortex flow. The characterislic is gradually constructed in lhe form of a broken line according to the known values oCthe number M and the angles Jl along the straight !lecond family characteristics issuing from contonr points F. G, etc. At. point II of the c.ontour. which is simultaneously on ,haracteristic JR, the velocity is found from lhe expression -(OK "-' Wo - (PD - PH). At. point K adjacent to H. the parameters are ('alculalcd according to the equations for the characteristks taking into a('count the vortex nature of the flow behind a shock. To determine the velocity at this point. it is su(ficient to know the velocity and its direction (the angle P) at point 7 Ileal' point K on -element. 7-K of t.hesecond family characteristic. To fllld this velocity. it is neccs!{ary to c.alculate the curving of the shock behind point J on element J ...1' and lind t.he parametcl'S on the shock at point 3'. To do this, we ha\'e tlle following data at our disposal: the number M = MI = Mp and the angle Jl = JlF of deflection of the now at point 1. and also the parameters M'I. "= MJ and Jls = Ps.J on the shock at point J. By Ilsing formula (5.4.46) and assuming that I! = O. we obtain the following expression for the change in the auglo Jl along -characteristir- 1-3:
~~I=[( ~L-1r'
(wt-CU.,-
~~I
.¥n-c.)
(7.5.5)
whtlre .I1~I=P3-~1t .I1x1=X'-%I;
!!
(S3;,Sl)Z~3S~~I~~I)
Cj
=
~~32(~:~~~
The derivative (dwldPh is evaluated by (5.4.39) for the values of the relevant parameters at point 1: the angles CUI and WJ are determined from (5.3.30) according to the numbers M at points 1 and J, I'espectively. Point 3 in Fig. 7.5.2 is at the intersection of the normal shocl< and -characteristic 1-3. Consequently, its coordinates are found as a result of the simultaneolls solution of the equations YI = (X3 XI) tan (PI + Ill)' Y3 = X8 tan (0 •• 0 - a) (7.5.G) The intersection of the characteristic with the shocl< at point 3' is diRerenl because of curving of the shock. We shall find the new shock angle 6 S,J3' on J-3' according to the flow deRection angle behind the shock ~s = Jl3' = apt + ~I ann the number M; calculated by means of the expression w; = aWl + WI in which aWl is found from (5.4.38):
Ys -
aWl = cuJ
+ (dw1dlHJ apl -
(7.5.7)
WI
Consequently, the equation of the shock element J~3' has the form Y3' -
YJ
= (X3'
-
XJ)
tan (6 s,J"
-
a)
Ch. 7. An Airfoil in
II
Compumibla Flow
289
Owing Lo deflection of the flow at point 3', tho characteristic on element 1-3' changes its direction. The equation of the characteristic on this element has the form .fIa' -
Yl = (xa' -
Xl) tan (~~.
+ !J.3')
By sol ving these two equations simullaneoll::;ly, we rInd the coordinates X a, ann Y3' of point 3'. LeL us consider point 5 at tlie intersection of characteristic elements 2-5 of the rlrst and 3'-5 of the second family. We determine the coor· dinates Ys and Xs of this point from the solution of the equations for the clement.'; of the corresponding charac.teristics: Y5-Y2=(XS-x~)tan(~2+JJ.2)
y,_ y•. ~ (x, -x,.) tan (~,._~,.)
I
(7.5.8)
We determine tIle change in the direction of the flow when passing from point 2 to point 5 along characteristic element 2-5 from Eq. (5.4.22) in which we assn me that e = 0:
~~2=t[ k~r' ~! (liX3'l3'+liZtCt)-·(~-ro3·)-(~t-~3.)J
(7.5.!)
where li~2=~~-~~: liX30=XS-Z3'; l'lS
(SJ,-S2)MS(~=+"-)eos(~3·-,,.d
Tn
I:
we assume that
f -:::
(xs -
X2)
sin !la' cos (~2
+ )J.~);
e = (xs - x.~,) .~i II ~12 cos (~a' - fla') We determine the values of C z and t a, by (SA.15). We lind the number M at point 5 from (5.4.23) llsing l<.~2' Similarly, according to the known values of the gas parameters al point 5 and at point H on the wall, we determine the velocity at point fl at the intersection of the characteristic elemenL.;; II-(j of the fir!;t family and 5-6 of the second one. We fwd the coordinates Xl[ and YH of point II bv solving the eqnations of the contour Yu-= III (XH) ano of elemell't 2-H of the characteristic YH - '12 -:: (XH- X2) tan (~2 - 1-'-2)' Let us now choose arbi trary point 7 with the coordinates X7 and Y7 on characteristic element fl-6. We shall determine the parameters at this point by interpola.tion. For example, the angle ~, = ~8 - (~6 - BH) la_1Jl"_fl, where l._1 and ie-u are the distances from point G to point 7 and to point H, respectively. Similarly, we can find the numher M, and the corresponding Mach angle 1-'7' We choose point 7 so that c·haracteristic element 7-K drawn from this point at the angle ~1-fl7 will intersect the contour at point K at a small distance1from point H. We determine the coordinates XK. YK of this
pt. I. Theory. Aerodynllmics of lin Airfoil lind
290
II
Wing
FIg'.7.S.]
Supersonic Isentropic' flow over a curved sharp-nosed airfoil: I-contour 01 the body In the now; 2-sccond family characteristic
point as a result of solving the equation VK - V7 = (XK - x 7) X X tan (~7 - 117) for the characteristic clement 7-K and the equation
YK -'" IH (xld of the contour. Since the flow at point K is a vortex one, Eq. (5.4.27) must be used to calculate the velocity at this point. Assuming in the equation that e = 0, we obtain (7.5.10) where 6Cil 7 :-:; CilK - Cil 7 and 6~7 = ~K - ~7' the angle ~K = = tan _1 (dvHldx)K' We determine the parameter t7 from (5.4.28) for the values 117 and ~7 and evaluate the entropy gradient hy the formula (5.4.29),
6S/6n = (S7 - SK) cos
(~7 -
f!7)/[(XK -
x 7) sin /.t71
We flOd the entropy SK at point K as a result of calculating its value at point 0 directly behind the shock, and the entropy S7 at point 7 by interpolation between its values at points II and 6. Similarly, by consecutively solving earh of the three problems considered in Sec. 5.4, we determine the velocity Iield in the region between characteristic DJ, the contour. and curved shock J-3'-4'. We flOd the shape of this shock gradually in the form of a broken line, and fill the indkated region of the flow with a network of characteristic curves (characteristics). We can determine the pressure at tJle nodes of the network of characteristics according to the Mach number with the aid of formula (3.6.28). We calculate the corresponding stagnation pressure Po = needed in this formula by interpolation, using formula (5.4.19). We find the pressure at points on the contour Ilsing the corresponding Mach numbers and the stagnation pressure Po evaluated by (4.3.22) for the angle 8 6.0 and the number M "".
Po
""0
Po
Ch. 7. An Air'oil in a Compressible Flow
2m
Calculation of the flow on the bottom side of an airfoil is simplified when a second family characteristic drawn from point J does not intersect the contour, and, consequently, this flow can be considered as an isentropic one (Fig. 7.5.3). We use Eq. (5.3.38) to calculate the Mach number at arbitrary point [, of the contour: (7.5.11) Here we determine the angle ~L by the formula ~L = tan -l(dYH/lkk with a view to the sign: at the leading edge of the contonr it is negative, at the trailing edge, posithe, the angle ~n is negative. We determine the shape of the shock by its angles as at points 3' and 4' at the intersection of the strAight characteristics drawn from points F and G. We find the shock angle Os at point 3' approximately by formula (4.3.25), using the values of B3' = ~F and 1'd 00' We calculate the shock angle at point It' ill a similar way. We have considNed the calculation of the flow at snch angles of attack Ct ~o.u when there is an expanding flow all over the upper side. If this condition is not observed (ex < ~o.,,), the flow oyer the upper contour occurs with compression of the gas at the leading edge and the formation of a compression .~hock. Hence, such a' flow should be calculated in the same way as fol' the bottom surface.)
>
Hypersonic Flow over a Thin Airfoil
U a thin sharp-nosed airfoil (sec Fig. 7.5.1) is in n hypersonic flow at such small angles of attack that the local :\.fach numbers on its upper and bottom surfaces at all points c~onsi
>
hlf';:'~[:co _k;1(a_~N)rl
(7.5.12)
obtained from (7.4..1). Since everywhere on the upper contour there is all e:rpanding [Ww, the number MN > M Flow over the hottom surface is attended by the formation of a shock and, therefore, by compression 0/ the gas. We c.a1clliato thE! pressure coefficient at point 0 directly behind the slwck with the aid of formula (4.6.12) as follow~:
co.
p,/(a-Bo.,)'=I/(1-6)+Vl/(1 where K = sign.
..
,
Mco (a -
~O.b);
6)'-,4IK'
(7.5.13)
we take the angle ~O.b with a minus
~
Pt. I. Theory. Aerodynamics of olin Airfoil and a Wing
Pb
From the formula =-=: 28~.0 (a ~O.b)' we can calculate tile shock angle 8 s.0 , and then lIsing the valuo of Kg = M 008 5.0 we call find the Mach number at point O. IIsing formula (4.6.16): M1,.b/M~ ~
K:/(lil
+ 6) K: -
61 (I - 6
+ 6Km
(7.5.14)
Behind point O. expansion of the flow occurs, therefore to deter· mine the 'Velocity at an nrbitrar~' point I., we can ;Ise the relation
(7.5.15) Hypersonic flow over a thin airfoil call be calculatod approximately by using the method of tangent wedg(>,S. According to this method, we calculate the now at an arbitrary point of a contollr by the relevant formulas for a nat plate assllming that the lauel' is in a flow with the number Moo at an angle of aUack equal to the angle between tile vector V 00 and a tangent to the contour at the poillt being considered. Hence, for a point N on the llpper side, the pressure coeffir.ient by (7.4.6) i, PN/("-~N)'~l/(1-6) -VI/(1-6)' +4/K,
;(7.5.10)
where KN = Moo (a. - ~N)' Foran arbitrary point L on the botto:n side, with a vi!:)w to (7.3.1.:1), we hav& pd("-~,)'~ 11(1-6)+ VI/(I
6)'-i-4/Kl.
(7.5.17)
where KL = JlI 00 (a - ~Ll. In formula (7.5.16), the angle ~'" has a pillS sign at tlle leading edge of the contour and a minus sign at the trailing odge, whereas in formula (7.5.17) the angle ~L hai a millns sign at the leading edge and a plus sign at the trailing one. At the limit, when K _ 00, we have
iN PL
~ 2 (" -
.= 0
~Ll'/(I - 6)
(7.5.18) (7.5.19)
With a zero angle of atla(;k, formulas (7.5.16), (7.5.17), and (7.5.19) acquire the following form: p,/~k ~ 1/(1- 6) -
V 1"'/('1-6""),"+--'4"1K",
Pr/~L~I/(I-6)+ VtI(1 6)'+1,[(1. ~ 2~U11 - 6)
PL
In formulas (7.5.l6') and (7.5.17'), K", = l"oo~N
~Moo~L
(7.5.16') (7.5.17') (7.5.19') and KL =
Ch. 7. An Airfoil in ~ Comprenible Flow
293
Nearly Uniform Flow over a Thin Airfoil
At this case, we consider a flow over a thin airfoil at small angle! of attack. This flow is featured by shocks of a finite intensity being absent and by the eharactcristics on the upper and bottom sides being straight lines with an angle of indin;:ttion of fL = sin _1 (11M",,). To determine the pressure coefficient for the alrfoil, we shall use Eq. (7.4.9). According to this equation, for (In arbitrary point N on the upper side of the airfoil 00
,7.5.20) and for a point L on the bottom side
;;L~2 ("-~cl/YM!"-1
r or a
(7.5.20')
zero angle of attack PN~2~,/VM!"c=j,
PL~-~Jl/"M!"-1
(7.5.20")
An increase in the small angles of attack lead~ to a growth in the error when calculating the pre>,snre on a thin eirfoil in a nearly uniform flow. The aecuron! of these calculations can be increased by usillg lhe second-order aerodynamic theory. Ac('ording to the lattPl', the pressure coeRicient is (7.5.21) where C1
= 2(M!, -
1)-I/z.
C2
=
0.5 (JU;'" -
1)-~
[(;11;" - 2)2
+ kIlI!:.1 (7.5.22)
The pIllS sign in (7.5.21) relatcs 10 the bottorl1 ..,ide of the plate CPL; 0 "-~ a - ~iJ, and the minus Sig:ll, to the upper one (PN.; e = ~,,- ~N)'
Aerodynamic Forces and Their Coetllclents
To determine tile llerodYllflmic prc!;!;urc forcos. we sha! I use formulaa (1.3.2) and (1.3.3), relating them to the body axes x, y (seo Fig. 7.5.1) and assuming that c,.x ,--- O. In this condition, formula (1.3.2) determine.., the 10llgitudinal force X for an airfoil. and (1.3.3), the normal force Y produced by the pressuro:
X = q",,8 r
.~ (5)
peos (n",'x)
ff-,
Y = - q«>Sr \ (8)
pcos (~:y)-¥r
294
pt. I. Theory. Aerodynamics 01 an AIrfoil and eWing
Flg.7.U
Aerodynamic forces for an airfoil in a body axil and flight path coordinate systems
Adopting the quantity S r = b X 1 as the characteristic area and taking into account that dS = dl X 1 (b is the chord of the airfoil, and dl is an arc element of the contour), we obtain the following expressions for tho aerodynamic coefficients: XI(q..,Sr)
Cx =
=
c, ~YI(q~Sc)~
~ peas (h,i> dT.
-fiicos (,;','y)
dT
where dt = dUb, while the cllrvilinear integrals are taken along the contour of the airfoil (counterclockwise circumvention of the contour is usually taken as positi vel. Let us introduce ...............
-
into
-
this
-
-
expression dl = dx/sin
/'-. x),
(fl,
-
cos (n, y) dl -= dx, where dx = dx/b. Next passing over from curvilinear integrals to ordinary ones, we obtain 1
1
c,~ -.i Pb (*)b dx + Jp" (*)"dx o
1
(7.5.23)
"
j (Pb-pJdz
(7.5.24) o where Ph and Pu are the pressure coefficients for the bottom and upper sides of the airfoil, respectively. Using the formula for con version [see formula (1.2.3) and Table 1.2.11, we obtain the aerodynamic coefficients in a wind (flight path) coordinate sy.<:tem (Fig. 7.5.1): (7.5.25) (!I'a = Cx cos Ct + CII sin Ct, clla = CII cos a c'" sin a C II
=
298
Ch. 7. An Airfoil in e Compressible Flow
y
'114I.l.U Determination of Ihe moment
of the forces for an airfoil
With flight angles of auack not exceeding tbe values of Ct we have
~
10-12.
(7.5.25')
For the coeUicient of the moment about the leading edge of the airfoil due to the pressure force (Fig. 7.5.5), by analogy with (1.3.6), we deriyc the formula m,= q:fj1r b = =
q~rb (~xdY- ~
1 {(-'
b
.' px cos (n, y)
(s)
ydX)
~-
dS
..., dS} py cos (n, xl s;
s; (
or
(7.5.26) where Yb '--' Yb'b and II -:; y,/b. We determine t.he coefficient of the cell Ire of pressure for the condition that the point or application of the re.
Y
(7.5.27)
For thin airfoil:;, we may disregard the second integral on the right-hand side of (7.5.26) and in the numerator of (7.5.27). Accord-
296
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
ingly, I
mz =
J(ilb-P~)xdx
-
(7.5.26')
o
cp =
I
I
o
0
J(PIt-Pu)zd.x[l (Pb-Pu)dxf
t
(7.5.27')
By comparing the first and second terms on the right-hand side of (7.5.26), we can estimate the order of the discarded infinitesimals. I
,y
It is determined by the value of \
(dYtdX) dX ~ All, where X = Mb
is tbe relative thickness of the airfoil. Ul'ing relation (7.5.21), we ohtain the longitudinal-force coefficient (7.5.23) corresponding to the second-order aerodynamic theory: C x = cl K 1 2c"K2<X (7.5.28)
+
where I
K. ~
I
J
j (~t·:
(7.5.29) ~:) di, K.~ (~,-~:)d; o o The normal force coclIicicnt in accordance with (7.5.24) is cy = 2c1o:.
+ c'!K'!
(7.5.30)
Ac('.ording to the results obtained, the drag and lift coefficients are c Xn = 2c 1o:.Il 3C2K2
+
+
+
In the particular case of a
symmetr~c
= {dyldx)2 and, consequently, Kl = 2
airfoil,
~t
'-=
I ~2 dx and Kz
~~ =
~2 =
= O. Accord-
o
iagly,
r I
c:ra
=2c t a 2
+2c t
~2ix,
cu.=2c ta
(7.5.31')
o From these two relat.ion.'l. we find c:ra =
c~,
t~-
2c 1 ;-2ctJ ~-d.x
(7.5.32)
o
Equation (7.5.32) determines the relation between the drag and lift coefficients-what is called the polar of an airfoil.
Ch, 7, An Airfoil in " Compreuible Flow
'297
We find the coefficient of the longitudinal moment abont the leading edge by inserting (7.5.2t) inlo (7.5.26'): (7.5.:33)
where
, , A,- o\ (~b'i ~")zd': A.- jo (~b-~") zdz
. B'-.i o
,
For a symmetric airfoil,
Az = 2
j pi eG.
~u
Accordingly,
o
(7.5.34)
(~t-~:)zd'
= -~(I
,...",
~.
(7.5.35)
therefore Al = B1
.
m%= (4cz ) ~idx-.Ct)
=
0;
(7.5.33')
ct
o The coefficiclll or the ('entre of cp = -(clA1 - C2Hz
+ (2cv12 -
For a symmetric airfoil c p =O.5 (1 -4
prcs~llre.
by (7.5.27'), is
'r-
Cl) o.j/(2c 1o.
czKz)
*J. o
pxd:;)
(7.5.36)
(7.5.36')
For a linearized now (low supersonic veiocitie.o;;), lhe cocffident Cz should be taken equal to 1·cro in the abo\'c relations. With hyper.wnic l.:elocUiea or II thin airfoil. formulas (7.4.6) nnd (4.6.12) can be used to calculate the pressure coeflicients ill (7.5.2a). (7.5.21i), (7.5.26')and (7.5.27'). The pre!":~ure ("oeftici('nt 101' the bottom side in ac("ordance with (li.6.12) i!; Pb-(ex-~b)'[I/(I-6)+Vl/(1
6)'·· 41KJ,]
(7.5.37)
while for tlle upper side, the \'alue of this coefficient by (7kG) is
p"-(ex-p",'[1!(1-6)-1/1!(1
61'-i-4IK;.1
(7.5.38)
where Kb = M.., let - ~b) and Ku = M_ (ct - ~u). Let us find the relalions for the aerodynamic coefiicients of a wedge-shaped Airfoil (Fig. 7.5.G). Since for the bottom contour we have (dyldx)., = tan ~b. and fOl' the upper one (dyldx)u = tan ~u. and taking into account that the pressure coeiTicicnts Pb and Pu
are ('onstanl, wo obtain from (7.5.23) and (7.5.24):
c.•. =
-Pb tan ~b
clI
""
+ Pu tan
~u
(7.5.39) (7.5.40)
Pb -Pu
The angle ~b should be considered negative in formula (7.5.39). We fmd the moment coefficient. from (7.5.26): mr = -0.5 (Pb -
Pu) -
0.5
tan i ~b
-
Pu
tan 2 Pu)
(7.5.41)
while the coefficient of the centre of pressure by (7.5.27) is cr = 0.511
+ (Pb tan! ~b - P. tan2
~u)i(Pb- Pu)1
(7.5.12)
For a symmelric airfoil, we have
Pb
In the above relations, the pressure coefficient is determined by accurate relations obtained in the theory of an oblique shock. We also lise this theory to fmd the pressure coefficient Pu for ex. < ~u' For ex. > Pu, the rmding of Pu is as..,ociated with calculation of the Pralldtl-Meyer flow on the upper side of the airfoil. The relations obtained for the aerodynamic coefficients of the airfoil relate to arbitrary values of the nose angle (h, ~u) and tho angle of attack. If these values are not largo, the second-order t.heory can be IIsed to calculate the coefficients. According to this theory, in formulas (7.5.31) for c!l.a and c lla ' we must assume that K, - ~, +~.
K, - ~, -
~
When calculating the moment coefficients (7.5.33) and (7.5.36), wo must proceed from the fact that
A, - 0.5 (~b -;- ~").I A,
= 0.5 (~b -
~").
B,
= 0.5 (~, -
~:)
Ch. 7. An Airfoil in
II
Compressible Flow
For a symmetric airfoil, we have c:I;.=2cda;Z~~Z),
clIa =2c j a.
t
m,=(2cz~-cj)a;,
cp=O.5(1-2--2-~)
J
299
where the angle ~ i~ chosen with a minus sign (for the bOltom side). The use of formulas (7.5.37), (i . .'J.38) allows HS to determine the aerodynamic coefficients (7.5.39)-(7 . .').43) corresponding to hypcrsonic velocities of a thin airfoil at a small angle of attack. 7.6. Sideslipping Wing Airfoil Definition of II Sideslipping Wing
...... Let 1\S assume that an inlinite-span rectangular wing performs longitudinal motion at the velocity VI (normal to Lhe leading edge) and lateral motion aiong lhe spall at tho velocity V 2 • The resultant velocity V = VI + V2 is directed toward the plane of symmetry of the wing at the sideslip angle ~. A rectangular wing performing such motion is said to be a sldC!:llipping one_ The flow over a sideslipping wing does 110t change if we consider inverted now in whicll a rectangular wing encounters a stream of air at the velocity - Y I perpendicular to the leading edge, and a stream at the velocity - V 2 in the direction of tbe wing span (Fig. 7.6.1). The nature of the l10w is the same if the resultant velocity V"" is parallel to the initial velocity - Vi (soc Fig. 7.6.1) and the reclangular wing is turned through the angle x = ~ (fig. 7.0.2). Such a wing, which is also a sideslipping one, is usually called an infmitespan swept wing. The angle x is determined as the sweep angle relative to the leading edge. Let us consider some features of Hie flow over sideslipping wings. The Dow about such wings can be divided into two nO\Vs: a longitudinal one (along lhe span of the wing) characterilcd by the velocity V"oo = V 00 sin x parallel to the leading edge, and a lateral ooe depending on the magnitude of tile normal velocity component to this edge Vnoo = V 00 cos x. The distribution of the velocities and pressures along the wing does not depend on the longiludinal Dow, and is due only to the lateral Dow at the velocity V n _ = V ... cos )to The nature of this flow and, therefore, the pressure distribution change depending on the configuration of the airfoil in a plane normal to tho leading edge, and on the angle of attack measured in this plane. Accordingly. the aerodynamic characteristics of the airfoil are the same as of a rectangular (unswept) wing airfoil in a Dow with a free-stream velocity of V n00 at the indicated angle of attack.
300
Pt. I. Theory. Aerodynamics of lin Airfoil and a Wing
I c:::=J (')
V,
(b~)_ _V'_-----,
I
Rt·7.6.t
Motion of a wing with sideslippiog: o-Iongiludlnal motion at an Inflnlte-apan rtctangulaf wins: b-Iatenl motion; _ ...ultant moUon at tbe sldeallp angle,
~:'~:i~p~ing I-wing
wing:
6urtac~:
2-nlrfoU In a Sf'ction along 8 nomal; ,,_alrloilin a eectlon along tbe Oow
This is exactly the content of the sld(>5lip elTed which to a considerable extent determines the aerodynamic properties of finitespan swept wings. The term swept is conventionally applied to a wing in which the line connecting the aerodynamic centres (foci) of the airfoils (the aerodynamic centre line) makes with a normal to the longitudinal plane of symmetry the angle x (the sweep angle). In aerodynamic investigations, the sweep angle is often measured from another characteristic line, for example, from the leading or trailing edge (Xl' x s), from lines connecting the ends of selected elements of a chord (XI/4' Xlii' ••• ). from the lines of the maximum thickness of the airfoils. etc. (Fig. 7.6.3a). If the leading edges are curved or have sharp bends, the sweep angle will be variable along the span. The flow over swept wings in real conditions is distinguished by its very intricate nature, due primarily to the partial realization of the sideslip effect. Figure 7.6.3b shows a possible scheme of such flow over a swept finite-span wing with a sufficiently large aspect ratio.
Ch. 7. An Airfoil in
III
Comprenible Flow
301
ra!
Fig. 7.6.3
Finite-span swept wing: a-dcalgnatloll of the sweep angles: b-sllb8Ol\lo now oYer a swept wing
The middle region of the wing (region I) is characterized by the mutual influence of neighbouring airfoils reducing the sweep (sideslip) angle. Here the mid·span effect appears that detracts from the aerodynamic properties because of the decrease in the sideslip effect. In region III, the tipeiTect acts. It is due to an appreciable deflection of the streamlines in comparison with a sideslipping wing. And only in region II, distinguished by a more uniform flow, is the curvature of the streamlines relatively small, and the sideslip angles ~ are close to the sweep angle x. In aerodynamic investigations, this region of the wing can be represented in the "pure" form as a region of a swept (sideslipping) infinite-span wing arranged in the flow at the sweep angle. It will be shown in Chap. 8 that the conclusions relating to such wings of a small thickness can be used to calculate the flow over separate parts of swept finite-span wings at supersonic volocities . ...,odynamlc Ch.r.derlsflcs of • stcIullpplng Wing Alrfo[l
When determining such charactoristics, a sideslipping wing is considered as a straight one turned through the sideslip angle x. In this case, the airfoil of a sideslipping wing in a normal section will evidently be the same as that of a straight one. The airfoil and the angle of attack in a plane normal to the leading edge differ from the airfoil and anltie of attack in the section along the flow (see Fig. 7.6.2). The chord in a normal section bn and the chord b along the flow are related by the expression bn = b cos x. The angle of attack Ctn in a normal section is determined from the expression sin aD
= hlbp. = hl(b cos x)
= sin Ct/cos x
(7.6.1)
302
Pt. I. Theory. Aerodynilmics of iln Airfoll ilnd
iI
Wing
where a is the angle of attack in the plane of the (low. It is evident that at small angles of attack we have
= WCOS x
an
(7.6.1')
If at a point on a wing airfoil in flow without sideslipping at the velocity V"" the pressure coefficient is F, then when the wing is turned through the angle x the pressure coefficient Pn at thei corresponding point is the same, Le. 2(P. - P .)/(p. 1"'; cos' x) = 2 (p - p .)/(p. 1".) Accordingly, for the airfoil of a sideslipping wing, the pressure coefficient calculated from the velocity head q ... = O.5p... V:, is
P
p" = 2 (P. - P.)/(p.1".) = COS'" (7.6.2) With a view to (7.6.2) and in accordance with (7.5.23), (7.5.24), and (7.5.26), the aerodynamic coefficients for the airfoil of a thin sideslipping wing are c:r,>< = Cr cos! it, = cos! x, m~,;( = m~ COS·" (7.6.3)
C",.
C,
Evidently. by (7.5.25'), we have c~a'" =
(c:c
+ c,a) cos! " = crr, cos
2
x
Since the drag force is determined not in the direction of the velocity component V"" cos x, but in the direction of the free-stream velocity V... , the coellicient of this drag is
c:c:;( = c~a"
COS" =
c:c. cosS "
(7.6.4)
All these coellicients arc determined for a velocity head of q"" = = O.5p""P"". Inspection of formulas (7.6.3) for c,.,. and mz,l< reveals that the coefficient of the centre of pressure c p = -m,.,./c,.y' corresponding to small angles of attack does not depend on the angle of attack, i.e. C p = -m~/c/l' Compressible Flow. According to the linearized theory, the pressure coefficient for the airfoil of a sideslipping wing in a subsontc compressible flow can be obtained from the corresponding coe1licient for the same '.... ing in an incompressible flllid by the Prandtl-Glauert formula (7.1.14), substituting the number Moo cos x for Moo in it:
p,.=P",teIV1-M!.cos2x
(7.6.5)
or, with a view to (7.6.2)
p,.
=
Pic cos
2
xlVi
M:" cos2 it
(7.6.5')
The relevant aerodynamic coefficients are obtained from (7.5.23). (7.5.24), and fI.5.26), and are found with the aid of formulas (7.6.3)
Ch. 7. An Airfoil in .. Compressible Flow
:~ (M""~~W':~--V_:_;___ i",
303
(b)
~
'iC-Si,2-,u_
~i~~:itp'Ping
wing with slIhsonic (a) and supersonic (b) leadintt c(lgros
and
(7.5.4) whose right-hand sides contain the qnantity Vi M!o cos2 x in the denominaLor. Particularly. the coerricieillsof the normal force and the longiludinal moment. are Cll.~ =cu cos2.xf1/1
M:" cos2.x; m1 , >! = rn; cos2.;;JV 1-· M~ cos:! % (7.6.6)
It follows from t.hese relations that for thin airfoils, the coeITicient of the centre or pressure c p = -m 1 ,)I,lcy,)I, depends neither on the sideslip (sweep) angle nor on tho eomprr.ssihility (the number ill ",,). The usc of a sideslipping wing produces the same flow efT('ct that appears when the frcc-stream velocity is lowered from F to V ... cos x (or the Mnch number from M ... to M"" cos Yo). HC're, Ilalurally. the local velocities on an airfoil or the shlcslipping wing also decrease, and this, in turn. le8tls to diminishing of tho rarl'r;lction and. as n result-. to an increase in the critical Milch nnmber. The Jatter call be dc.termined from its ]mown v31ue JH eo C r for a straight wing of the same shape and flngle of aLtack as 11m airfoil of Ill(' sideslipping wing in a normal seclion: <&>
Moo cr,"
= M ... cr/cos'fl. x
(i.G.7)
Supersonic Velocities. Let us llSl'umc that aL a ~upcr!'=ollie frcestream velocity (V ... > a .... Moo> 1), tbe sweep angle satislips the inequality )t > n/2 - ..... , according to which cos x < sin 1-1"" and, consequently. Vn"" < a,., = V"" sin ]l ... , Le. the normal component to the leading edge is subsonic. lIence. the flow over the seclious of a sideslipping wing is subsonic in its nature. In the case beiug considered. the swept edge is called suhsonic (Fig. 7.6.1a). At increased Dow veloeitie!'=. the normal velocily component may become higher than the speed of sound (Vn... > a ... = V"" sin 1-1 ....). so that x < n/2 - 1-' ... and cos x > sin 1-'00' In this case, the Dow
304
Pt. I. Theory. Aerodynamics 01 an Airfoil and a Wing
over the airfoils of a sideslipping wing is supersonic. Accordingly, the leading edge of such a wing is called supersonic (Fig. 1.6.4b). Let us consider the calculation of the supersonic Dow over a sideslipping wing in each of these cases. Supersonic Leadlng Edge. The Dow over such a wing can be calculated by the formulas obtained for an infinite-span thin plate provided that the free-stream velocity is V n"" = V"" cos x > a"", and the corresponding number M 0. "" = M "" cos x > t. The angle of attack of the plate aD is related to the given angle of attack a of the Sideslipping wing by expression (7.6.1). or at small angles of attack by (7.6.1'). Using formula (7.4.9) and substituting aD. = a/cos x for ~ and MD"" = M "" cos x for M "" in it. we obtain a relation for the pressure coofficient in a plane perpendicular to the leading edge:
j;-I± 2a/(eosjx V M'.eos'x-I) In this formula. the pressure coefficient Ii is rolated to the velocity hoad qn = O.5kp....M1D oo. To obtain the value of the pressure coefficiont rolated to the froe-stream volocity head q"" = O.5kp...,M1"", we must use formula (7.6.2) according to which p,. _ ± 2a eoslxtV M"':'-:Cco:-:s"'.c-'I (7.6.8) In (7.6.8). the plus sign determines the pressure coefficient for the bottom sido of a wing. and the minus sign for the upper side. In accordanco with formulas (7.4.20)-(7.4.22) (replacing a with aD and M ... with M"" cos x in them). and also with a view to relations (7.6.3) and (7.6.4). wo find a rolation for the aerodynamic coefficients of a sideslipping wing airfoil cll
.,.= 4(111 cos2 x/V M!,cos:l.x-1
C=.1C=
4a.~cossx/VM!. cos2 x-1
') (7.6.9)
m"a>l= -2a"coszy'/V M!,cos 2 x-1 Upon analysing these relations. we can establish a feature of swopt wings consisting in that in comparison with straight ones (x = 0). the lift force and drag coefficients, and also the coelliciont of the longitudinal moment of airfoils (in tlteir magnitude) are smaller at identical angles of attack an along a normal to the leading edge. The physical explanation is that in now over a Sideslipping wing Dot tho total volocity head g... = 0.5p ... V!. is realized, but only a part of it. gO. = g ... cos2 x, and flow in the direction of the oncoming stream occurs at a smallor angle of attack than in the absonce of sideslip (a < an. aa = a/cos x). Subsonic Leading Edge. The Dow over sections (',orresponding to the motion or a straight wing with the number Mn_ < 1 is investi-
Ch, 7. An Airfoil in
II
Compressible Flow
305
gated with tlle aid of the subsonic or transonic (combined) theory of flow over an airfoil. The drag and lift forceg arc determined by the laws of subsonic flows cllar;lcterizcd by interaction of the flows on the upper and bottom sides of a wing that manifest.s itself in thE! gas flowing ovcr from a region of lligll pressure into a zone \\'itll reduced pressure values, V\'U\'C losses may appear olily in sllpcrcritical flow (Mno<> > M «> cr) when shocks form on the surface, If M" '" < lW 00 c r, then shocks and, therefore, \\'ave drag are absent. This conclusion relates to an inlinite-span wing, For lillite-span wings, wave losses are always prescnt because flow O\'er their tips is allccted by the velocity component V sin %, A result is the appearance of supersonic flow properties and of a wayc drag, The three-dimcnsional theory of supersonic flow has to be llsed to study this drag. 00
Suction Force
As we have established in Sec. 6.3, a ~mctioll force appears on the leading edge of an airfoil over which an incompressible fluid is flowing. The same effect occllrs when an airfoil is in It subsonic flow of a compressible gas. The magnitude of tile snction force is affected by the sweep of the leading edge of the wing. To calculate this force, we shall use expression (o.il.25), which by meallS of the corresponding transformations can be made to cover the more general case of the tlow over an airfoil of a wing \vith a swept leading edge (Fig. 7.6.5). Let us consider this transformation. In au iu\'iscid flow, the free-stream velocity component tangent to the leading edge of a sideslipping wing docs )lot change the Iield of the disturbed velocitics, and it remains the sume as for a straight wing in a l10w at the velochY V n "" ,.., V"" cos %. The forces acting on the wing also remain unchanged. Therefore, the following suet ion force acts on a wing element dz o with a straight edge (in the coordinates .1'0' zo): dl'o = ;tpc! dz o (7.G.1D) where in accordance with (6.3.28') c~= lim lu:(xo-x~.eo)l Xo-Xs .eo
A glance at Fig. 7.6.3 reveals that dT o = dT/cos = u/cos~, and Xo - Xs.eo = (x- x~.e) cos ~. Insertion of these values into (7.6.10) yields
~,
dz o = dz/cos
~,
Uo
dT/dz=npc:z.V1-i- tan:z.x
(7.().ll)
c:z.,._", lim fu 2 (x-xs,e}J
(7.().12)
where Z-X s . r 20-01715
FIg. 7.6.5
Suction force of a sidellipping wing
Expressions (7.6.11) and (7.6.12) can be generalized for compressible Dows. For this purpose, we shall use expressions (8.2.4) relating the geometric characteristics of wings in a compressible and an incompressible Dows. It follows from these relations that all the linear dimensions in the direction of the x-axis for a wing in a compressible Dow are 'V 1 - M!, limes smaller than the relevant dimensions for a wing in an incompressible Dow. whereas the thickness of the wing and its lateral dimensions (in the direction of the z-axis) do not change. Accordingly. we have
x=xlc'Vl-M!... bdz=b1c dz 1C 'V1-M!, tanx=tanx!c'V1-M;' From the conditions that x find
= XIC 'V 1 -
M~ and
(7.6.13)
=
we
i.e. (7.6.14)
'therefore, the pressure coefficient for a compressible Dow is determined by the Prandtl-Glauert formula: (7.6.15) ~he suction force T in its physical nature is a force produced by the achon ?f the normal stress (pressure) and at low angles of attack is determmed from the condition T ~ aY a. The corresponding suction
Ch. 7. An Airfoil ill 11 Comp.eS$ible Flow
307
force coefficient is C;(.T = TI(q"S".) = ac y,,' Since the dimensions of a wing in the direction of tho y-axis do not Ch
With a vie\,,· to the relations for an airfoil of a sideslipping wing, we ha.ve c,~. T
P<>o1';,
p..r;,
Cx,T---y-bdz= VI-,ll!, .~bjc
V--, I-Jf;,dz
1e
dT = dT,c
But since dz = dz lc , then (7.6.17)
dT/dz = dT1e.tUle
The right-hand side of Eq. (7.6.17) corresponding to an incompressible flow is determined by formula!; (7.6.11) and (7.6.12): dT\cldz}c.
'-=
:tP""CI~
Vi -+- tanl! i(IC
where
cre=
lilll [UTc "Ie-Xs.e. Ie
(.rIC-.Ts,e.
Icl]
In accordance with (7.6.17), (7.6.13), and (7.6.14), we obtain cfc= lim [u2(x-x~.e)lV1-J1~; x ......s.e
Vl-
tan 2 %IC
Consequently, dTldz
= npaoc2V1 + tan:! x
M';.
(7.6.18)
where c~=
lim [u:!(x-xs.e)j
(J.6.19)
X-Xs_e
I t follows from the abo .... e formulas that at a given value of the number JU co, the suction force depends on the sideslip angle and on the nature of the change in the velocity u within a very small vicinity of the leading edge. We can assume that a finite-span swept wing is also characterized by a similar relation for the suction force.
8. t. Linearized Theory
of Supersonic Flow over a Finite-Span Wing Llnearlzatfon 01 the Equation for the Potenflal Fundlon
Let us consider a thin slightly bent fmite-span wing of an arbitrary planform in a supersonic Oow at a sIllall angle of attack. The disturbances introduced by such a wing into the flow are small. and for investigation of the Dow we can use the linearized theory as when studying a nearly unirorm now near a thin airfoil (see Sec. 6.2). The conditions for such a flow are given for the velocities in the form of (6.t.t). If we are considering a linearized three-dimensional gas flow. these conditions are supplemented with a given velocity component along the z-axis. Accordingly, the following relations hold for a linearized three-dimensional disturbed Oow:
V. = V
lSI
+ u.
V..,
= V,
V:: = w
(S.Li)
where u, v, and UJ are the disturbance velocity components along the z, y, and, axes, respectively. In accordance with the property of a linearized flow, we have
"<:
V _.
v
<
V _.
w
<
V_
(8.1.2)
Theae conditions make it possible to linearize the equations of motion and continuity and thus simplify the solution of the problem on a thin wing in an inviscid steady flow. In the general form. the equations of motion of such a flow are obtained from system (3.t.t7) in which .... '.:: 0, aV~/8t = aVlllat = aViat = 0:
V.
a;:. +V" a:u. +V:: 8~:c =
V.
8~:
V. 8Jzr
+VII
_+.~
a:; +V:: iJ~u =-f.* ::r = - i-. ~
+ VII a:,~ -~ 1':
(8.i.3)
Ch. 8. A Wing in
ill
Supersonic Flow
309
We shall adopt the continuiLy equation in the form of (2.4.4). Calculation of the partial derivatives yields
p(a~% + 8:;, + iJ~2 )~'V:O:~'r'Vv* :"V:*=O
(8.1.4)
It was shown in Sec. 5.1 that the equations of continuity and motion can be combined into a single equation relating the velocity components to one another. By performing transformations similar to those made in Sec-. ;i.f, we can write this eqnation in the form (Vi-a:!)
iJ~~x
..
(V~-Q:!)
::" .
+V:o:VII(iJ~x + iJ~" ).~.ll'xVt
+ l-'yV,
(V~-a:!) o;~. (0;;'(
+ ~;:?)
((J;:" -7- iJ/~~z ) ... 0
(8.1.5)
Taking into account relations (2.:l.2) for the potential function, and also the condition of equality of the cross partial derivatives (a 2fPlox ay = a2ff,'()Y ox, otc.), from (R.i.;'}) we derive an equation for the velocity potential:
(Vi - a2 ) ::~ -T- (V~ - a:£) :~
+(Vi _ a2)
~-2V:o:V!1 iJ~~:~ -:-2V:o:V: :::: .~2VyV,
8;:
:;:1
=0
(8.1.6)
Equations (8.1.5) and (8.1.6) are the fundamental d(tjerential equations of gas dynamics for three-dimensional steady gas flows. The first of them relates to the more general case of a vortex (non-potential) now of a gas, while the second is used to investigato only vortex-free (potential) flows. Since the flow over thin wings at small angles o[ attack is potential, we can use Eq. (8.1.6) for the velocity potential to investigate this ftow. To linearize Eq. (8.1.6), ,... hieh is a non-linear differential equation, we shall introduce into it expression (7.1.::n for the speed of sound, and also the values
Vi-
V z = V"" +u, V!I =V, V l =w = l'!. .!- 2V""u, V::::::::: v 2, 'Vl w2 ;
=
"'~"'-+'I'. After analysing the order of magnitude of the terms in the obtained equation in the same way as we did in Sec. 7.1 when considering a
310
Pt. I. Theory ..... rodynamics of lin Airfoil .nd a Wing
f!l
(oJ
"
Fig. 1.1.1 Thin wing In a linearized floVo"
nearly uniform plane flow, we obtain a linearized equation for the "elocity potential of a three-dimensional di5turbE'd flow in the following form:
louncllry ConciHlcns
Investigation of the Dow over a thin finite-span wing consists in solving the linearized partial dHIerential equation (8.1.7) of the second order for the velocity potential
)s =
+"*" COs(~)+ :~
::
cos
(~)
C05«(i)=O
(S.1.9)
Ch. 8. A Wing in " Supenonic Flow
311
Here
The direcLion cosines of an outward normal to the surface are found by formulas of analytical geometry: /'--
cos (n, x) "'" -A iWtJJ',
/'--
cos (n. y)
/'--
A;
cos (n, z) = -it
araz
(8.1.10)
in which ,I ~ II ,- (a//8x)'
+ (8//0,)'1-'1'
(8.1.11)
while the fUlIction I is determined by the shape of the wing surface [the fllIlelion y -= lu (.1'. z) for the upper surface, alld y = Ib (x, z) for the bottom ollei. A linearized (nearly uniform) [low is reali~ed provided that the wing is thin and, consequently, alliJx <: 1 and iJlliJz <: 1. According/'-.
/'--
/'--
-afax, cos (n, y) = 1, and cos (n, z) = -iJlia:,. (Utp':ax) allax <: 1 and (fflp' /iJz) iJ/liJz
Iy, cos (n, x) = The conditions observed in the can be written
b
, '\' (x) dx equals tile circulation
-,
r
in the section z being considered,
the lift coefficient for an elementary surface equals the "alue = (2/V cob) r (z), whence the circulation in the given section r (,) ~ O.5c;.V ~b (8.1.13) According to the conpling equation (8.1.13) Isee (6.4.8)], UpOIl moving to a neighhouring section with a dillerent lift coefficient. the velocity circulation also changes. This change is dr (,) ~ (dr/d,) d, ~ O.5V ~ Id (c;.b)/dzl d, (8.1.14) In accorclauce with the vortex model of a wing treated in Sec. 6.4. an elementary bound vortex belonging only to the section being con~
a12
pt. I. Theory. Aerodynomics of on Airfoil ond ., Wing
sidcred passes within the contour enclosing t.he adjacent section. This vortex turns and is cast off the trailing edge in the form of a pair of elementary free I)()rtices forming a vorU'x she
(aq>'iiiy), __o
(8.1.15)
in which the left-hand side corresponds to the velocity Vn directly above the vortex sheet (y = +0), and the right-hand side, to the velocity under it (y = -0). Hence, condition (8.1.15) expresses the continuity of the function orr-'lfJy when passing through a vortex sheet. In addition, the condition of the continuity of the pressure is satisfied on a vortex sheet. According to (7.1.5'), we obtain the relation (8.1.16) that also expresses the continuity of the partial derivative fJcp'18x when passing through a vortex sheet. Let us consider the fiow over a wing with a symmetric airfoil (yu = -Yb) at a zero angle of attack. In this case, there is no lift force and. consequently, no vortex sheet. Owing to the wing being symmetric, the vertical components of the velocity on its upper and bottom sides are equal in magnitude and opposite in sign, i.e. v (x, +y, z) = -v (x, -Y, z). On plane :cOz outside the wing, the component v = 0, therefore a~'liiy ~ 0 (8.1.t7) Let us assume further that we have a zero-thickness wing of the same planform in a flow at a small angle of attack. The equation of the wing's surface is y = f (x, z). A glance at (8.1.12) reveals that the vertical velocity components Vn = 8cp'/oy on the upper and bottom sides of the wing are identical at corresponding points. Since we arc dealing with a sufficiently small angle of attack of the wing, the same condition can be related to the plane y = O. At the same time, such a condition can be extended to the vortex sheet bebin" the wing, which is conSidered as a continuation of the vortices in the plane y = O. Therefore, at points symmetric about this plane, the
Ch. 8, A Wing in
Supersonic Flow
components Vn are the same, i.e. D<{l' (x, -y, lay. Consequently, the additional polential relative to the coordinate y, Le.
'1" (x, -y, ,)
~
= (hr'
313.
(x, +y, z)!
an odd function
- , ' ix, - y, ,)
(8,L\8}
Accordingly, the partial derivative uq//Ox on tile bottom side of the vortex sheet equals the value -chp'lux on the upper side. But the equality of the derivatives U(v' 'OJ.' \\'as estaLdi.-;IJed from the CODdition of pressure continuity. This equality C
(8.L20}
Components 0' the Tol
The solution of Eq. (8.1.7) for the lHJditiollal potenlial I.p' must correspond to the considered bOllnuary condiliOlls. Tlli:> solution can be obtained for a wing with a given plan form by summation of two potentials: the first, ((I;, is fOHlld for an idealized flat wing 1 (Fig. 8.1.2) of the same plan form ;1.:; the given on(', but ,yith a symmetric air/oil, and at a zero angle of attack (r,l. = 0). The second potential, ((I~, is evaluated for a different idealiwd cambered wing :: 0/ zero thickness, but for the given angle of attack a. The surface of nn idealized symmetric airfoil ,\·ing can be given by the equation
(S.UI} and of a zero-thickness airfoil wing, by the 1!(luation of the surfaceof the mean camber lines of the airfoil
y =O.5Uu+/b)
(S.L22}
Hence, the total potential of the given wing is (~' =-
q; .r'
(I·;
(8, 1.23}
314
FIg.
Pt.
r.
Theory. Aerodynamics of an AirFoil and a Wing
a.u
Linearized supersonic now over a fmite lhickness airfoil at an angle of attack (I.: 0= 0, symml'tric alrroil, and givl'n thickness distribution; 2-a =!' 0, u'ro. th.tckne~ (tb" airfoil colncid~s with thp mpan cltmbl.>r line); 3-a 0= 0, z~ro thickncss (the alr'oll coincides wLth the mean cambt'r lm~); 4-a oF 0, zero thicknc$$ (the pl~lc airtoil colncid~s "'ilh the chord)
I-a
The flow past wing 2, in turn, at a; *0 can be represented as a flow past wing 3 with a uro angle of attack and a surface equation y = 0.5 (fu + fb) and an additional flow superposed onto this one. The additional flow is formed near wing 4 in the form of a zero thickness plate coinciding with the chord of the initial wing having an angle of attack a; (Fig. 8.1.2). In accordance with this flow scheme, the total potential for the given wing is ~' = ~; + ~; +~; (8.1.24) We can use this value of the velocity potential to calculate the -distribution of the pressure coefficient
p = p; + Po + Po
(8.1.25)
and then fmd the drag and lift forces. By (8.1.24) and (8.1.25), the total drag of the given wing is composed of the drags produced by wings 1, 3, and 4, Le. Xa = Xl + X3 + X, (8.1.26) Introducing the notation Xl -T- X 4 = XI and going over from the forces to the corresponding aerodynamic coefficients, we have
q:'~w
=
q:;". + ~~~:4 q. .X~". + q:;". =
(8.1.27)
Ch. 8. A Wing in ~ Supersonic Flow
315
In accordance with (8.1.27), the wing drag coefficient c"a consists of the drag coefficient CliO of a symmetric wing at cVa = 0 and the additional drag coefficient cx.; due to the lift force and calculated for a zero-thickness wing at c y " O. The coefficient C;r.h in turn, consists of the induc(>d wav(> drag coeffi.cient calculated for the case when the influence of vortices is absent and an additional induced ,·ol·tex drag cocfficie-nt due to the span being Hnite anel to the formaliOIl in this connection of a vortex sheet behind the trailing edge. By analogy with expression (8.1.2G) for the drag, let us gh'c in the general form a relation determining the total magnitude of the lift force of a wing }' a = Y 1 .•- }':, I' Y4' A glance at Fig. 8.1.2 reveals that wing I baving 11 symm('tric airfoil and arranged at a zero angle of attack does not create a lift foree 0'1 =O).Consequentiy, the total lift foree of the wing: is 1" a = }r:1 .!. Y'~ (8.1.28)
*
and the correspollllillg coefficient of this force is
(8.1.29) Hence, according to the approximate Iinearizerl theory of Oow, the thickness of a wing docs not aHect the lirt force. Wing ~ pro{luces a constant lift force that docs not depend on tiw angle of attack. It corresponds to the "alue of this forc(> at a zero angle of attack and 11 given concavity of a wing. A lift force due to the angl!' of at.lack is p~oduced by wing 4- and, therefore, depends on the planform of the wlIlg.
The iindillg of tile pressure distribution, the resultant forces, and the releyant aerodynamic coefficients with a ,'ie\\, to their possible resolution into components according to formulas (8.1.27) and (8.1.29) is the basic problem of the aerodynamiCS o[ a finite-span wing in a nt'ariy uniform supersonic now. Features 01 Supersonic Flow over Wings
When determining the aerodynamic characteristics of wings, we mnst t.ake acconnl of the features of the supersonic flow over them. These features are due to the specifiC property of supersonic flo\\'s in which the disturbancps propagate only downstream and within tile con(ines of a disturbance (Mach) cone with all apex angle of f-l "" = sin -l (1/Jl1 00)' Let us consider a supersonic flow oyer a thin wing having an arbitrary plan form (Fig. 8.1.3). Point 0 on the leading edge is a source of disturbance!' propagating dow.nslream within the conlines of a Mach cone. The Macll lines OF and OG are both ahead of tile leading edges
318
Pt. I. Theory, Aerodynamics of an Airfoil and a Wing
Supersollic flow over n wing: G-wlng with subsonic
rdgr~;
b- wing with
sll~l'$onlc
cdg\"s
(Fig. 8.1.3a) and behind them Wig. 8.1.3b). The arrangement of the Mach Jines at a given wing planform depends on the number M .... In the Jirst case, the nnmbcr M ... is smaller than in the second one, and the angle of inclination of the Mach line ""OIl > nl2 - x (x is the sweep angle). The normal velocity component to the leading eelge is Vn ... =V""cosx. Since cosx<sin 11 ... = 11M"" and V OIl/a ... = = M ... , the normal component V n... is evidently smaller than the speed of sound. The flow of a gas in the region of the leading edge of a swept wing for this case was considered in Sec. 7.6. This flow corresponds to the sllbsonic flow over an airfoil that is characterized by interaction between the upper and bottom surfaces occurring through the Jearling edge. Such a leading edge is called 8ubsonic (Fig. 8.U.). Upon an increase in the now velocity, wben the tone of disturbance propagat.ion narrows and the Mach lines are behind the leading edges as shown in Fig. 8.1.3b, the normnl velocity component becomes supersonic. Indeed, examination of Fig. 8.1.3b reveills that the angle of inclination of the Mach line /1 ... < rr./2 - 1(, bence sin ~ ... = = 11M ... < cos Y., anti therefore V n_ = V <» cos X .> a-. Such a leading edge ucalled 8uperBonic. The now over airfoils in the region of the leading edge is of a ~upersonie nature whose feature is the ab~ence of interaction between the hottom and the \lpper surfaces. If the l-loch line coincides with the leading edge (x = ';[/2 - ~ ... ), such an edge is sonic. It is quite evident that in thiscilse the magnitude of the normal velocity componen t to the edge equall'! the speed of sound. Let us introduce the leading edge sweep paralllf'!ter n :"'- tan xl Icot /1 .... For a supersonic. Jeading edge, cot ~l"" > tan x, therefore
Ch. 8. A Wing in a Supersonic Flow
317
1. For suhsonic. and SOllie lending edges. we have n > 1 and It -= 1. respectively. hecause! in the first rase rot !J."" < tan x and in the second one rot J.l "" -- tan x. By analogy with thli' J(lading edges, we can inlrodllce Ihe ronrept n
<
of Sllbsoni('. "'onir. and sllpc>rsollie lips (side edgcs) and trailing edges of a wing, Tip CD wilh all allale of illclinaion Vt to the direclion of the free-stream relocity smaller tlutn the Mach anale (Fig. 8.1.3a) ig called subsonic. The velorilycompollent normal to a lip CInd equal to = l' "" sin Vt is lower thnn the speed of SOl1nd ill the giYen case. Indeed. sincu a x ,V .'>in p"" nnd Pox» Vt. we ha\'c Vn < a"". It is ohvious that the leading edge sweep parameter II > 1. The part of the wing surface with a sllhsonic tip is inside !.he region cut ()[( by the 1I.-[ach rones is.' J.l.,., (Fig, Fl,1.:1b), In this case, t.he normal eomponent V" '" ,_ 1'..., ."in '\\ i.'< hight:'r than the speed of SOHnd a"" - VoosiJl 1-1"". Similar reasoning can be related to the trailing crlgc of the wiug. Fig, X,1.3a shows a ."ubsonir trailing edge ('Vtr < J.1 ""; 1'" < a",,), alld Fig, R.1.~/)-a supersonir on(' ('Vtr > 11 ... ; 1'" "" > 1I",), The above analysis allow." one 10 e:-tablish the qnalilati"e difference between supersonic 8mi stlh:-onlr nnw over wing.". This difference manifests itself in the diHerl'nt influence of the tips and trailing edges on the now over thli' entirl' wing surface. If in a ~upersollir flow. the tips and trailing edges do 1I0t afiect the flow o\'er the wing (Fig. 8.1.3b), or this innuenr,e i.-;. limited to the part of the surface adjoining these tips and eugt:'s (Fig. 8.1.3a), in a subsonic. flow the action of the tips aud trailing edges manifests itself on the entire surface because the di."turbances can propagate both downstream and u pstrt:'am.
"n""
N
«0
<&>
8.2. Method of Sources To sol"e the problem on determining the aerodynamic characteristics (Pl' cxo) of a thin symmetric airfoil wing of an arbitrary pI aliform in a Hearly uniform silper."onic flow at a zero angle of attack (c v -::: 0), we shall usc the method of sources. Sources ill an incompressible fluid are treated in Sec, 2,9. The velocity potential of the incompressible now from a point source at the origin of eoorriinates of the system Xlc. Ytr. Z(C' aecording to (2.9.14), is
Ifle = -qte/ (4;r. Yrte
+ Yfe
: zle)
(8.2.1)
818
Pt. I. Theory. Aerodynamics of an Airloil and a Wing
where qlc is the flow rate of the source, i.e. the volume of fluid flowing out of the source in unit time. The method of sources deals not with individual point sources, but with sources continuously distributed over a part of a plane. usually the coordinate plane xOz. Let dqlc be the elementary volume flow rate of the fluid produced by the sources 011 the small area dalc = df.lc d~IC in the plane xOz. I-Ienc.e the derivative dqlc/dole = QIC' known 8S the density (or intf'nsity) of !';ource distribution, determines the strength of the sources per unit area. If ±v is the vertical component of the velocity on the area dUIe (the plus sign signifies that the fluid is discharged upward from the sources, and the minus sign-downward), the elementary volume flow rate is evidently dqlc = 2v dalc and, therefore. (8.2.2) QIC = 2v The following potential corresponds to an elementary source: dqJlc= -
QIC
du,c/(4n V zic -~- yfc + zlc)
(8.2.3)
using this expression, we can obtain a relation for the elementary potential of a source in a subsonic compressible flow. To do lhis, let us consider Eq. (8.1.7) and introduce the new variables xlc=xtV1-M~,
YJc=Y,
Zlc=-Z
(8.2.4)
With the aid of these variables, Eq. (8.1.7) is transformed as follows o21Jl'loxlc + o2rp'loyfc + a2!p'lazfc =-: 0 (8.2.5) This expression coincides with the continuity equation (2.4.8') for an incompressible flow. Consequently, the problem on the compressible disturbed flow in the coordinates x. y, and Z can be reduced to the problem on an incompressible disturbed flow in the coordinates Zlc' Ylc. alld ZIC. both systems of coordinates being related by conditions (8.2.4). Accordingly, we ('an go o"er from potential (8.2.3) for an elementary source of an incompressible fluid to the relevant potential for a subsonic source of a compressible fluid. To do this, we shall flOd the relation between the small area dolC on the plane XICOZIC in the in('ompregsible flow and the area du on a corresponding plane in the compre5SibJe flow. Using (8.2.4) (with the substitution of ~ for Ilc and t for ZIP) and lhe expression dolc :-: d~,C dt, C' we fmd that dUlc -= d£ dt (1/V1 - M!.). whence, taking into account that ~ do, we find (8.2.6) Let us transform the expression for QIC in (8.2.3). Thecomponent Vic = {)!P'I{)YIC' or with a view to (8.2.4), Vic == {)qJ' /(jy. It thus follows
d, d,
Ch. 8. A Wing in " Supersonic Flow
319'
FI,.I.l,t
Disturbed flow due to a supersonic source: "t the right-a )lnch (oislurbantf!)c:onl' In a rt'alsupt>rsonic now, at till' lell-an "inverte$ Math
~"olIe"
that in a compressible flow. the vclocity componcnt v equa.ls thecomponent ViC in an incompreflsible flow. TIencc. the density of sourcedistribution in a compre.c;sible and incompressible flows is the some,
i.e. (8.2.7)
With account taken of the relations obtained for A ('ompressibJeflow. (8.2,~) is transformed to tho following expression: d~'-Qda/[4n]/'z' ,(1
"l~)(y', z'))
(8.2.8)
We can COllvince ourselves by direct substitutioll that the huu·tion q/ is an integral of Eq. (8,2,5). It does not matter whether the velocity is subsonic (M ... < 1) or supersonic (M ... > 1). III Ihe latter ca!'le. it
is convenient to write the expression for the elemeutary potential as. d~'- -Q da/[4n]/,z'--a"(y'·c "JJ
(8.2.9)
where (Z,'2=M!.-1. Examination of expression (8.2.9) re\'eals that whell M 1 it has real values in the region of space where ;r: ~ (1:: (y~ .-:- Z2). This signifIes that the region of i'lonrce influence, i.e. the region of disturbed flow experiencing thE'!' actioll of the~e MUf('es, il'l inside the couical surfaco represented by the equation .1'2 - a.'2 (yZ -;- ZZ). If a poillt is outside this surface, the sourCE'!'S have un influellce 011 it. Hert' thereis no disturbed Row from a gh'en source. Formally, tbe equation x,2 = Ct'Z (yl! + z~) dE'!'lermincs the Sllrfaces of two aligned cones (Fig. 8.2.1) with their \'ertice!'l at the origin of coordinates, and. consequently, the strength of a !am aod only inside one cone (the right-hand one in Fig. 8.2,1). The disturbed flow in such a cone is determined by a potential double that given by (8.2.9) because the entire strpngth of the source.
320
Pt. J. Theory. Aerodyn"mics of an Airfoil and a Wing
Fig. 8.2.2
Region of influence of supeI1lonic sources
and not half of it, is realized in the flow confmed within the Mach cone. Accordingly, (8.2.10)
s
In the considered case, the elementary area da ~ d~ d with the .sources is at the origin of coordinates. If it is displaced with respect to tile origin of coordinates to a point with (he coordinates x O.=: Ii and z ,.", ~, Eq. (8.2.10) has the form d~' ~ -
Q dol (2o< V'(x---,"')"--a'''-''[yO'-+-(,-z----.,=)'1J'
(8.2.11)
When studying the flow over a wing, its surface is replaced with a system of disturbed SOUfCC5. To obtain the potential due to these sources at arbitrary point A (.:c, y, z) (Fig. 8.2.2), we must integrate (B.2. "11) over the region a in which only part of the sources are located. Eacll of these sources inflILen('es point A (x, y, z) if it is inside a !\[ach COlle with its vertex in the source. Hence, the region affected by the sources (the region of integration) is in the zone of intersection with the wing surface of an "inverted Mach cone" with its vertex at point A (x y, z) being considered. In a simpler case, point A alld the source are located, as can be seen from Fig. 8.2.2, in the same plane y ----' O. In this case, the affected zone coincides with the region of intersection of the wing and the ~Iaeh lines issuing from point l'vl (x, z), while the region of integration 0" is on the wing and is the intersection of the wing surface with the im'erted plane Mach wave having its vertex at point t
A (.1", z).
Having determined the integration region cr, we r.an evaluate the total potential at point A (x, y, z): (8.2.12)
Ch. 8. /II Wing 'in a Supersonic Flow
321
By t',alculating the parlial ucrh'ative of If' (8.2.-12) with respect to oX, we fmd the additional axial velocity tomponent: ~-u-~ QZ - 2n
j).
>
l/{lz
Q(~, t)(z-ild~d; S)l a;':I. (:
[II~:
;PW
(R.2.13)
We use this value to determine the pres~ure coefficient = -2u1V_ at the eorresponding point. Let us introduce the new ("oordinates
Ii =
(8.2.14)
In these coordinates. expression (8.2.12) bas the form 'P
'(z
,.
y,. . ) 1 -
1 -2:[
rr J,,J
Y(ZI
Q(",)"": !)I lit (III
In a particular case for points on the wing additional potential bl
~'(z.. O. ")~
_-..!.. r 2tt
J0
I ,l
~)'
~urface
lO.)t
{II
••
Wl = 0), the (8.2.16)
Qt<, ')",d, (rl
(8215)
QI
where by (8.:L7) (8.2.17)
The expressions obtained for the potential fUllction allow us to calculate the distribution of the velocity Ilud pressure over the surface of a thin wing if its plnnform. airfoil ("onfiguration, and the free stream Ilumber M are given . ••3, Wing with • Symmetric Alrloll and Trlangu. . P.antorm (%=.:0. cy .. ~O) Flow over _ Wing '_nel wHh • Subsonic leading Ed.e
Let us consider a supersonic flow at a zero angle of auack over a wing panel with a symmetric airfoil that is a triangular surface in which one of the side edges is directed along the x-axis. while the trailing edge is removed to infinity (such a panel is also called aD infinite triangular half-wing, }<'ig. 8.3.1). If such a surface has a subsonic leading edge, the Mach line issuing from \'ertex 0 is ahead of this edge. The now parameters at small angles of attack a; can be determined by replacing the surface in the now with a system of distributed sonrces on the plane y = O. Let. us consider arbitrary point P on the surface and evaluate the velocity potential at this point by arld21-017U
322
Pt. 1. Theory. Aerodynamics of lin Airfoil lind II Wing
Flg.I.l,t
Triangular wing with a subsonic leadin, edge
ing up the action of the sources in region OAPIJ conlined by the leadiugOA aud side DB edges, and also by the Mach linesAP aud BP. We determine the strength Q (s, ~) of the sources by formula (8.2.7) in which according to the condition of now \.... ithout separation we have v AV where J,. = dy/dx is the angular coeflicient of the wing surface. lIenee '0
0<),
The velocity potential at point P is determined by formula (8.2.12). Performing the substitution Q :-: 2J,.V and assuming that y = O~ we obtain ,'v~ll (83 Cf = -~ (I Y(XI' ,)1 o:'l{zp ')~ . . 1) 00
d,d,
where Xl' and Zp arc the coordinates of point P. This integral takes into account the acHoll on point P of 80111'c.es located on the area 0 equal to region OAPB that can be represented as the slim of two sections: OAPJ{ aud HPH. Accordingly, the integral !fl' (8.3.1) can he written as the sum of two illtegrals:
A~~
r II f(" ()~\P}f
,)d,d,+ II 1«. ()d,d,]
(8,3.2)
HPB
where
II" ')_[(x._')'_~"(z._,)'r'i2 (8.3.3) On section OAPH, integration with respect to for each value of must he performed from S = ~ tan x to S = SD = ct' (zp ~), and integration with respe(,t to ~ from 0 to Zp. On section BPB, integratioll with respect. to S, for which the value of ~ = ~2' must be performed from S - ~F ;..: ~ tan x to S = SE "" = Xp - a' (I; - zp), and integration with respect. to ~ from Zp to ~ --= ~1 = Xp -
sc -.
s
Ch, B, A Wing in ~ Supersol'lic Flow
Zn
= (Xp -+- a'zJ»'("X'
-J_
[J d,
)
Ip
'r'~
323
tnn x), lienee,
- ;.~~
.Tp-tt'(JJ.>-;l
o '1)
-'- Jd,
;
t~n
ft!,
)
'1'
,)d,
)(
.Tp-,;r.'(t-ll'l
1("
,)
(8.3.4)
ttl'lnx
Tho indcnrfiLe integral
Jf (s. ~) d~ = J 1/(xl'
~)~ ~fl.'~ (zp
(8.3.5)
;)i
Using this exprp~sion and introducing the main \'alue of the integral, we obtain the following formulll for the potential fUllction:
,
l_lzp-~tanxd" a'lzp-",1 ...
1I.l''''%i
836)
II' .-"",~ .. cos I
( ..
By rairulating Ihe partial Ileri\'ali\'c Qlp':OX, we lind t.he component of the additional velocity at point P in the dirertion of the .T-axi!l: 11.1"...
i'lj/
1l=7j"i""
'. f
c-=~ ~
V(zp
Taking into aeroullt t.hal. Lan x
>
d; ;tany.)S
a'l(zp
~)2
(8.3.7)
a' (cot !!""), we lind as II result
of integration that :t
V~ CO~h-1 ;r(~~it:;~~:z:)
(8.3.8)
To facilitate the ("lirnlations. we shall inLrociure the angle e del('l'milll"ld from the condition Lau 0 . 41' '.(;>, Clnd the nose lingle Ilf the leading edge y . ~ ,1(':2 - x (Fig. H.3.2). In addiHon, let. liS introdurc the Ilotlltion n - tan Kia' .. " tan I.l ,..,/tan '1. Zp tan x/,t'p . tan O't.an l' With this taken into accounl. we havc Il= -
and tho
presSUl'il
:ta'
~=z_t
cosh-I
1!1I(~-=~)
(8.8.0)
coefficicnt is
p= - ~:
= :ta' ::3_1 cosh- t 1111(21-=.°(1)
(8.3.10)
324
pt. I. Theory. Aerodyn"mics of IOn Aidoil "nd " Wing
Now let us consider point N outside of the wing between the Mach line OK' and the x-axis (.see Fig. 8.3.1) and calclliate for it the velocity that is induced by the sources distrihuted over the wing surface. To do this. we shall \l~e fonmlia (S.:1.1) determining the velocity potential. Taking into a,connt that till.! action of the sources on point N is confined by the region rJ = O/.!. we obtain the expression
~'~ - ),~~ ll/(U)d,dc
(8.3.11)
OLJ
where the function f (£. s) is determined hy relat.ion (8.3.3). Integration with respect to; for each value of S .= ~3 lllust he performed from £ = £R = S tan x to £.= £T = X;-.1 + Ct' (ZN - s), and integration "..-ith respert to C, from 0 to ZJ. Hence, ZJ
~'~
"'NHt'(lN-t)
- '~~ Jo d, tunx J
I (;,
c) d,
(8.3.12)
+
where ZJ = (XN a'zN)/(a.' + tan x). By integrating and using the main value of the integral, we obtain cp'= ),V"" n
"
r cosh-1
~
.t;;-ttanK d a IZl'>-~1
s
This expression i~ similar to (8.3.6) with the djfference that the coordinate zJ is taken as the upper limit of the integraL By calculating tlte derivative UIfJ'/ux and integraLillg, we obtain relation (8.3.9) for the component of the additional velocity. We must assume in it that cr < 0 because the coordinatezN is negath. e. In calculations, the coordinate ZN may be assumed to be positive, and, consequently, 0> O. If we take the magnitudGS of tan x, then to determine the
Ch. 8. A Wing in
f~'riu~~~
oBI
Supersonic Flow
S2ij
"
of sources on the velocity oulside a wing
induced ... clority we may use Eq. (H.3J) in whidl 0 ~hoilltl he taken with the opposite sign. The \vorking relalion 1I0W be('ome;<:
u,..,. -- .,'X'
~,72_'1
cosh-
l ~1;1~;~cr)
Thc IlreSSlll'C roefficicllt ill the region being
(~onsiderell
(8.3.13) is
(8.3.14) The sources (listl'ibuled o,'E'r the wing also induce n \'t'Jorily in the region between the "Iar.h line OK and the leading subsonic edge (Fig. B.3.a). Tht! magnitude of this velority at a point L is determined by the sources distributed on region DUG. We fllld the corresponding potential fUII('tion by ('xpres.o;ioll (8.3.12) in which we must iJllrodllce the ('oorilillatc %\. instead of ZJ. and replace the quantity IX ~ ex' (:1" -~) with Ihe valut· xL - -:L' (ZL - ;) equal to the longitlldinal coorllinate or point R (Fig. 8.3.:i). lIence. q.' --' -
A~""
~?
1(l~
XL -O\'(:L-~)
11
J
t tan
l (s. ;) ds
(8.3.15)
K
where :e ~ (XI. - cx'zd:'(tan x - -x'). Integration yieJds
'f' .," ~ ;~ cosh-l
., J
.t'L -;;. tan K dt ,=,'
\~I.-~I
~
By evaluntillg the del'i ,·ath·e d'F' ;{).x and then perforUling integrnt.ion pro\'ided that 0 ::"" ZI. tnn x/xp > 1, we obtain the following
326
Pt. I. Theory. Aerodynamics of oSn Airfoil and a Wing
FIg.. I.U Pressure field for a trianllular wing with a subsonic learling edge
formnla similar to (8.3.9) for the additional vclor.ity component: u= -
:la'
~~
(8.3.16)
cosh- t n"(:-=-G1)
where n>o. We lISC thc value of this vclocity to fllld the pressurc coeflidcnt:
p= - ;:
=
;1'];'
:~
cosh- t
:Ita--:.at)
(8.3.17)
Figurc 8.3.4 shows the ficld of pressur('.<; for a \\·ing witll a trianglllar planform having a sub!;()nic Icading edge. The pr('ssure roefficient along the Mach lines is zero. On the leading edge, the theoretical pressure coefficient equals infinity. The physically possible pressure can be considered to be sufficiently high, corresponding Lo the stagnation pressllreatasubsonic vclocity whose directioll roincides wilh a normal to the Icading edge. Triangular Wing Symmetric about the .x·Alls with SubsonIc Leading Edges
The velocitr at POtut l' of a triangular wing symmelric aDout the ,x-axis and having snbsonic leading edges (Fig. 8.;~.5) is determined by snmmating the action of ::'OUf(',CS ill region OHPA' confmed by leading edge::. Oil' and ON and hy Mach linesPA' and PB. The \'Blocity induced by t.he sourccs distributed on section OAPE is detel·· mined by formula (13.3.9). Tile sourccs distributed in region OilA' produce (tt point P a \'elocity tllat is ('alcuialed by exprcssion (8.3.13). The total Yalue of tile velocity is
u= -
:1(:1;'
~~:2_1
[cosh- J
:It;·~_aG)
-i-cosh- I
nll(zl·~.GG)
J
Ch. 8. A Wing in a Supersoni(; Flow
927
Fig. L15
Triangular wing
s~'mmetric
about the z-axis with subsonic leading edges
This upre.!=sion can be transformed a$ follows: It=
_~('OSh-l1/ II~-(f~ na.'~ r t-o~
(8.3.18)
Formula (8.:1.18) is suitable for tlte rOTldition~ 11 > 1 > o. We C(I;Il determine the pressur(! cocllident with the aid of (6.1.5) according to thc .known Y(I;luc of the additional velocity component: p~_' _~= VOO
1IU'
41. cosh-t'l / yn~-1 V
1/
2 __
a2
i-a!
(8.3.19)
Point /. located bet.ween the leading edge and the Mach wave (Fig. 8.3.5) is inl1uence(l by t.he sources distributed on section OUG' of the wiug. The ,'clodty induced by these !'OlilTes ran be calculated as tll(l slim of the velocities induced by the sources in region aUG (formula (8.:1.16)1 and by the sources distributed in triangle 000'
[formula (8.3.13)[. Con::;equentlr, U= -
111];'
~=2_1
[cosh-I
,;,~~(1)
+cosh- I
1:1;~~~(f) J
or nfter transformations
u= _
211....
00sh-11/ tI~-1 0 2 _1
Ita' yal-I
(8.3.20)
The corresponding pressure coefficient is
p= where n
> a>
L
U.
:1(1.·yn~-l
eosh- 1 1/r 1I~-1
r
0 1 -1
(8.3.21)
328
Pt. I. Theory. Aerodynllmics 01 lin Airfoil lind
II
Wing
Seml·lnfinPe Wing with • Supersonic Edge
For such a wing (Fig. 8.3.6), Mach line OK issuing from a vertex is on its surface. Consequently,
n/2-x>ll.." taox a"". By expression (i.G.S) and the formula p = = -2u/V"'" the additional velocity component is U
= -AI'"" cos xiV.lI!, ros 2
iC
Since
+1
M!,. = a.'z
and
1 - L'cos z x = -tan 2 x
we have u = -AV""IV
a.'2 - tanz)(
whence
" ~ -/-V ~/(a' Vl=n')
(8.3.22)
The relevant value of the pressure coefficient is
p=
-2uIV"" = 2A/(a' Vi _ nZ)
(8.3.23)
Formulas (8.3.22) and (8.3.23) may be applied for n < 1 and 1>o>n. Let us calculate the \'elocity and pressure at point P between Mach line OK and the side edge. If we assume that this point belongs to a wing whose vertex is at point C (Fig. 8.3.6), the velocity would be calculated with account taken of the influence of only the leading edge and, consequently, of the sources distributed in region PCH. By (8.3.22), this velocity is "PCII ~ -)'V~/(a'
VI -
n')
(8.3.24)
To find the actual velocity at point P belonging to the wing with its vertex at point 0, we must subtract from (8.3.24) t.he velocity induced by sources distributed in triangle ACO and having a strength of the opposite sign. The magnitude of this \'elocity is determined by means of formula (8.3.i). Substituting Zc = (xp - a'zp)/(tan iC-(t')
Ch. 8. A Wing in
II
Supersonic Flow
329-
~1
!
Flt·8.U
TriaDgular wing with a supersonic edge
for the upper limit ZB of the integral, aftI'I" integriltioll we obtaill. M'...
'~
UAOC=~ ~
d;
V;2(tan2x_a'2)
:t
va~:~tan2x
2~(J·ptanr.
COS-I
:t'tZI'J . .c!'
::~:: ::~:~:)
Taking into account that n = tall x:-x' and
0
=
Zp
a'2=i,
(8.3.25)· tau x/xI', we
nnd (8.3.26). The total value of the additional velocity at point P is u=UPCH-UAOC":: -
,=,'
V';-n
2
+
COS-I
II(J(~-~:J
]
lS.3.27)-
[1-+co.,,-1
"o(;-:~)
I
(8.3.28}'
[1-
and the pressure coelTicient is
p=- ~:
=:t'
~f.l_!!2
where 0 < n < l. The indu('ed \'clol'ily at point .V, bl'\we('l1 :\111("11 \\·il\·(' OK' and the side edge is determill('d b~' 51l1nmatiOll (If thl' adioll of the : ("ooJ"(linl1l£'s Xl' and z""
.330
Pt.
r.
Theory. Aerodynamiu 01 an Air/oil and a Wing
Pig. 1.3.7
Pressure Held for a semi-infinite triangular wing with a supersonic lending edge
witll the rele"ant "alues Xx and
uOEF~
Introducing the symbols UOEF
= -
cos- j
~:\~.~n_:~~~:::J
(8.3.29)
and n, we have
(J
:la'
Inlegration yiclds
ZN'
- ., l/::~':tan:)(
\';7'_11
2
cos- j
IInt,~.(J(J)
(8.3.30)
where (J < 0, n < 1. and I a I < n. If we adopt positive values of Z:-; /lnll (J -'" ZN tall x/xt-> and take absolute values for n, the additional \'clocity is u=
-
:let'
~~; __ II:
COS-I l"t,"':"'..(JO)
(8.3.3!)
and the pressure coefficient is
p= - ~:
= na'
~
cos- J
:'tl_':_(J(J)
(8.3.32)
The pressure field for a semi-infinite triangular wing with a supersonic leading edge is shown ill Fig. 8.3.7. Between the leading edge and the internal Mach line, the pressure is constant, then it lowers, and on the external l\-lach line reaches the value of the free-stream pressure (jj = 0). TrlaJtlulu WIng 5,......rfc about the ,x.AliI with Supenonlc LeitClllJtl Edges
The velocity and the pressure coefficient at point L (Fig. 8.3.8) between !\lach wave OK and the leading edge are determined by formulas (8.3.22) and (8.3.23), respectively, because the now at this point is affected only by edge OR. These formulas may be applied for the conditiolls n < 1 and 1 > (J > n.
Ch. 8. A Wing in " Supersonic Flow
331
Fig. 8.].8
Triangular winq symmetric about thl' .r-axis with Supl,!f$onic leading etlges: J-/lfacll hrw: 2-mnximuln thickness I",,,
The v('locity :prcssion Ui.:J :11). Adding (8.3.27) and (fLLH). we oblain the total n']orit.\· (It point ') of a .~ylll metric wing: It'-_ -
0;'
~:~~ "~ __
r1
+cos-
"n(l
J
JI~) ++co_~-! 1I1\'I-~<J(J)
J
or after Iransfonnlltiolls,
u .. -' -
ct'
The c.orresponding
::};~J'~ (1 Ilt'CS):il\l'f.-'
-+
sin-!
1
(8.:U3)
cocf[icirnt is
p= - ~.:. _ ~. ';~_II~ (1 -
-+
"in-!
1/ - ';l~a~')
(H.3.o1)
8.4. Flow over a Tetragonal Symmetric Airfoil Wing with Subsonic Edges at
OJ
Zero Angle of Attack
By u:o;ing the formulas for dell'rmining the yclodtr And pressure on the surface of a II'iangular willg, we {"an Cah'\llate the now at a z(,ro angle of .. Ilark O\'cr willg:-; with II symml'trir ail'foil and nn arbi-
332
Pt. I. Theory. Aerodynamic$ of an Airfoil and a Wing
FI,. B.4.t Tetragooal wing with a symmetric airfoil nod subsonic edges
trary planform. Let 115 cOllsider the tetragonal wing shown in Fig. 8.4.1. A left-hand s1'stem of coordinates has been adopted here and in some other fIgures to facilitate the spatial depicting of the ,.... ing and the arrangement of both the sections being considered and the reqnired notation. We shall assume that for SItch 11 wing the leading aud trailing edges, and also Tntlximum thickness line CBC' are subsonic. Acr.ording11', the sweep angles Xl' Xa of Ihe leading and trailing cdgc!> and the anglc X2 of the maximum thickness line nre larger than 11./'2 - fLoo. The distributioll of the vclodtr and pressure O\'er an airfoil depends on where the latter is along thc wing span. i.e. on the laterAl C,OOfdinate z of the section. Airfoil FL (z = z,). Four Dow regions, namely, FG, GH, HJ, and J L, should be considered all thc airfoil. Region FG is confined by point F Oil the lcading edge and point G at thc intersectioll of a Mach Jine with the coordinate plane % --' %1' Point G is considered to be on plane zOx and is determined, cOllseqll('ntly. as th(' point of intersedion of the :'tIach line issuing from the projection fl' of point Jj onto plane zOx and of the straight line z . %1 (Fig. 8.4.1). The "elocity and pressure eoefficient in region FC behind !\Iach line OKo on the sllrfa('c of the wing are dE'termined wilh the aid of the distrihution of Ihesources in IriangieOCC' hy the relevant formulas (8.3.18) and (8.3.10).
Ch. 8. A Wing in
o!I
Supersonic Flow
aaa
Since the inclination of the sUl'fact:" is "1 in accordance with (8.3.19)
we
h.\\·(\
PPG= :ta;' ~~COSh-1 V' ~f~o~r
(8.4.1)
where lit ~ tan x, '''-' •. O't .-c z. tan x, .1"1' and It is the l'unning coordinate of the point. The drag coefficient of the airfoil corresponding to region FC is Xu
cx. ~'G =
~
(Pu.L Pb) d·.l.'1
)
(8.4.2)
'F
where b is the local chord of the profile. Since
pu.+Pb=2PFG. dO', =
_
z\
l:~
. :,
0',,..;
:\I:~)tl,
=1 ::~ . :\ cri
dJ.', == -
we have
where O'IP = Zl tan XI/XF --,:: Land O'IG ".---= :::1 tan x.JxG· Region Gil is acted upon by the di:;:tl'i\.Jution of the source:;: in triangle OCC' with a strenglll of Q = ::!t'll".", alHl in triangle BCC' where the strength of the sources Q = 2 (A.~ - I-I) 1-coo (the sign of the angle 1.,2 is opposite to that of I_I)' Since region GH is hehino ;'o,Iach line OKa within the limits of the wing, the pressure coefficient due to tllC distrilliited sources in region OCC' must be calculated with the aid of formula (8.3.19). The influence of the distribution of sonrces in triangle BCC' on the pressure coefficient should be taken into acconnt with the aid of relation (8.3.21) because region GH is outside triangle BCC' between Mach wave B' KB and edge nc. Hence,
PoH= __4_A'__ cosh-t' I ;ta;'~
-1-
4 (A.s-).t)
mx' Vn~-t
V
COSh-I'
nl-oi t-ol
I nl-1
V
(8.4.4)
01-1
where n2 = tan X2/a.'; O'~ = z. tan y.~Ix~, and X2 is the coordinate measured from point B nllo equal to X2 = Xl - XB.
334
Pt. I. Theory. Aerodynamics of an Airfoil <'Ind <'I Wing
Using formulas (8.4.2) and (8.4.4), let cient corresponding t.o region Glf:
ItS
oe-termine the drag coeffi-
O,lI
__
J cosh-
b:~,z';:'~i ~\ I
C,.;. OH =
J
V/ '~i~:l
. do?
(II!>
(lzil
80';:~\~A,:;~~n ~2
__
CO.~h-1
)
V :~=! . ~~2
(8.4.5)
°:oG
where 0"2H and 02G are cvaluated relative to point II.
Slimming (8.4.3) and (S.Ll) and tnking into account that 0IP'"""' z,
02G = :, 1;(:
%2
~~~x' "-'
c-,-, L
om
tan 11.", tan )(2
-=
_~
III t;:X\
== X:~~:~l~'XI
I";.XB =~,
(J~H"""
z. ;; x~ = 1
we obt~in Cx, }'H
-= Cx, Fe: ·rex. all II tlln)(l
.Tn+71 tlln )(2
8J·fz, tall XI
b:ta'
Xd:r, Let us
Ylli--l
r
~
,
coc;h-I
'V '~i-=-;.f
--
8(A:.'1-:~!~n)(1) casICt}/'" :~=~
a~<;urne
.~~z
(8.4.6)
that part of cho;;; HL (wc presume that point H is 011 line JJC) equals rb. where r is a dimensionlegg coefficient of proportionality determined from the condition T = B'Dib o (here /Yo is the central chord). Therefore. for Sllrface ORe, the part of chord FH will be (1 r) b. Since a part of centrlll dlOrd R'D {'l]uals Til". the remaining part OR' equals (1 - r) boo The angle.<; arc "-I = 'X/2 (1 - r). "-2 -;1.'(2T) (R.4.7) WhCrC X ~ IJ./b o is the relative thid;ness of the airfoil. Taking int.o tlcc()unt the values of AI and "-2 we elln write formula (S.1.6) a.<;:
Ch. S. A Wing in '" Super~onic Flow
335
The velocity on line HJ is induced hy :-OllfC('S of the strength Q = = 2~1 V"" distributed in triangle 0('(" flnd by :-:ollrces of the ."trength Q '-' 2 (~2 - "1) V"" distributed over region /ICC'. Tile lir:-t di.~tri bution of the !'ollrces gi"es risc to tllf' pres.'illrC {'oeffident nt\clliated by formula (a.3.H.I) in which we 11111,-.t aSRHHle thill iI. ;"1,1/ =- 11\. and (1 -~ (11' The pressure coefficient (lue to the inOIlt'IKe of the sprond distribtltion of the ~otlrces is 1'1150 found with thr aid of fOl'tnula (8,3.HI) ill which we musL as!'>llllle lilat I. ;"2 - i'l' II f/2, alld (J '"'" 0"2' Summation of the pressun' ('oeJ1iriellls yie\(ls -
_
I _I
4AI
na.'V ni- 1 COSI
['HJ-
:t::A0J~~)
-i
V
V
1 COSh-I
nl-of
.. /
r
I-('f[
lit ~~;
(8.4.9}
C!'>illg forJllllla (8.4,5) in which w() mllst replace the "alne of A} with I.: and taking into account that
A] = 6./12 {1 - T)I, 1..2 - I.] n l = tan x/a', =
Oz
22
=
-
K~1I2r (1 - r)1. 1..2 = - ~(2r)
n2 = tan x.ja'. 0'1
tan xz!J-~,
J'H ~ Xl .:&;
.t'.l>
= %1 tan xlixi
x; =
Xl -
/
JJi-o.r
J'u
we obtain __ 24tzl
Cr,nJ
Gj1J cosh-I
[tanxi
"r(l- r):l'x'
¥Jll-'! !Jill
-
~
j
V
1-0,
il(1,
--or-
__
(J~.J
eo~h-1
V nl~-:l' dO~2
]
{8,4_10~
(J~I[
where (12 i!' C,'"llIlILcri l'ellllivc to poillt /l_ On J/•. we take inlo UCCCIHlll lhe illfillC'nce of 1II1'l'e di5trilHlliOIl,~ or the SOlll'('(!S, lIamely, 011 trillngular Sllrf,l('CS OCC' _ flCC' ali(I neG', The Hr5l. two distl'ihl1l.iolls give ri~0 10 111(' pre:-:slll'C' coefficienl determilled with the aid of formula (8.".~J) ill which (1\ (';\Iell' lflted relative to points () and n·_~rectiH·I~'. TII(' (-ueffi· rienl of pl'essII1'c iuilu('cd by the ~ollrces II istri hlllpd 0\'('1' l'I'g-iOll DCC' with a ~LL'el1gth of -A2 I!'> fOlllH1 wilh till,' nid of rurlllllill (S.:t:!t) in which we 1Il11."l aSSIIIII(> that ;,' A~. u "3' illHI (1 (1:1' Silmmilig 1I1(' pre:-sllrc coefficients due to all three soul'ce disll'iblltiolls, we obtnill
n.
Poll =, •
yeos
4AI
:w;' y'1I~-1
h-I"/
V
cosh-!
V" 1'11-
(Iii -t (11
~ ().,~-).,1)
"o;t' ~
n~-fJi .!i).,l 1-1"'/"'iii'=T t_()'I-l'1a,~cosl V 0:-1
8411
( ••
)
336
Pt. I. Theory. Aerodynlmics of an Airfoil and a Wing
where n:') = tan ;(:,)/(1.': the values of 01' 02, and 0, are calculated nllative to points O. Band D. Introducing the value of P-;L into formula (8.4.2) and substituting A!! for AI in it, we obtain _
cr ,
JL = -
r
:r~~:~
(/IL
(1-
~;n~
•
d:~l
~I ~1::a~; . ~~'
oosh-(
'u
+ ;. V:J)(~l
V ~~I1~t
--
j
1
__
COJlh- 1
',J
'IL
t~n;; n~
r (1
j
',L
__
)" COSh-I V =:=: .d;.a J
(8.4.12)
',J
Summing (8.4.10) and (8.4.12). we have _
[
!:~. HL= - :t.~:I,
(/IL
(I
_~n~
j ',"
(/2L
r(1
__
cosh- t
V ~f~:l- da;1
,/_. __ •
)
t~;~ co.~h-I V ~'-=-:i' ~2 ',L '," __
+ ;. ~~ J cosh-t V :i=! .~ ]
(8.4.13)
'oJ
In this expression 'I tanK 1 • (t-;:)bO+.31 tanxa
0'2H-1, C13J=
1:1
x,
rbo~::~:~xa
0'2L
tan X, zj
0'11a= ~l bO+=1 tan
I
=n,.
0'3L=
SI
tan x,
zi,
I }
=
1.
(8.4.13')
where xJ = Zt cot fl ... = ZI(1.'. and xi, = Zt tan )(3' We take the integrals in formulas (8.4.8) and (8.4.13) by parts. These formulas hold (see Fig. 8.4.1) when (8.4.14)
where zlh
=
(1 - r) bo/(tan
Xl -
ex'}.
Zo.
= rbJ(tan
X2 -
a') (8.4.15)
Ch. 8. A Wing in a Supeunnic Flow
SS7
or (Fig. 8.4.2) when (8.1.16) The drag cocfficicnt of th(' airfoil dctcrmined by formula (8.4.13) has been relntcd to the local chord 1,. We calculatc the value or the drag ('oellicicnt. Cx.o relatcd to t1l(' rcntral cliord lIo by thf' fornmln C:l',fI = C:r: (blb o)· Airfoil in the :\1id-Span Section (z = 0). We determine the \'alucs of the aerodynamic coefficients for this airfoil ns 10110\\,5. 011 nirfoil sec·tion OB (sec Fig. 8.4.1) with a 1f'ngth of (1 - ii boo thc "clarity is indllr.ed hy ~ollrc.es having' i1 strcngth of Q.-, 2i'Il"." distrihutrd in trianglc Accordinglr. we evaluatE' the pressure coerti<'icllt II)' ramuli" (S.3.1Sl). A!':'lullling that IJ ''''' Z, tan Xl/Xl - O. we ohtain
vec'.
Pos=
:la'
~
(8.4.17)
COSh-I ",
Airfoil scct.ion l)fl with a length of rbo experienc.es thE' ac,lion of sources with a ~trength of Q = 2;'11'... distribute(1 over rE'gioll OCC' and with n strength of Q = 2 (A2 - AI) V... dil'=tributptl in trianglc BGC'. J n a('.('ordance with I.his. we dl'lermine the pI'C!!!!lIre coellkielltF. Ul'Iinl? formllin (8.S.H) nt IJ 0, we find
PaD =
~~' ~;:f-1
COSh-I nl
+ n:~~
COSh-I
I/~
(8.1. '18)
The drag coeHidcllt of the airfoil wilh account tnken of the. upper and bottom snrfnees relaLe-d to the ('entral ("hord 110 i~ (1-;,1>(1
cx=.-fo-
,J
Pon]'1 dx-:,
~
rbo
.\ PnD"2 dr
,
(H.4.1!1)
Introdudng the value of POD from (8..1.17), UrpDD rrom (S.It.iS). and having in vicw that "': = -M(2r). A2 - i'l' -£\/12( (1 --,)1 and also thai (i - r) Al = Xl2 ;l.Iul ;"2 ~ --3:/2. WI.' obtllin (8.4.20)
where ('oRh _1 "2 .-: 111 (nz ..!. 1/ n~ - 1). Airfoil FI~" Lpt us calculate the lirag coeffiCient or an airfoil (see Fig-. 8,4,1) Wh05C coorliinate %1 satisfies the ineqllality %D, < < %1 < '0,· Sect ion FI", or the aidoiI expericnces the action of sourcc!' with Ihe strenltth Q .-:- 2A I V"... distrihntC'd in region OCnG', and with thL' !'Irc-nglh Q -= 2 (i'2 - "',) V ... dil'ltribntl:!d in IICG'. TherefOl'r. WI:! Illay use formula (8.104) to calculate thc pJ'('~!'IlIre 2::-0171$
aas
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
Fig. B.U
ArrDngement of :\Iach lines coefficient, and I'elation (SA5) to determine the drag coefficient. For airfoil !ie('tion F 1 H j • relation (BAS) has the form "111 1
c.,".r1Il t ·= -
/.>~~i,Z\~:tKI1 ,\
--
cosh-I
"1F t
8().::~t~n}(2
-
•
"2H1
.\'
V '~'-='a~~' (~att
V ~:=: .d:i~ --
cOsh-I
(SA.21)
"21'1
where 0"1 and 0"2 nicient c.",,",!., from expression (8,4.-13) in which we replace the limits 0ln and <JIL with the quantities <J1H, and O"IL" the limits 0"2H and 0"21, wilh the quanLities 0"2H, and 0"21.., and the limits 0"3.1 and 0"31. with the quantities 0"3J, and <JJL" respectively. determined by formulas (8.4.13'). The over
Cl:, F,R,
+ ex, H,L,
Airfoil F 2L 2 . Lel us consider the airfoil b(>lween points D. and B. (Fig. 8.4.2) with the coordinate ZI satisfying the inequality zD, < < Zt < Zn,. The flow over this airfoil is of a more intricate nature. The velocity on section F 2G2 is indneed by sources having the strength Q ,.-.:: 21l, V distributed in triangle OCC'. Consequently, the pressure coefficient on this section is determined with the aid of formula (8.4.1), and the corresponding drag coeITieient Cx • l'-.G, from expressioD (8.4.3) in which the integral is taken between the limits O"IF , and OIG,. 00
Ch. 8. A Wing in e Supersonic Flow
aag
The pressure OIL scction G2.H z depends on the influencc of sources having a strength of Q -"- 21..1 Y 00 distributeel oyer section OCC', and also of sonrces having a strength of Q = 2 (Az - 1..\) Y 00 in region BCG'. Conseql1ently. we cakulHte the pressure roeITicient by formula (8.4.4), and the drag coefficient c,\:. ("H, from ('xpre.!'sion (8.4.:)) in whkh we take the first inlrgral hetween the limits OIG, anti OtH,. an(l the second between (J~G and O"~II • Section II zJ2. experiellrcs'lhc si~ultaneolls actiou of sources distrillllted in region OCC' (Q . . , 2"11"00), and also in triangle RCC' IQ --: 2 (A2 - AI) Fool and 011 sl1rfflre [JCC', where the strength Q = = -2A21'oo. The first. distribution re!'.lIils in a pressure coefficient determined by formula (8.:tHl), alill thr se('ond two distributions resulL in the roefiicienl calcl11ated by expression (8.3.21). The total vallie of the pressure coerfLcient 011 this section of the airfoil is PHzJz
,
~
:vx'
1-1
',COSI
~;:ti-1
c05h- 1 1-/
'~i~:/ .~ .1~/'~-;:;·~1
1.~.2 :t~' l/n~-1
/III=T J; o~-I -
]-1' /
COSI
V
n:f= I
(J~-I
(8422) ..
where 01' 02' and 0"3 arc determined relative to points 0, 8, and D, respectively. By using the formula
') Xft _
C.", 112J, =
i;:.\
P1I2J2AZ
dx
(8.4.23)
'H,
we can calculate the drag coefficient ror lhe sectioll of the airfoil being considered relate!l to the length bo of the centre chord. The flow over the last airroil sect-ion J 2L'J is the result of indllction by the sources distribnted in tlte sallle regions of the wing as for airfoil section II ~J~. lIere accouut must be taken of the feature that the velocity induced by the sources in region ReG' is determined by formula (8.3.1R), where A is replaced by the ,",ngular coefficient A~ - At. Therefore, the pressure coefficient should be calculated by formula (8.4.11). and the drag cocrncient ex, J,I .• trom expression (8/1.12) in which we take the integrals between the limits onJ, and (Jnl., (n = 1, 2, :~). We obtain the total drag coefficient by summation of the coeffi~ dents for all four sections of the airfoil: ex.
F,L,
= ex, F.G,
provided that the value of tao
+ cx. G,H, -!- ex .II,J, + c:t, J.l. Zt
satisfies the inequality
~~O_ct' < Z1 < t~~-;:~b~,
340
Pt. I. Theory. Aerodynamics 01 an Airfoil arld a Wirl9
Airroil FaLa. Let us consider seclion F3La (Fig. 8.4.2) with the coordinate Zl satisfying the inequality (8.4.24) where (8.4.25) Tho pressure on section F aJ a of tho airfoil is due to the action of sources distributed on the triangular surfaces OCC' (Q = 2A1 V .,.,) and BCC' lQ = 2 (All - A,) Vool. We shall therefore calculate the pressure coefficient by expression (8.4.4), and the drag coefficient by formula (8.4.5) in which we replace the limits all, G and an, H with the values On,!'. and on, J. (n = 1, 2). Airfoil section J 3H 3_ in addition to the indicated source distributions in regions OCC' and BCC', also experiences the action of sources having a strength of Q = -2A\I VL distributed in triangle DCC'. Consequently, the pressure coefficient equals the value calculated by formula (8.4.4) plus the additional value calculated by formula (8.3.21) in which we assume that A = -At. Flow over section H 3£3 is characterized by the induction of sources distributed in three regions of the wing, namely, OCC', BCC', and DCC'. Accordingly, the pressure coefflcient on this section should be calculated with the aid of formula (8.4.11). By calculating the relevant components of the drag coeflicient for all three SGctions and summing them, we obtain the total drag coefficient: c%:. F.r.. = C:c. P,J. C.'C.J.H. + C%:, H,I.. The relevant expression is suitable for calculating the drag coeffI~ cient of the airfoil between points DI and D2 (Fig 8.4.2) with the coordinate Zl that satisfies the inequality
+
tao
y.:;~~, < Zt <
tan )1.,1):..0.'
(8.4.26)
F~L"
(see Fig. 8.4.1) with
Airfoil F,!.. ... Let as consider Sf'ction coordinate ZI> zD,
(8.4.27)
Three source distributions simultaneously act on this section, namely, OCC' (Q - 2~, V ~), BCC' [Q - 2 lA, - ~.) V ~l. and DCC' (Q = -2A2V ooJ. Airfoil section F,H, is behind Mach line OKo within the confIDes of the wing, therefore to calculate the pressure produced by the distribution of lhe sources in OCC' we must use formula (8.3.19) in which wo assume that A = AI' The second distribution of the sources in BCC' acts on section F ,H, located beyond the conilnes of the surface between the Mach line and edge BC. Therefore, to calculate the additional pressure dlle to the influence of the distribution of the sources in BCC'. we must use formula (8.3.2t) with the substitution of At - A)' for A..
Ch. 8. A Wing in a Supersonic Flow
a41
Fi,.8.4.3
Distribution of the drag coerrl-
~~~~a~~e~h~hred S~~~g o~\"i~hs~~~~ sonic edges (tbr dasbeLl
Un~
sholl's an unswrJ,1
~·lng·
The second seclion of airfoil H4L~ is located within the confine., of the wing surface, i.e, at the same side from the Mach lines and the relevant edges OC and BC, Consequently, to determine the pre~~ure coefficient resulting from the dislribntions of the sources in OCC' and BCC', we use formula (8.3.19) in which the value A. = ;.., corresponds to the distribution in OCC', and the value A. = 1.2 - Al to the distribution in BCG'. Section H4L4 is at both sides of Mach line DKo and trailing edge DC, Le. it is beyond the contines of triangular surface DCC' where the strength of the sources is Q = -2A. 2 V"". Hence, to calculate the pressure coefficient produced by these sources, we should use formula (8,3.21) with the substitution of -1.2 for A., Using the obtained value of the pressure coefficient, we can determine the relevant drag coefficient for airfoil F4L4: Figure 8.4.3 shows the results of calculating the distribution of the drag coefficient c:r/lJ.2 (3. = I1lb) over the span of a swept constant chord (b) wing (Xl = X, = X3 = 60°) with a symmetric rhombiform airfoil (r = 112) for M"" = 1.8 and 1.9. A glance at the figure reveals that with an increase in the distance from the centre chord, the drag coefflcient riist grows somewhat, and then sharply drops. For purposes of comparison, the figure shows the value of the func~ tion c,/ t;, for an airfoil belonging to an unswcpt wing. To determine the total drag coefficient of the wing, we must integrate the distribution of the drag coefficients Cx ." of the airfoils over the span, using the formula It'
cX
=7- Jc:r,,,dz
(8.4.28)
342
Pt. I. Theory. Aerodynemics of en Airfoil end eWing
Influence of a Side Edge on the Flow over a Wing. If a wing has a wide tip or side edge (Fig. 8.4.4), its influence on the pressure distribution and the drag coefficient must be taken into account. The flow over such a hexagonal wing is calculated as follows. First the velocities and pressures in region 00' D' D produced by the distri· bution of the sources on triangular area OCD are calculated. The calculations in this case are performed similar to those of a wing with a tetragonal planform (see Fig. 8.4.1) having no side edges. Next the calculated velocities and pressures are determined more precisely with account taken of the influence of side edge O'C', which is oquivalent to the action of sources distributed in triangle O'CD'. The strength of these sources has a sign opposite to that of the sources corresponding to a wing of area O'CD'. The action of the sources distributed in triangle O'CD' extends to the wing within the area O'TnD' confined by Mach line 0' K', the side edge, and the trailing edge. For example, for airfoil F 2L21 the action of the sources is confmed by section F~L2 (point F~ is at the intersection of chord F2L2 and Mach line 0' K'). Let us see how the pressure is evaluated on section J 2L2 of this airfoil. Taking into consideration only the distribution of the sources in region OCC', we can determine the pressure coefficient by for· mula (8.4.11). We can introduce the correction IJ.p-for the action of sources of opposite signs in triangle O'CD', so that (8.4.29)
When fmding the correction IJ.p we must take account of the posi· tion of airfoil section J 2L2 relative to Mach line O~K; that passes through point 0; belonging to the opposite side edge (Fig. 8.4.4). If this line does not intersect J 2L2' the latter is influenced only by the distribution of the sources in region O'CD' of one side of the
a.
Ch.
A Wing in ~ SupersoniC Flow
343
wing. whereas the influence of the sources ill O;C'U; is excluded. The induced velocity is calcuh\led by formula (8.3.13), and the corresponding additional value of the pressure cocITieient b~' the formula !!.p = -2u/V .... We shall write this a(lditional nlluf' in the form of the slim (8.4.:l0) where !!.Pl depends on the distriblltion of the sources ill O'CC' (Q = 2"'1 V 00), while IlP2 and IlPa depend on the distribution of the sources in B'CC" IQ = 2 (!,~ - AI) v ... 1 and [)'CC~ (Q = -2"'21' ,.,). Hence, by using formula (8.a.14). we obtain
D.p=
.10:'
~::f-I
cosh-
l n;l~~~(f~I)
+ ~~~}.; ~;'l~ 1 cosh- n:'l;~_(f;~) t
_ :-(0;'
n~ )1"1-1
cosh-!
1I~·;-1]3 " 3 {1-i- 0 3)
(H.Io.:311
°
where 1, 02 and 03 arc calculated relative to points 0'. /J', and [)'. If Mach line O~K; intersel.'ts chord F2 L l • then simult.aneously with the action of sources 0' CC account must also be taken of th.e influence of t.he sources distributed in triangle O;C'C; at the opposite side of the wing. The same formula (8.:·U:~) is used to c~lc\llate tile induced velocity. N
B.S. Flow over a Tetragonal Symmetric Airloil Wing with Edges 01 Different Kinds (Subsonic and Supersonic I Leading and Middle Edges Are Subsonic T'llIing Edge Is Supersonic
The disturbances from the trailing supersonic edges of a wing (Fig. 8.5.1a) propagate downstream \\'ithin the confines of the Mach cone with the g(lneratrix DKD and therefore do not affect the flow ovor the wing surface. The velocities and pressures depend on the influence of the leading and middle subsonic edges. Let us consider profile FL with the coordinate ZI < 200,. The pressure coefficient on section FC, which depends on the action of sources having a strengtll of Q = 2"'lV.... that are distributed in triangle OCC'. is determined from (8.4.1). and the corresponding drag coefficient c"'. n. from (8.4.3). On t.he following section GH experiencing the influence of the sources distributed in triallgles OCC' (Q = 21.1 V..,) and BCC' IQ..." 2 ("'2 - AI) V .... 1. the pressure
344
Pt. I. Theory. Aerodynamics of an Airfoil a"d a Wing
«)
(h)
Fig. '.S.t
Tetragonal wing in a supersonic Dow:
rng~h:.!cr;d~L:~rc.u:t~::I~:f:~:~t~lm~l~i~e a~a!~~~~~;I:~~\1~~1':'a~es~~
Ionic
coefficient is calculated by (8.4.4). The con'esponding value of the drag coefficient c~. GH for this section is determined from (8.4.5). The action of the same source distributions is observed on section HL as on section GH. But taking into account that section HL is below Mach line BKB (on the wing sudace), the pressure coefficient PHI. must be calculated from (8.4.9). and the drag coefficient C~, HL. from (8.4.10). The total drag coefficient of the aidoil is c~, FL = cjC, PO c~, GH c~, HL (8.5.t)
+
+
When considering section FILl with the coordinate %1 > %8,. account must be taken simultaneously of the source distributions in OCC' and BCC'. For section FlHl • the drag coefficient c~, P,H, is determined by formula (8.4,21). For section H1L" the drag coeffieient Cit, H,L, is found by expression (8.4.13) in which the third term in the brackets is taken equal to zero, while the limits Gn.H and a p ,1,
Ch. 8. A Wing in a Supersonic. Flow
34lS
are replaced with the values Gn, H, and Gn, L,(n = 1,2), respectively. The total drag coefficient for airfoil FlLI is (8.5.2) Leading Edge Is Subsonic, Middle and Trailing Edges lire Supersonic
A feature of the flow over the wing (Fig. 8.5.1b) consists in that the sources distributed in region BDD t do not affect the distribution of the velocities and pressures on the remaining part of the wing above Mach line BK D• Let us considor airfoil F L with the coordinate Xl < ZD,. Section FH of this airfoil is acted upon hy the sources distributed in triallglo OCC' (Fig. 8.5.ia); consequently. the pressure distribution can bu found from (8.4.1), and the corresponding drag coefficient C:E.l<'H, from (8.4.3) in which G t " is substituted for the upper limit GIG' The second section HG is influenced by subsonic leading edge OC (and. therefore. the distribution of the sources in OCC') and by supersonic middle edge BC. The corresponding pressure coefficienl is determinod as the sum of two coefficients, the first of which is evaluated by expression (8.4.1), and the second by (8.3.23) wherl~ we assume A = A2 - AI and n = "2' Consequently,
p,.G-- __ 4l._'_ _ cosh-I"; na' Vnl-1 V
n~-or ...:, 20.~-AI) i-oj . a: ~
(853) ..
On section GL, the velocity is induced by the sOllrces distributed in triangles OCC' and BCC' (Fig. 8.5.1a). By using formulas (8.5.3) and (8.3.34), we obtain the following working relation for the pressure coefficient:
The drag coefficient of the airfoil is Xli
C:E.l'L=f(J Xl'
PFlIA1dx+
Xc;
Xc
"'H
"c
J PHG~dx+ JPCl.A dx) 2
(8.5.5)
Airfoil FILl is below Mach line OK o• therefore it is influenced by the distribution of sources with a strength of Q = 2AI V in triangl& 00
PI. I. Theory. Aerodyn,mics of ,n Airfoil lind
346
II
Wing
·OCC'. In addition, section HILI is acted upon by the distribution of sources with a strength of Q = 2 (A: - AI) V 00 in triangle BCC' that produces an additional pressure evaluated by formula (8.3.23), where we assume A = At - AI' Accordingly. the pressure coefficient
PP.H.
on section FIHl is evaluated by expression (8.4.1), and the pressure coefficient PH,I.• on section HtL I • with the aid of formula (8.5.3). The drag coefficient of the airfoil is Cs;,
~'ILI
=+ (j
"'Ll
"'HI
Pt'lu\Ajdx -+-
"'PI
J PHILl~ dX)
(8.5.6)
"'HI Wing with All Supersonic Edges
For such a wing (Fig. 8.S.1c). Mach lines OKo• BKB • and DKo drawn from points O. B, and D are below the corresponding edges OC, BC and DC, therefore formulas (8.3.23) and (8.3.34) are used to calculate the pressure coefficient. Let uS consider airfoil FL with the coordinate 0 < Zl < zo•. Section FH is between leading edge OC and Mach line OK o• Therefore, the other edge OC' (Fig. 8.o.1a) does not affect the flow on this section, which is considered as plane supersonic. Taking into account that section FH is influenced by the sources in triangle OCC' having a strength of Q = 2Al V 00, the pressure coefficient PFH can be determined by formula (8.3.23) in which we assume A = At and n """ nl' The additional pressure on section HG is due to the influence of edge OC'. The pressure coefficient PHe on this section is found from expression (8.3.34) in which it is assumed that n = nl and (] = 0'1' Section GJ. in addition to the distribution of the sources in OCC'. is affected by the source distribution in BCC' having a strength of Q = 2 (A2 - AI) V 00. Using formulas (8.3.34) and (8.3.23). we obtain an expression for the pressure coeO'icient: PGJ
=
211
a'
J-'i-ni
(1-2sin-1'/ IIt-a~)+ 20"1-"-il 11 V 1-at a' ~
(8.5.7)
The last airfoil section J L is influenced additionally by opposite edge BC'. Accordingly. the pressure coefficient on section J L on which sources having a strength of Q = 2 (A2 - At) V 00 act is
PJJ.= a' ;~'-lIl
(1-
+
sin-I
V ~l~:l )
Ch. 8, A Wing -in
/I
Supersonic Flow
347
The drag coefficient of airfoil FL is C",.
FL =~
+(.\
3:u
XI;
PI'HA1 dx+ ) PUGA , dx
"'p
"-..
"'J
"'L
"'G
"J
+ ) PGJ).,. dx + J PJLA2 dX)
(8.5.H)
The pressurr coefficient PF,II, on section FIlii of airfoil FILl with the coordinate Zn < ZI < zu, is determined by formula (8.3.23) in which I.. = AI and n = n l . To determine the pressure coefficient PH,O. for the neighbouring section H]G I , we use formula (8.3.34) in \vhich A "-' AI' n = Ill' and a ='" a 1 . On the last section GILl' the pressure is due to the influence of source distributions in OCC' (Q = 2)'1 V 0<) aud RCC' IQ = 2 (1.. 2 - AI) V",,]. Consequently, to calculate the pressure coefficient Fe.L,. we can use formula (8.5.7). The drag coefficient of airfoil FILl is
CX.F'li.I={-(
"'HI
"'V\
"'Lt
jl;F]
:rHt
:to l
J PFIH,A,dx--:-- J PHIG1Ajdx+) PG,LIA.-zdx) (8.5.10)
ThE! flow near airfoil 1-':[,2 in the section zn, < ZI is plane supersonic, therefore the pres."ure coefficienl for it is determined Ly formilia (8.3.23). The sources in triangle OCC' having a strength of Q = 2Al V x act on section F 2G2 • therefore the pressure coe[ficient Pf,G, is found by formula (8.:1.23) in which I.. = Al and It = Ill' The second $ectioll G:L: is additionally influenced by the sources having a strength of Q =-= 2 (A 2 - AI) V 00 distributed in triangle BCG'. Therefore, the pressure coeHicient on this section is -
2}·1.
PG!I.,=
Ct'
1/1-1l~ --t-
2P'2-1.\) 1/1-'1l!
Ct'
(8.5.11)
The drag coefficient for airfoil F2L2 is
+(
XGI
C:t.F t LI ""'"
~.
jl;F~
PF2GzAI dx ~
"'Ls
J PG2LI~dx)
(8.;).12)
:to.
If a tetragonal wing has an extended tip or side edge (b t *0), the flow parameters are calculated with (lccount taken of the influence of the side edge on them. The part of the wing where this inl1uenc.e is ohsen"ed is below the Mach line issuing from the froot point of the side edge (see Fig. 8.4.4). The velocity and pressllre on
348
PI. I. Theory. Aerodynamics of an Airfoil and a Wing
Fig••.5.2 Drag coefficient of wings with asymmetric rhombiform airfoil:
Xc
= ·~clb •
.,
0.5; 11"....
b.lbt
= 5;
~2/1:1:~:'~1)/~~:/;1~ ~a: :~~ IIh~
dashpd curve 19 lin pxperlmcn.
talon(»
this section are calculated by the method set out in Sec. 8.4 with account taken of the kind of the leading and middle edges (Le. depending on whether they are subsonic or supersonic) by means of the relevant relations similar to (8.4.31). By integrating over the span, we can determine the wave-drag coefficients in each of the cases of flow over a hexagonal or tetragonal rhombiform airfoil wing considered in Secs. 8.4 and 8.5. According to the relations obtained, these coefficients depend on the number M ... and on the configuration and relative dimensions of the wing: (8.5.13) Here in addition to the known notation. we have introduced the quantity Xc = xc1b-the dimensionless distance to the spot with the maximum airfoil thickness. The number of independent variables in (8.5.13) can be reduced by using the relation C%/(Aw6. 2) = 12
O,w V M'oo -
1. Aw tan
;1.
Tiw.
xc)
(8.5.14)
where tan x, is the tangent of the sweep angle along the maximum thickness line (i.e. of the middle edge). Figure 8.5.2 shows a family of curves constructed in accordance with formula (8.5.14) for the following conditions: llw = bolb t = 5 and Xc = 0.5. The salient points of the curves correspond to sonic edges. Particularly. for the curve corresponding to the value of Aw tan = 3, the salient point for the smallest value of 'AwV M~ - 1 corresponds to a sonic trailing edge. and the second and third points to a sonic middle (the maximum thickness line) and trailing edges.
x,
Ch. 8. A Wing in ~ Supenonic Flow
349
:;r':.:-:-::1~=---;---'--ri--,---, I
---1- ,-
0.2
C5
t:r.ll
FIlii. 1.5.1 Drag of triangular wings wiLh a symmetric rhombiform airfoil in a supersonic
flow
(n, _ Un x,/«')
A comparison snows that the experimental and theoretical vallles of the wave-drag coefficients differ, especially near values of A".V" M!. - 1 ~ Aw tan x 2 , i.e. when the maximum thickness line becomes sonic. In this case, the linear theory cannot be applied. The discrepancy between the indicated values diminishes when this line is supersonic (I..wll M!. - 1 > I.. w tan )(2)' From the relations found for a tetragonnl planform wing, we can obtain the relations for the aerodynamic characteristics of a triangular \'ling as a particular case (Fig. 8.5.3). The trailing edge of such a wing is supersonic and straight (xs = 0). The leading edge nnd the maximum thickness line (the middle edge) may be either subsonic or supersonic and, consequently, inclined at different angles Xl and x 2 • Depending on this, we determine the locnl velocities nnd pres.~ures, and also the total drag coefficient c",. Figure 8.5.3 shows the results of calculating the function cxa'/(4L\2) for triangular wings with a subsonic (n] > 1) and supersonic (n) < 1) leading edges depending on 1 for varions vnlnes of n l (the sweep angle Xl)' The values of c",a'I(4K2) at n1 = 0 correspond to a straight wing with a symmetric airfoil. Figure 8.5.3 can be used \0 appraise the influence of the location of the maximum airfoil thickness on the drag. We can indicate the valne of which the minimum drag coefficient corresponds to. The salient point!; of the curves correspond to valuE'S or ~t which the maximum thickness line becomes sonic (n! = III ~nd r- = O.
r
r
r
350
Pt. L Theory. Aerodyn ..mic:s of
"1'1
Airfoil lind II Wing
Genefal Relation tOf Calculating the Dfal
The total drag coefficient of a wing can be written in the form (8.S.15)
where A is a coefficient determined by the kind of the leading edge; if it is supersonic, the coefficient A equals the reciprocal of the derivative of the lift coefficient with respect to the angle of attack [A = (C:,)-l]; with a subsonic edge A < (C~)-l because a suction force appears that decreases the drag. The quantity AC~a is the induced drag depending on the lift force. The coefficient of friction drag cx. r can he evaluated for an equivalent rectangular wing whose chord equals the mean aerodynamic chord of the given lifting surface. We may consider here that theboundary layer ncar the leading edge of such a wing is laminar and on the remaining part is turbulent. Let us consider the second component of the total drag coefficient, cx. wo, which is the wave-drag coefficient of the wing at 0: = O. We shall use formula (7.5.31) to evaluate it. According to this formula, at 0: = 0, the wave-drag coefficient of an airfoil is Cx. VI =
,
=
clK I = c1
J
(~L
+ ~~) dx,
therefore
for a symmetric wedge
we have Cx. w = O.5cliS~. Such a relation, characterizing the challge in the wave-drag coefficient proportional to the square of the relative thickness of the airfoil [iS Z = (6.lb)zl, canbeapplicd for a thin airfoil of an arbitrary configuration. By comparing the above relation for cx. w with (8.;).1 /1). we can determine the ratio (8.5.16)
in which cx. w is the wave-drag coe[[icient of the airfoil oriented in the direction of the oncoming flow (see Sec. 7.5), Aw and llw are the speed ratio and taper ratio of the wing, respectively. The function 13 in (8.5.16) is similar to the (unction 12 in (8.5.14) and is determined with the aid of the method of sources set out above. The quantity x may be taken equal to onc of the characteristic angles of the givcn wing (the angle of inclination of the leading and trailing edges Or of the maximum thickness line). Figure 8.5.4 shows a cmve characterizing the change in the function Is for a swept wing with a taper ratio of llw = :> ~ rp!~~.ive- coordinate of Xc = 0.5 and the quantity A.w tan Xl -= 3.67. The tirst
3~1
Cho 8, A Wing in II Supe'lonic Flow
,';~~v~_~~~~ ~),: /
r1
r-
I
___
~.
j
Mar(ll/llm UUCkf/CSS IHW
0.8
Fig.
'
8.S.~
Change> in drug {unction I, by Tormuln (8.5.t6) saliant point corresponds to the tl'ansformalion of the I.railing edge,. and til(! second. of the leading edge into a sonic one.
8.6_ Field of Application of the Source Method The method of sources, as we already know. is used to calculatu· the flow in order to fmd the drag forc~ of a wing with a symmetric Airfoil at a zero angle 0/ aUack, i.e. in the absence of a lift force. In,,-e1ltigations reveal that the fiCld of application or this metilOu ill ap.rodynamic research call be extended. Let \loS consider case::; when the method of sources can be used to determin0 a nearly uniform flow over a thin wing at u non-zero angle of attack, and we CMI thus lind the lift force in addition to the drag. Let us take two wings with different leading edges. One of them lias a curved edge wi til a nnite sllpersonic section (Fig. 0.Li.l), while the other has completely subsonic leading edges (Fig. 8.6.2). In Fig. 8.6.1, the supersonic section is bounded I;y points E and t'r at which a tangent to the contour coillcides with the genera trices of the Mach cones. Let us con~ider tlw velocily potential at a point .H in the region confined by euge!; ED and the Mach lines dmwn in plane xOz from points E, J), (Jr, and E'. By formula (8.2.16) in which the integration region f'1 shouleJ bo taken equal to (J " - SI S~, the velOCity potential at the point being considered is
+
JJ y (~1 ('i):) d~z~~ t)2 --irt JJ V(Z~I(~)II;)~ld~
cp' = -
2~
5,
t;)i
5,
(8.6.1)
atl2
Pt. I. Theory. Aerodynamics of lin Airfoil and a Wing
~~~n;'~1tb
a finite section of a supel'3onic leading etlge
'F1,.I.6.l
Wlng witb subsonic leading
edges
In (S.6.1), the function Ql (x. z) = 2 (OI.p'!oY)II'-'o' This follows from (8.2.17). This function is determined from the condition of flow over the wing surface without separation (8.1.12). Since the equation of this surface is given, the function Ql (x. z) is known. In the particular case of a wing in the form of a plate in a flow at the angle of attack IX, the function Ql = 2V""IX. Hence, the determination of !p' by formula (8.6.1) is associated with the finding of the unknown function Q2 determining the intensity of sonrce distribution on section S •. To find this function Q" let us take an arbitrary point N (x 0, z) in the region between the Mach lines issuing from points E and D . At this point, according to (8.1.20). the velocit}' potential is zero, therefore in accordance with the notation (sec Fig. 8.6.1), we have t
0= -
it- JJ y' (~I (ti);) d~J:~~ S,
+ -it .\ .f -V (~2 (~t)a~) ~~ ~)1 S,
t)! -
(R.6.2)
Ch. 8. A Wing in " Super~onic Flow
3~3
The ftrst term on the right-hand side of this int.egral equation is a known function of the coordinates of a point because the strength Q1 on area S3 has been determinerl from the boundary conditions. We can therefore use the equation to determine the unknown function Q~ that is the strength of the sources in region 8 4 , Hence, if the leading edge of a symmetric airfoil wing is completely or partly supersonic, the metlwd of sources is suitable for investigating tM flow over tM wing at a non-zero angle of attack. The same conclusion evidently also relates to a wing with similar edges and a nonsymmetric airfoil in a Gow either at the angle 0: =#:0 or at ct = O. I\ow let us consider a wing with subsonic leading edges. We can derive the following relation for a similar point N (Fig. 8.6.2):
0= -
2~
.\.\
s,
y(~3(~)2~)'~:ld~ I;l~
-
~ JJ 1/(~d~)2{:)~~~
1:)1
~-
(8.6.3) We have obtained an equation with two unknown functions Q, and Q2' Similar to Q2, the rnnction Q3 is the strength of the sources on area 8 s belonging to the region located between the left-hand leading edge and the Mach line issuing from the wing vertex. Hence, if a wing has a subsonic leadirl{! edge, the source method cannot be used to investigate the flow oeer a thin symmetric airfoil wing at an angle of attack, or the flow over a non-symmetric airfoil wing at a zero angle of attack or when 'J. =#: O. '.7. Doublet Distribution Method It has been establishod that the application or the source method for investigating supersouic flow is restricted to wings with completely or partly supersonic lending edges. In other cases associated with the investigation of the supe-rsonic aerodynamic characteristics of wings with subsonic leading edges wit.h a non-zero angle of attack (or of similar non-symmetric airfoil wings and at a = 0), the doublet distribution PlE:lbod mllst be llsed. Let us consider a doublet in a supersonic flow. To do this. we shall determine the velocity potential of the Dow produced by an elementary source and an elementary sink of the same strength Q having coordinates x = ;, z = t y = 8, and:x = ~,z = t. y = -e. respectively. The chosen source is located above the plane 11 = 0 at the small distance I:: from it, and the sink is under this plane at the same small distance -e. Expressing (8.2.11) as a differenceequation, we shall write the potential produced by the source and sink
354
Pt. I. Theory. Aerodynlmic5 of on Airfoil lind 0 Wing
in the form A
~~o
+
V(l:
{Jill:
~)~ Ct"~(:
~)!
Ct"I\"-l.jnZ-t-(s
~)IJ
ep+(s
Q1J }
Introducing the symbol p = V (x GrOl a'il [gil disregarding the quantity a'le t , we obtain Aq:=
+
(z
e)'] and
~:I~ ( 1/1+~\ge!!l1 + Vl-:a.'2ge/p'I)
Using the expansion of the square ro~ts into a series and deleting the second and higher order infinitesimals, we fmd ..\.rp =
Ct'I~::6a
:l
U~ Ct~~i:!~ (z
tt.!:
t):lJ}3/'i
By calculating the limit of Arp at E __ 0 and assuming that the quantity ;vl = Qe called the moment (or power) of the doublet remains constant. we obtain an expression for the differentia) of the potential function of a doublet
This expression can be written in the form d(Jldoub=
~'lda .-!y{
r{Z
~)! Ct~~IY~+(Z ')~J}
Integrating over the region a which the influence of the doublets extends to, we obtain for the potential function 'Pdoub=..!.... Oil
ii 0
V{.!:
M(~. ~ldtd~
~I-
':A.-~IY~+{s
t)~1
(8.7.1)
where the number n; has been included in the doublet distribution function. It ean be shown that funetion (8.7.1) satisfies Eq. (8.1.7). To do this, let us differentiate (8.1.7) with respect to g: (M!.-1)
fu-( ~:~' )-i (::~. )- :11 (0;';' ) =0
Wo can write this equation as (M!.-1)
!2~
( DO:' ) _ :, 1. ( °a~' ) _
:Z2: ( D8~' ) =0
(8.7.2)
Now let us consider expression (8.7.1) for the potential or a doublet. If we include the quantity -1/(211:) into the expression for this
Ch. 8. A Wing in a Supersonic Flow
355
potential, then in accordance with (8.2.12) the double integral can be considered as the potential of sources whose distrihution over area o is set by a function M. Consequently, 18.7.3) Comparing (8.7.2) and (8.7.3), we see that the funclion (fdoul) does indeed satisfy Eq. (8.1.7) for the velocity potentiaL Knowing the configuration of the surface in the flow, the freestream velocity, and the angle of attack, we can find the doublet distriblltion function J/ and thus determine the potential Ifd(lub of the douiJlels. The derivative of the fUllction (fdo"b with respect to x yields the additional longitudinal componellt of the disturbed velocity u = alPdOllbiaX that is llsed lo c;llculate the prpssure coefficient ;; = -2u/V;.o on the lifting surface and the lift force proullced by this surface. 8.8. Flow over a Triangular Wing with Subsonic Leading Edges A plane triangulaf wing with subsonic leading edges is inside a Mach cone (Fig. 8.8.1). To find tho lift fOfce of such a wing, we shall use the doublet distribution method and the relevant relation (8.7.1) for the velocity potential of doublets. H is general knowledge that the additional velocity u induced by sources distributed over an inclined triangular surface with subsonic edges depends only on the function cr = zu.'lx [see, for example, formula (8.3.18)1. This signifies that along the fay issuing from the salient point of the leading edge of the wing at the angle 'I = = tan -1 (zlx) the velocity is constant. This ray can be considered as the generatrix of a cone with its vertex coinciding with the salient point of the leading edge. A flow retaining a constant velocity anci, therefore, other constant parameters along a generatrix of such a conical surface is called a conical flow. We can consider that the additional velocity u for such a flo\v in the plane y = 0 is a function of the ratio z/x, i.e. u = f (zlx). It thus follows that the potential produced by the sources for a conica.l flow also depends on this ra.tio. Considering the flow set up by doublets near a triangular surface as a conical one, its potential can be written for point P in the plane y = 0 (Fig. 8.8.1) as follows: IPdo"b
= xFdollb (zlx)
!(8.8.1)
where Fd""b (zlx) is a function :c1epending only on the angle of the conical surface 'I = tan- 1 (zlx).
Pt. 1. Theory. Aerodynamic:s of an Airfoil and a Wing
866
fig.
a.l.t
Plane
triangular wing
with subsonic edges
fig. 8.a.2 Region of doublet influence on the flow over II. wing with subsonic leading edges: I-region of doublet InDuence; . trlaniu1ar wlnr
For a point with the coordinates x, y, z we shall write the potential of a conical flow in a more general form:
= ZFdoub
(zlx. y/x)
(8.8.2)
In accordance with Eq. (8.8.1), the relation characterizing the distribution of the doublets in the plane y = 0 (00 the wing surface) is such that AI (S. ,) - ~ (h) (8.8.3) where h = ~~, and m (h) is a function of the argument h. Let us transform Eq. (8.7.1). We shall express the elementary area occupied by a doublet in the form da = ds d~ = dh because d~ = S dh. Therefore
s ds
tl
eotl<
CPdoub =
) -cot~
m
(h) dh-f;) 0
Y (z E)i
!~I~!la+(s
11;)'1
(8.8.4)
Ch. B. A Wi"g ;" ~ Supe'$or'lic Flow
3a7
where ~l is the coordinate of a doublet that may influence the flow at point P (x. y. z). This doublet is on the C\1Tve confIDing the region of doublet influence (Fig. 8.8.2) and obtained as a result of the intersection of the plane y = 0 with the Mach cone issuing upstream from point P. This curve is parabola AIBI described by the equation (x - "~I' - ,," iy' -'- (z ~,)'I ~ 0 (8.8.5) in the coordinates the form
61' ;1'
We shall write the rUtlicand in (8.8.4) in
where
a=
1_r:t.'2h 2•
b=2x (a.'Zh+ -
1). (8.8.7)
(8.8.8)
r
Taking into account that a = 1 - a,'2 and h 2 integral on the right-hand side of (8.8.8): ~1
>
0, we lind the
~
•
\ ~ '"+lnIZVa(as'c bO"): 2aO·'-bll"
~
«;2+
-;-C
"a
0
By (8.8.5), the value of
?~ -+
+ C = 0, = ~Vc
as~ -i- b~l
~2d;
v~t,
b,.:-c
3:~2-:~C JIn (2a~t -!- b) -
18.8.9)
consequently,
'la-
hi (2
va;; + b)j
(8.8.10)
Solving Eq. (8.8.5) in the form a~~
we obtain
+ b~l + C =
(8.8.11)
0,
(8.8.12) Accordingly,
In (2a~1
+ b) =
In Yb~
-
4ac
(8.8.13)
3!58
Pt. I. Theory. Aerodyntlmics of tin Airfoil .nod • W,ng
is The difference of the logarithms in (8.8.10) with a view to (8.8.13) In (2a;, : b)-lo(2V;IC,b)=ln lfb Z.=4ac =...!....In 21'ac,b
=...!....In b-2~ 2
=
2
b~:..4ac
(2 Y
-th-'-'-
(8.8.14)
2 Ira;:
b·j-21'ac
Therefore,
}t ~
~2d~
_ """' 3b
l'asz+b~~c
,r;: _ 3b~-4ac
~
8a 2
th-' _ , _
1'a
2
vac
(8.8.15)
We ca.lculate the derivative with respect to y from (8.8.15):
y r' aS~+b~--:-C ~tds
.*( iJ
N
8ii 0
:...: 2a
~'a
4
J~;;C • 4a~b2
-i-Ue' 2
Vac)
Introducing the symbol
'" ~ M(2 ViU) and llaving in view that i}cioy = _2y(1.'2 (see formula. we obtain
A'=
~:~'2
(
1~\"2 +th- I v)
(8.8.16) (8.8.7)1. (8.8.17)
We introduce tlds expression into (8.8.4): (8.8.18) To determine the form of the doublet distribution function, wo shall use the condition of flow without separation, in accordance with which
~ WI"" (.!!!...) m (h) dh ( ,.''''") iJg y=o -cOl iJy y=o
(8.8.20)
y.
Introducing this expression into (8.8.19). we find
cOI>< V,.,a= -1>< (°0: )y=om(h)dh
(8.8.21)
Ch. 8. A Wi:->g in 5 Supersonic Flow
Se,9
Differentiation of (8.B.1i) wilh resp('ct to y yields
( u:;
)11_ II~":"
-
,::1.;;.
(I-':V1 ;- th- I v)Y;oOQ
(8.8.22)
Let liS calculate the partial derivative with respect to z/x from (8.8.20) with a view to (8.8.22): (8.8.23) We calculate the value of the derivative ov/o (ZiX) 1,1 "'"'0 by (8.8.15) and determine the quantity (1 - v·!)~ 111 -0 by using (8.8.7) and (8.8.16). After the corresponding" substitutions into (8.8.23), we obtain the equation (8.8.24) We shall show that this equation has the same Corm as the one obtained when using the doublet distribution method to soh-e the problem on the flow of an incompressible nuid over a flat plate of inrlDite length installed at right angle:.; to the direction of the freestream velocity. We can determine the velocity potential for a plate that is a part of a wiug in section AH with the coordinates ZA = -c. ZB = c, and over which an incompressible Ouid 80\\'s in a lateral direction at the velocity v = V ...a (Fig. 8.B.:i) with tbe aid of Eq. (2.9.16). We shall write the latter in accordance with the coordinate system chosen in Fig. 8.8.3 in the form 'fdoub = y/[(z -"Il)2 -\- y2J
(8.8.25)
This expression determines the potential at point P (y, z) produced by a two-dimensional point doublet with a unit moment
a60
Pf. I. Theory. Aerodynamics of
M
Airfoil and a Wing
(M = i), If the moment of the dOllblet differs from unity and the
doublet distribution ovcr the span of the platc is set by the function M(ll), the velocity potential induced by the doublets on the wine; sE'ction dll is ~ (8.8.26)
The \'elocity potential at point P due to the influence of the douhlets located alollg the span of the plate on the section from ZA = - r ; tozn=cis (8.8.27) The vertical component of the velocity v face of the wing is
v,.-_o=(OlJlrlOUh) y
iJ!I
'/~O
,=[~ l~ {/y
Since the component plate, we have
M(II)Y,dl l
_c (Z-Tll:Tg~
vI/CoO
~-
]
= ,1-",0
CM(IJ)d~1
!c (z-J[)-
sur-
(8.8.28)
docs not change over the span of the
(k) "__ " ( -d· •. __ , ~
= ()ffdoub,'{)Y on the
\~
,11 ('lld11 .-... 0
(Jz.:: c (z· -
I!J~
(8.8.29)
Differentiation yields (8.8.30) A comparison of Eqs. (8.8.24) and (8.8.30) reveals that they are both of a single type, therefore it is possible lo take the dou blet distribution function m(h) in a .supersonic linearized f10\\' of the same appearance as the corresponding function M(ll) in an incompressible flow, To determine the form of the function i11('l), we shall us!:' the solution of the problem on ftnding the potenthd function for a plane plate over which an incompressible fluid nows in a laleral direction (see Sec. B.2). According to this solution, the velocit.y potential on the plate is determined by formula (6.2.7). The relevant potential difference on both sides of it is l\cp = 2VJfall - T. The now of an incompressible fluid near the plate is considered as the result of
:~;::ao~~~~o~:: ft~~~~~: f6.3~~)rd ~ns~~~,!~~~~et:he°ndt~trt!~~t~~di!i doublets lor a plane plate in a non-circulatory floUJ of an incompressible fluid is equivalent to the potential diflerence l\ff. Having in view that
Ch. S. A Wing in
C!I
Supersonic Flow
361
the doublet distribution function m(h) in a compressible flow hasthe same form as for an incompressible one. we can consider theexpression for this function in the form m(h) ~ L ]f H' - It'
(8.8.31)
where 112 = cot".! x and L is a proportionality coefficient. Wc shall lise Eq. (8.8.21) to fmd the coefficient I... Inserting (8.8.22) and (8.8.31) into it, we ohtain o;V... =-o;'2/,
.. a;/a
"'
~
-cot
('1-':,,2 +'lh-l'·),,~.. nVeot~~-h2dh
y.
(8.8.32) To simplify the calculation of the coefficicnt L. we can int.egrate· (8.8.32) along a longitudinal coordinatc on the wing. Having in view that along the x-axis the coordinates z ~ 0 and y = 0, we determine the value of I v/(1 - v".l) .+- t.h- 1 v 11/ .... 0 in accordance with (8.8.7) and (8.8.16), and the value of a Yaby (8.S.7). Integration yields :fl,').
, V1
aV ... ",_nL ~
(1
a'zcoV"X)sin2rpdrp
(8.8.33)
The integral n/'
~
Vi
(1
o:''J.cot.Zy.)sin2q:d
(8.8.3·\)
is a complete elliptic integral of the seeond order with the parameterk=]fl-a"cot'x
(8.8.35)
The values of inLegral (S.8.34) arc determined with tlte aid of Lile previously calculated tables depending 011 the parameter k. The doublct distribution can be expressed by function (S.8.3) provided that m(h) is replaced by (8.8.31) and with account taken of the value for L determined from (8.8.33). As a result, we have ill (I, t) ~, V cot' x - h'
;;'7.)
(8.8.36)
Let us find the component of the induced velocity pro(Juced by the doublets on the surfaco of the wing for y = O. From (8.7.3), we have (8.8.37} i.e. the potential of the doublets on the wing is determined by the magnitude of the vertical component of the velocity induced by the
362
PI. I. Theory. Aerodynamics of an Airfoil and a Wing
"Sources. A comparison of (8.2.12) alld (8.7.1) reveals that .11(£, 1;;) can be considered as a function similar to the function Q(t 1;;) determining the distribution of the sources over tho surface of a wing having a given planform and angle of attack. Consoquently, if we take into account that ill (8.7.1) the number It is included in M(s. 1;;) in accordance with (8.2.7), (8.8.36), and (8.8.37), we can compile the equation
(crdoubl,=o=llM(S. ~),..,(L\--'",~ l;.~
(8.8.38)
where h=:~/; (or h ".oZIX). The component of the induced velocity due to the doublet$ is
u.=( a!f~;uh },_o=
~I:~ .~(x
Vcot2K-*) (8.8.39)
The corresponding value of the pressure coefficient with a view to the possible signs in front of the square root is
p=
2(:",-;{")
= _
~.: = E (k~l~::::x
h2
(8.8.40)
where the plus sign determines th.e pressure on the bottom, and the minus sign 011 the upper side of the wing. The field of pressures corresponds to a conical flow relative to the pertex of the wing at which the pressure coefficient p = const for all the values of z/x = const. TIle lift force acting 011 a triangular willg consists of the force produced b~.. the pres'mre on the hottom surface, and of the suction force eqnal to it in magnitude and caused by the rarefaction on the upper side. The elementary lift force acting on area dS = O.5xdz (see Fig. 8.8.1) is dY, _ 2 (p -
p~)
dS -
(p -
p~)
x d,
(8.8.41)
The total force is obtained as a result of integration over the entire surface of the wing S ..... = Xli cot x: Y,_ ) 2(p-p~)dS
The lUt coefficient is
(8.8.42)
'w
(8.8.43)
Ch. 8. A Wing in • Supersonic Flow
963
After integration, we have C'a =
2a:t cot x,'E(k)
(8.8J.4)
For a conical flow, the centre of pressure o[ each triangular element issuing from the apex is at a distance of two-thirds of the altitude from the apex. Therefore. the centre of pressure of the entire wing is on the root chord at a point that is at a distance of t\\'othirds of the chord froID the apex. Accordingly, the coefficient of the pitching moment about the wing apex is mi. = -c'acp = -(4/3) an cot xlE(k) (8.8.4S) For a triangula •." low aspect ratio a ' cot x
wing he _ a/2). th.e quantity value of the elliptic integral wing we have -(4/3) em cot i( (8.8.46)
l':xpressing cot x in tcrms of the wing aspect ratio All' = 4 cot x, we obtain (8,8,46')
If the sweep angle x is chosen such that i{ """ n/2 -11"" and, consequently, the Mach linc coincides with the leading edge. we have cot x = tan ~ "" and 0.' cot i( = cot !l"" cot X = t In the given case, til(! elliptic integral E(f..:) '-, :t '2. therefore for a triangular win~ with a sonic leading edgc, wt' luw(> (8,8,47)
Since coLx=lanll ... ",1,J!M;,-1 we obtain
C,. -"
4a. V M;" -, 1 (8,8,4g) This value coincides with tile lift coefficicnt for a thin sbarpnosed airfoil in a linearized snpersonic flow. We shall show that the lift coefficient of a triangular wing with. supersonic leading edges is expresf:led by the same relalion (8.8.48). In accordance with (8.3.:14). the pressure coefficient for the wing in the region between Mach cones is
PI . .- a'
(1--...!. sin-I" /-
± 20x )/1
_Nt
:1
V
112 __
~~
1_0 2
)
(8.8.49)
and on the section between the leading edge and a Mach cone (see
(8,3.23)1 it is
i, ~ where n = 18nx/a.' t and
±2a/(a' V 1 - n') = Z t8nx/.z; = h tan x.
G
(8,8,50)
364
Pf. I. Theory. Aerodynollmk5 of In Air/oj! Ind I Wing
FI".8.U Wings with supersonic trailing and side edges (zo = hoI: "-"'1raKona) wing will,
hl'orut
pl"t~):
dov~ta'J;
b-h'lragona) win!: wtlh a vee-sbaped appendagf' (rhom-
c-pl.'nh.gomti wing; d--hna,onRI wll1g
According to (8.8.43), (8.8A.9), and (8.8.50), the lift coefficient is Cll
8
=2
'j'n Iplldo-2 1Ip'l.ldo= v
V
4c
./_ 0:' r t-n¥
;, [1 (1-+,'..-. V n;=:: )00-1 dO] o
I
Integration yields the relation clia = 4ala' that coinci(les with (8.8.48). The methods of calculating the flow over triangular (delta) wings can be used to determine the aerodynamic characteristics of lifting surfaces in the form of tetragonal, pentagonal, and hexagonal plates with supersonic trailing and side edges (Fig. 8.8.4). The flow over them is characterized by the absence of zones of mutual influence of the tail lind side regions confined by the intersection of Mach cones with the wing, i.e. when the flow at the side and trailing edges is supersonic. As a re~;ult, the coefficient of the pressure on the \ving surface is the same as at the relevant point of a triangular plate. The formula for calculating it is chosen with account taken of the kind of leading edge (subsonic or supersonic). The lift and moment coefficients can be determined by integration according to the pre9sure distribution. We shall give the results obtained for a tetragonal wing: with a subsonic leading edge
ells = m'a '-
(:'~:;)~x/k)
[(1- e)-J/2cos-le-eJ
~;40:e~O~. ~l) [(t~-:;:I' cos-Je- 8~4-=:~)2]
(8.8.51) (8.8.52)
Ch. 8. A Wing in a Supeuonic Flow
1"'":+
I
SSG
.
(=a I-t:~ !a.~
FIg. I....
Curves characteri~iDg the change In tile derivative of the lift coeflicic.>nt (a) ami centrHlf·pressure cocfllcient (b) for a tetragonal wing
with supersonic edges
cv~",--- (1-~nCl' (1/1~,tt mz. =
cos- t nl-
vb cos-in)
3(t=~:1Cl' [tt~~)~f(i1.!~;;!/~ (t+:~V-~:~nt
(8.8.53)
COS-Inl
cos-1n- 1':",t1 ]
(8.8_51)
where £ = {Xt - Xo)/Xt = tan xJtan XI; n 1 = tanx3/a.' = e.n; Xu and Xt are tbe dimensions shown in Fig. 8.8.4. For dovetail wings (Fig. 8.8.4a), the quanti.ty e is positive, and for rhombiform plates (Fig. 8.8.4b). it is negative. The relations for C"II and m~a allow us to determine the centre-ofpressure coefficient c p = xp/xo = -rnz/c ,I •• Figure 8.8.5 sho\,,-s curves characterizing the change in c:a and cp in accordance with the above relations. The dashed lines in this ligure determine the values of the aerodynamic coefficients in the limiting case when tIn = cotx1a.' = e (a sonic trailing edge). A salient point appears on the curves (Fig. 8.8.5) at a valua of n = 1 (when cot )£Ia.' = 1). i.e. when the leading edge is !:Iunic. The coefficients c"a and cp behind the salient point (with a snpersonic leading edge) diminish with an increasing M "". Let us consider another limiting case corresponding to sufficiently high values of cotx1cx'» 1 (n 4:: 1, "t
366
Pt.
r.
Theory. Aerodynamics of an Airfoil and a Wing
wings in Fig. 8.8.5) alters the lift force of the wing almost in proportion to the change in the area of the plate. It is not difficult to see that for e = 0 formulas (8.8.51)-(8.8.54) yield the values of the relevant coefficients for a triangular wing: cVa = C""4. and m:ra = m'a4.. The calculations by these formulas can be simplified if the tetragonal wings differ only slightly from triangular ones. In this case, provided that I £ I <: 1, we have (8.8.55) •.9. Flow over a Hexagonal Wing with Subsonic Leading and Supersonic Trailing Edges
Let
115
find the aerodynamic characteristics in the more general
!~~~ o~ti~e~~n~t kad~~la=dg~~p~~s!~~~cta~ii~g aeli:rnw~~~8~9.1)~ Such a kind of the trailing Adges excludes the influence of the vortex sheet behind the wing on the flow over it. To determine the velocity potential,let us use the results of solving the problem on the flow over a triangular wing with subsonic leading edges. Let us consider point A with the coordinates x, z in region 1 confined by the leading edges and the Mach lines issuing from points G and D. The velocity potential at this point can be expressed by analogy with (8.6.1) with the aid of the following formula:
~'~ _....!.... \ I
2:t.0'
j'
VC:r
Q,I" tld,dt
e:p
a'l(z
Q2
+&'1' '·A~
2'
3
(8.9.1)
in which (1 is the region of integration on the wing surface, dIP, and alP, are the additional potentials determined by expressions similar to the second and third terms on the right-hand side of (8.6.3) when integrating over regions (11 and (12 (hatched in Fig. 8.9.1). Taking into account that Ql = 2). V 00 = 2« V 00, we can write Eq. (8.9.1) as follows: , .v~g r r d,dt 92 tpl= - - n - J.,J Y(z e2 a'~(lI Q" (8 . . ) where in accordance with (8.9.1)
Q=1-(~IP2+d'P,)[
a:"" tJ
y(z
,)2
d'~'I(J: ~)l
]-1
(8.9.3)
For our further transformations, we shall introduce a charaew teristic system of coordinates whose axes rand $ coincide with the
Cfl. 8. A Wing in a Supersonic Flow
367
Fig. a,t.t
Hexagonal wing with subsonic ]Nding and supersonic trailing edges
directions of the Mach lines issuillg from the vertex of the wing (Fig. 8.9.1), r = (M",,!2a') (.r -
a'z),
S
= (M",,l2ct') (x
The characteristic coordinat;:>s of point A(XA' rA = (,v",i2a') (x,\ -
a'ZA). SA
+ ct'z)
ZA)
(8.9.4)
are as follows:
= (M",/2ct') (XA + a'z-\)
(8.9.4')
Let liS convert Eq. (8.9.2) to the characteristic coordinates rand s. From (8.9.4). we fmi!: r
+s =
Consequently, x - S = ;rA a' (z -
~)
-
= a' (z..\ -
(M ",,:a')x, r -
I =
(a'iM "") ((r.\
S
= -M ... z
+ SA) -
z) = -(a'/l"I "") !(r,\ -
(r
(8.9.4")
+ s)1 }
s,d _ (r _ s)] (8.9.5)
Introdncing these expressions into (8.9.2) and taking into account that. an area element in the coordinates rand s is do = dr ds X X sin (21l_) (Fig. 8.9.1), and that the integration limits are SB and S..\. rc and rA, we obtain an equation for the potential function:
(8.9.6) We shall express the coordinates 6D and rc in terms of tIle coordinates rA and SA of point A. Since the equations of the leading edges in the coordinates z and x are % = ±x tan x. then in accordance wiLh (R.D.4"') these equations in the coordinates rand s are
a6S
Pt.
I. Theory. AerodYl"lamics of an Airfoil and a Wing
transformed to the form: T = 8m for the starboard leading edge: z = +x cot x r ~ ,/m fo, the po,t I.ading edge: '~-x cot x
l
(8_9_7)
where m = (n - 1)!(n + 1) and n = tanxJ!a.'. Therefore the coordinates TC and SB can be expressed as follows: TC = scm = sAm and SB = 1'Bm = TAm (8.9.8) Substituting SAm and TAm for after integration we obtain q>i= -
4~~:Q
TC
V(rA
and
Introducing into (8.9.9) the values of also the quantity m ~ (n -
SB
in (8.9.6), respectively,
mSA) (SA
1)/(n
rA
and
m"A) SA
(8.9.9)
from (8.9.4') and
+ 1)
(8_9_10)
we find (8_9_11) where h = ZA!XA (or h = ~!~). By comparing Eq. (8.9.11) with relation (8.8.38) for the potential function at the point being considered, we see that for matching of the results, in (8.9.11) we must assume that (8_9_12) Hence, at point A located on the wing in region T, the velocity potential is (8_9_13) Accordingly, the equation that must be used when determining the potential function on the wing has the following general form:
,
«v"" (n+t)
q>I= 4nM ... E(k)
II a
dr ds VrA-r' VSA-S
(8.9.14)
Let us use Eq. (8.9.14) to determine the potential function at point A in region II that is conHned by the Mach lines issuing from points D and G, the tips, (side edges) and partly the trailing edges. We shall write Eq. (8.9.14) in the following form: q>II = B~'
~:;~~~k~)
rA
"A
rK
sD'
J vr~r_r J
Vs:S_s
(8_9_15)
analogy with (8.9.8): SB' =
TB,m = TAm
(8_9_16)
eh.
8. A Wing in e Supersonic Flow
869
Point K is on a tip whose equation is z = U2. In the coordinates rand s, the equation of a tip in accordance with (S.9.4W) is
r - S - -M .li2 Therefore, for point K, the coordinate is
(8.9.17)
rx - SK - M.li2 - SA - M.1I2 (8.9.18) Taking also into account that SB' = 'Am, after integration of (8.9.15) we obtain
fJI[I= 1X:;~~t:) J./[rA-(sA- M;')] (sA-rAm)
(8.9.19)
Introducing instead of rA, SA and m the relevant values from (8.9.4') and (8.9.10), we find
Til = I::~~)
V
n"';c,1'--(l---2-'-A)""("':"'~-T-'
-n-,."")'-
(8.9.20)
The velocity potential for point A in region III is ~A
r ..\
Till
~~:;::.;;;
J
II
"I('
r~r_r
I
l"s~:-$
(8.9.21)
'0'
Points K' and G' are on thn tips whose equations are z = ±1I2 (the plus sign is for the starb031'(1 find the minus sign is for the port tip). In the cooruinates r <'lid $, the equations of the~e tips in accord. ance with (8.9.4) have the form: r - s = -M colf:!. [or the starboard tip (8.9.22) r - S = 1)1 ""lI2 for the port tip
I
Consequently. for points K' and G'. we have. respectively: I':K'
= SK' -
SG'
rG' -
M_1I2 = SA - .lHCOll21 M.li2 _ rA _ M.1I2
(8.9.23)
After the integration of (S.U.21), with a view to the values of t.he limits (8.9.23), \\'e obtain ~----uo~---,",~
'Plu=
a:':;~::~k~) V(rA-sA~~'
Mil) (sA-rA M;')
We introduce into (S.9.24) the values of
SA
'PIll "'" 1X1;::~~~,t) 1!P-~4z~\ By calculating the partial derivatives witll respect to the components of the disturbed velocity It = u(P:J{h A 2~-Ot ~
I~
(8.9.24)
and 'A from (8.9.4'): (s.n.25) XA. we lintl ~II .'''' I, II,
pt. I. Theory. Aerodynamics of lin Airfoil and a Wing
370
Ib)
FI~.
8.U
/~' <, "~'.:. g
WlDgs with subsonic leading edges and tips and with supersonic trailiDg edges:
li
,,/ ", '" X
..
mo' ~ .
1--&--+---1---,
FIji!. B.9.3
Dlstribution of the quantity
:1'~ =p~~fa;la'i)/awr;:r ~:~ subsonic leading tion AA)
edges
(sec-
lJO
60 80 100r. X"(xjb)100
III) and determine the pressure coefficients in the corresponding regions on the bottom and upper sides of the wing:
PI~
-
Ir~'
::A; .. _± 2C1.:'('~2)X.-".~,;"",'='" l'.r~c"t";<-z}.
I
I
PI! "-' - v2". • ::.~I =- ± nt.~k) ,~/ 2;:\-!~1~~,d~ZnA~) ) (8.9.26) I Pll! -- - ~~". . 8::;1 :.-,0 J Hf're the plus sign on the right-hand side of the equations corresponds to the bottom side, and the minus sign, to the upper one. The above relations for calculating the pressure coeRicients can be lIsed not only for triangular wings with a V-shaped appendage (Fig. 8.9.1), but tllso for similar wings with a dovetail Hnd wings with a straight trailing edge (pentagonal plates shown in Fig. 8.9.2). The change in the quantity I1pla. =
Ch. 8. A Wing in ~ Supersonic Ffow
371
(h)
O.761--t-,ij'---l
Fig. 8.9.4 Curv~s chara('t~ri:ting lh~
coefficients fur
;1
change in the lill (4) and eentre-of.prl'ssure (b) pt'nlHgonal wing t
cllOl'(l) caku\atf'il by tile above formulas for a pl'ntagonal wing with 1.11 is showll in Fig. S.!.Li, On the surface of the tip Mach COlle,
M".
CII"
q~;,,"
---* ,\.\
(Ph-p',)d,nl;
(H.D.27)
~w
where Sw alld bl) are the area ilnd the root ChOftt of the wing, respcctiYcly,andph and Pit nre the codfjcienls of tllP prt'~slll'e on the holtom and upper sides determined by formulas (8.\I.:!\i). Arcording to the VAlues or nil;! and tv;,. we ClIli dclCfllline the centrf'-of-pressllre coefficient c" == dWbl) """ -nt z;, cy,,' Figure 8.9.4 shows theoreticnl CIIl'\'es charactC'rizing the Chl\11gc in t' ll a and t'p for a wing in the form of a pcniligolllli plntc. III this flgllre. the parts of the curves for which "",liM!, - 1 «: Aw tally. correspond to subSonic leading edges. A feature or the graphs chAnn:terizing the change ill c!J~ is the ahsC'ncC' noticeable salient point!l- 011 the corresponding curves in the transition to snpersonic l(>ading edges (Le_ whell AwV M;, - 1 = ')..~, tan y.).
or
372
pt.
r.
Theory. Aerodynamics of an AirFoil and a Wing
'.10. Ffow over a Hexagonal Wing wth Supersonic Leading and Trailing Edges
To calculate the now over arbitrary planform wings (including hexagonal ones with supersonic leading edges) in regions I and /I (Fig. S.iO.ia, b) we can use the corresponding results for a triangular wing with the same leading edges (see Sec. 8.3). According to these results, the pressure coefficient for point A (XA, ::::,d in region I between the leading edge and the Mach lines issuing r'rom the vertex of the wing and points D and G on its tips is determined by the formula (8.10.1) that has been obtained from (8.3.23) provided that a has been substituted for 11.. In the same condition, we find the corresponding relation for the pressure coefficient in region /I confinod by the Mach lines issuing from t1Hl vertex and the same points D and G. Dy (8.3.34.), we have
pu=± a'~ ~t-n~ (T-sin-'V~~-=-:~2)
(8.10.2)
The flow over the remaining part of the hexagonal wing (Fig. 8.iO.ie-h) can be calculflted with the aid of Eq. (8.3.1) for the velocity potential of the sources and sinks. We transform this equation to the following- form in the new varia bios rand S [see (8.9.4)] (8.10.3) where rA and SA are the coordinates of point A on the surface of the hexagonal wing. The coordinate lines rand s arc directed along the Mach lines. The wing surface is divided into eight regions by these Mach lines, as well flS by the lines of weak disturbances issuing from points G and D, and also from intersection points G' and j)' of the coordinate lines r anri s (the '1ach lines) with lhe tips. For each of theSE! eight regions. tile velocity poll'ullal is calculated with the aid of Eq. (8.10.3). Let liS conside(' arbitrar~' point ;1 in region 1/1 (Fig. B.10.1c). Wht'!n determining thc velocity potential for this region, we must take into accounL the sources both on the surface of lhe wing (on areas 8 1 = ADl/J~;I' and 8 2 = A'/J 2D) and olltsi(le of it (area S, ~ A'DD"),
Ch.
e.
A Wing in e Supersonic Flow
~
Fig. UO.t
Hexagonal wing with supersooic icadiDg aoil trailiog edges
T' .S,
373
874
Pt. I. Tnaory. Aerodynamics of an Airfoil and a Wing
t:'sing (8.2.13), let \IS write tbe expres.<;ion for A (x, o. z) in the form
(Pill
at point
(8.10.4) On the wing area SI + Sz. the sollrcestrength is known and equals Q = 2v = 2cx V.... therefore
IPtlI ;:'"
-:v. . ))l(X
~)~d!'2(t ;I~
S,
-
a;l.~"" 1
) ) V(x
s,
r
\
-", 's,J )/(.
~):;d;,~(Z ~}~ Q<;.
I)'
0 '
,)d,d(
(8.1 .4)
a"(x .,)'
where Q (G. ~) is a function determining the Law of source distribu· tion in region Sa outside the wing. We can perform integration if we know the function Q (G. ~). To determine it. let us use the boundary condition (8.t.20). in accordance with which the potential function on plane :d)z in the region between the tip and the Mach line issuing from point D is zero. For point A I belonging to this region, the condition of the equality to zero of the potential function by analogy with (8.6.2) has the form
O.
rr
-al'.
" - , - -'s!
1
\
l
-""2:( 'S3
V{.1"
d,d(
l(x -- ~)I
a;'t{1
;1 2
()(S. pd~d~ . ;)2 ·a;·'1.(t __ ~)t
A comparison of this equation with (8.10.4') reveals that to calculate the velocity potential at point A(x. 0, z), it is sufficient to extend integration to region S1 in the main formula (8,10.4), i.e. . ':tI' d;d:::
lr
By this formula, the overall action on point A of the sources on sections SI of the wing area and Saof the area outside the wing equals zero (see 1171). Using expression (8.tO.4~) transformed to the form (8.10.3) in the coordinates r. 8. we obtain (8.10.5)
Ch. 8. A Wing in a Sl.lper~onk Flow
37~
where in accordance with (8.9.8) for point D~ on the starboard tip, we have SP3 = rDim = rim (8.10.6) We find the lower limit rA' of the integral from the equation z = lI2 for the tip. In the coordinates i and s, this equation has the form of (8.9.17). Consequently. the coordinate rA' = SA- - M.1I2 = ' A - M.1I2 With a view to (8.10.6), the integral
where m = (n - 1)/(n Henel' ,
crIll
=
+ 1); m= -2cttroo :1..'1", V~
(1 -
'.
m)/(n
(8.10.7)
+ 1).
--
ri,r J,/r-s,vll rA-r
By calculating the integral, next inserting into it rA' from (8.10.7). and taking into accoullt expressions (8.9.4') for rA and SA, and also the val lies of m and ni, we lind the potential
qlill=-.:
:t
~(~:nz
{V/(l-2z A) l(IA,a'zA)-a'l(n+1)11'~,t
2(%A-~;,-tanXI)tan-1
J,/ 2(X'~~(:'zA~){I!('l2~\i)Cl',}
(8.10.8)
We determine tIle derivative 8q: IlI/8xA and nnd the pressure coefficient:
(8.10,9)
The velocity potential for point A in region IV (Fig. 8.1O.1d) is determined by the action of sources distributed over area AJOTR. We divide this area into two parts, namely, SI = OTRQ and SI = = OQAl. and find the potential at the point being considered as the sum of the potentials due to the action of the sources on areas St and S~. By using formula (8.10.3), we obtain
~iv = ~~;: s.""S:! 1j l't". d"~;'A
'l
3'16
Pt. 1. Theory. AerodynamiC5 of an Airfoil lind
o
'J.
= :~. '"
X
J v'~-' -
II
vr~\r_r -
J vr~-r 'J'J ll::-,
FA
:;:
'T'
Wing
'A
(8.10.10)
0
We determine the integration limits ST' and SJ' with the aid of (8.9.8). For point l' on the starboard leading edge and point J' on the port one, we have ST'
= T'T,/m = rIm,
SJ'
= rJ.m = rm
(8.10.11)
With a view to the:se vflJues of the limits, we calculate the inte. graIs:
Accordingly,
o
wiv=
J
-2«V~
nM. lim.
rR
v
__ r - sAm
r,4.-r
d,.
I
_2«:~.V-in '. }t/r~~~m dr o
where ra
is determined by formula (8.9.23):
ra
=
M.1I2 = SA - M.1I2 Integration and substitution of the values for SA and rA from (8.9.4'), and al •• m = -iii = (n - 1)/(n + I) yield ,_ -2a.v. {( _ t )[_ - I " /'(.:-+'t")7:(.".-=eT;'JA"'I, CJIIV- ,,'n 1"1-11.' XA z" anx1 tan V (1 11.) (z,,+o:'s,,) 8ft -
+ tan-1 V 2(z:~:;s~~ (l!'~!~1)J ++ V(l
2<.)(2(•• +,,'•• )
,,'I(n+I)] (n+I),,'
+(x,,+ ZA tan Xt) tan-I V (~1+1;){~:+~':~)}
(8.1012)
Ch. 8. A Wing in a Supersonic Flow
377
upon calculating the pressure coefiicient PI\' == -(2: V,.,) D~i\ .taXA and using here the formula tan-! x - tan-I y ,-" tan- 1 [(x - yH1 -T+ xy)l, we nnd
Let us consider region V. For point A in UIi:-; region (Fig. 8. to.1e), the potential is (8.10.11,)
By this formula, the velocity at point A is induced by t.he ::;ource~· on section ANPU of the wiug area. The iUller integral in (8.10.14) whose lower limit SN' = rm. is
Consequently. -:laV." y~ :t,U"",
!JA 1,i- r -
h·A,m
dr
" r .... -r
'u Since ru= SA- JI"",l':2, then. going over to conventiollal coordinates :c and y and taking into account that III = (n - 1)/(n + 1), and = -m, after integrat.ion we find
m
{M." y7(Z-A-----'~-,)-,[~(X-·A-·-.a-'-z,-,)-,,-,'(I!---'-)~-.'.!.',
M""(:~tzA:~D;tl)
tan-I
V (.rA-. a'z:)~o.,l{,:
After finding the derivative with respect to the pressure coefficient:
1).1/2} x.....
(8.10."15)
we determine-
Pv=-*·~=~ 1/2) (II (//2) (I.
1) 1) a.'
(S.10.16)
378
Pt. I. Theory. Aerodynamics of en Airloil and a Wing
In region VI, the velocity potential is determined as follows (Fig. 8.10.1f);
In accordance with (8.9.4) and (8.9.7), the limits or the integrals in expression (8.10.17) have the form
sE~""'Omr,
SE1=TEl-
M;l =rA,- M;l
r u;' =- r E { = sF-tim = sElim rS I
=
S .... -
= (11m.) (rA, -
)
M.l:2)
(8.10.18)
Moo1l2
As a result of integration, we establish the corresrondin~ relation for (f"I in the Yariable' TA, SA_ LSing th~ir dependence on XA and ZA. we find the velocity pot~ntinl liS ~ fUllction of thcse toordinates. By calculating the derivath'c 0'1 \·(i'u.r,\ and inserting it into the formula PVI = -2 (arp"./iJ;t· . d. V"' we obtain the following .expression for the pressure coefficiellt in the region bcing considered: -p,'! =
- -±4?: - = [ tan-I :ta' V1-n~ . (8.10.1 [I)
For point A in region VII (Fig. R.10.1g), we have
Ch. 8. A Wing in • Supenon;c Flow ~J'
379
'A
- .~~: jn
l/r:r,_r
J
1/8:'-S
'\'~
u
'A
- .~~: rwJ Jrr:r_r ~
'w~
BV~rA-Mool!2.
1/::-$
rJ.--(1!m) (rA-
(8.10.20)
.V;l) Jt
(8.10.21)
sv,,--=mr, sw~=r.:m, rw""'sA,-M..,':2 The following pressure coefficient corresponds to the potential (8.to.20) calculated according to the "allies of the integration Hmits (8.10.21): P"II ~ [tlm-t ) / t_11 ) / ' :ta.' l'1-112
.
1-11
.1.-\. -a.;:.!.
x'\'j'a.z,\
-tan-I
1/ z..\ ~~~~~ ti'~)(~,'(I~):"1l
",tan-I
V,r(:r,\->-IZ~'(~'\~" ~~~~:,~~._(;).'
-tlln- J
l/,r,~".;, 1'/'" ~:~=~:~:~
'II
J
(8.10.22)
LelliS now consider region Vll/ (Fig. S.W.tlt) for wbose points the region of integration simultaul'!Ousl}' intersects regions Q 1 and 9 2: disturhances manifest. tllemscl\,cs in these rcgioll~ t.hat are pro-
duc-ed by the flow ovcr both tips DE and GH. To cn,llIatc the velocity potential for point A t.hat bf'longs to rl'gioll VIII, it is sufficient to extend integration in COI'mIlIA (t'UO.3) to thl' region S = 8 1 + S2 hatched in Fig. 8.tO.th. Hf're the integral calculat(ld o\'cl'regLon S2 in formula (8. to.:l) must be tukcn wHh the oppo~ite sign. i.e. with the plus sign. Accordingly. we hR"(-'
_ ..;tJ·.... :f.JJ,,-
-l
Here
.~.~:
.\
J
S~
J" 51'
,/(r.\
drd~
l'.r,\
d:~~"A
r)t~,\ "SI
$/
(8.10.2:J)
880
Pt. I. Theory. Aerodynamic:s of an Airfoil and a Wing
j) 1/{rA d~)d;$,\_$) J v(rA_~~)d;SA .,. )f . d,," +\\ d,d, -*
=
01
o~
.
'y
y(r,\
d, yr,,_r
0
s)
)
s~
'y
r) ($A
d.
'03"' V(rA
$)
~1.1 1/ s,\-5
'r
d,
"rl;i }/r .... _r
r) (SA
5)
'r '" &KI
(8.10.25)
VSA-S
Tbe integration limits have the following values: TF~ = TFi
=
= SA- M fJ<JU2, SL1= tnr,~ SFi= SPI= rA- M ooli2)(B.10.26) tnsFi'o;ms" = m (rA- M ... IJ2), SKI = rIm
TH1
provided that SFI = rA- M""lJ2, THI = SA- M ... 1I2 (B.1O.26') With D view to these values of the limits, we calculate integrals (B.10.24) and (8.10.25), and then find the velocity potential (8.10.23). The following press1lre coefficient corresponds to this potential: -
_~[
P"lll-- C(.'~
-1an- 1
_I
i l · .. " ..
/(x,\
l/I-"~ tan~n V
V
~~.;: V'::~~::~~
- tan-Ill'
+1an- 1
et'sAH-(lj2)a'(n
i)
a~. ZA)(~
V ~-=~ V ~~~::::
!~;: 'II z.4.:~~~~'!.~'~~)(~::(I~)
i) ]
(8.10.27)
The above method of Cr.llcltlating the pressure distribution can be related not only to hexagonal wings with a V-t;haped appendage (see Fig. B.9.1), hut also to other planforms, namely. to a wing with a dovetail and a pentagonal plate (sec Fig. 8.9.2) provided that the tips a:re subsonic, while the leading and trailing edges are supersonic. The co1'l'esponding change in the quantity apia. = (Ph - Pu)/a. near a tip with the distance from the leading edge (in per cent of the chord) evaluated for a wing with a pentagonal planform at M..., = = 1.61 is shown in Fig. B.10.2. The coefficient apia. is constant in the region between the leading edge aud the Mach cones issuing from the vertex and tips of the wing, on the section confined by the tips and trailing edges, and also by the Mach waye reflected from a tip. A break in the curve cliaracterizing a change in the pressure drop coeRicient occurs on the Mr.lch lines and is a l'esult of the break in the configuration of the wing tip. The found pressure distribution sud formulas (8.9.27) flnd (8.9.28) are used to determine the lift, momeut and ccntre-ol-pressure coeffiden ts. Figure B.9.4 shows curves characterizing tIlc change in the values of C'II and c p for a pentagonal wing. These values should be
Ch. S. A Wing in
.!I
Supersonic Flow
3S1
fig. 8.10.2
Distl'ihuliul\ of the quantity
~l/~=le~;;:~~1 ~1~~ a\\"~th supersonic lea(liDg I'liges
(s~c
tioo :1.4)
determined provided that i ...·VM;' - t > I.", tlln XI' i.e. if the leading edge is supersonic. If a wing- has subsonic iealling edges (see Fig. 8.8.1), Lhe flow oYer Lhe section of the surface ilt'lweell the eligt·s and the Mach lines issuing from points E and }{ is aHect('d llr a \'ol'tex sheel. The calculation of lilis flow is associated \\·ith solution of integral equation (8.2.t6) anf! the u~e of boundar~- conditions m.1.15) and (S.1.1G). Such a solution is treated in drtail in Itil. 8. t t. Drag of Wings with Subsonic Leading Edges
Let us consider the calculation of the drag of SW('pt wings with subsonic leadin.g edges ill a supersonic flow at an angle of aUl\ci-.. We already kno\\' that the disturbed flo\\' normal to the I!'luling edge is subsonic near such wings. Such a no\\' is all('ntied I)y the o/;er,flow of the gas from the region of increased pressure inlo the region where the pressure is reduced and is the calise of the corresponding force action on the \ving. To calculate this acLion, we cun lise the results of investigating the disturhed Ilow of an incompl'essible fluid near an airfoil in the form of a nat plale arranged in the flo\\' at an angle of attacl, (sec Sec. G.3). The drag coerficient for a thin winq with subsonic iea(ling edges in a supersonic now is
(8.11.1) in which ex T is the coefficient of the \\'ing sllction force depen(ling on the sweep anglo Y. of the leadin~ euge and the number M ... : C X•T = T/(q ""Sv.-) (8.11.2) \\'here T is lhe suction force, q~ = p«>Vcoi2 is the \-elocity hea(I, anf!' S", is the> area of the wing planform.
3B2
pt. I. Theory. Aerodynllmics of lin Airfoil lind II Wing
~:. tt-:'~~leulation of the suc· tion force for a triangular wing with subsonic leading edgl!s To detarmine the force T, we can use tbe relations obtained in Sec. 7.6 for an infinite-span swept wing. This follows from the fact that in accordance with (7.6.18) and (7.6.19), the suction force depends on the change in the axial component of the velocity in the close vicinity of the leading edge of tile airfoil being considered, and also on the lOCK 1 sweep angle, and lloes not depend on the behaviour of this velocity far from the leading edge. t:sing (7.6.18), we obtain T-2:tp_ V Tl_7;.7'.=n"'''"x---;M''';''
'I" c'.!dz
(8.11.3)
where c is a coefficient determined from (7.6.19). Let liS determine the drag for a triangular wing. For this purpose, we shall lIse expression (8.8.39) for the disturbed volocity. Assuming that cot~ = z, xJ.e. we fi1ld
u where xJ.e is the
di~tallce
to the leading edge, z and x are the coor-
diI~i~~d~fci~:~i:~t :~u~h~f'~:rrn~:if7.g:1~)1~~d taking into
that
lim
c1
account
(xlxl.,) = 1. we have
r-%).e
=[ :~';,
r :c~~ll,.. ~:;:rl~;~~)~(~_~':'~:')
-+[ ;~~~
rzeotx
Introducing this expression into (7.6.18) and integrating. we find
T~ ~po.:'lt~ [~~.~
r2V'l-."7""","",'-".---;M;<.,, '"Jzdz o
= Jlpoo;tx
[:v~;)rfV1+tanaK
M!.,
Ch. 8. A Wing in 0 Supersonic Flow
383
Flg.8.tt.1
Cliange in the factor J. T in formula (B.H.5) for calculatin({ the suction force The suctioll force coeflicil'lll hy (8.11.2) is
C~.l'
.\'~/'~r:tV1-.
. [ 'I...
Iflu:!x-·,l1;'
18.11.1)
Taking into account expr('ssioll (8.8.-H) for cUa' WI! have
(8.11.4'). According to this formula, tlw suction forcc coefficient is proportional to c;a' i.e. CX,T ;:"- CT~.,. The proporl ionality coefficient,
V
equal to CT = (1 l4:t) 1 -;. tan~ Y. M-;" for U lriallgular \Yillg, ehanges when goillg O\'cr to another plan form and depeuds not only on x and M <>:>, but also on the t 1), ilnd also wh!'11 the leading edge becomes SOllie (tall x - cot!l ~" l/M~ - 1). \.he Sllction force coefficient equals zero. With a subsonic leadlJlg Nlge (Moe cos x < 1), the quantity C).'.T .- 0, hilt l'xpr.rimental in\pstigntions show that its /lctual value is lo\\"!:r than the c[lkui[ltrd one. This is especially noticeable at large angles of allack or sweep \\'hpil local sep[lration of the flow (bllrble) occlIrs ill the \'icinity of Ihe leading edge and no further growth of thl1 SHCtiOIl is obser\'l'Il. The decrease in the suction force can be taken iuto HCC01lllt by the correction factor d r in Accordance with \..-hich 000
Experimental data on the change in the qUlI.lltil.y.i T ar('- contained
in Fig. 8.11.2. According to these data. the sllction forc(' at. snhsonic velocitil's lowers less than in !'lupersonic flow upon an increase in the
384
PI. I. Theory. Aerodynamics of
.!In
Airfoil end a W;rtg
angle of attack because at such "elocities the separation of the flow from the leading edge is Jess noticeable and the suction is greater. According to experimental investigations, at subsonic velocities the correction factor in (8.11.5) is determined by the formula (see [181) (8.11.6) Xo suction force appears on wings with a sharp leading edge, i.e. the coefficient ex. T = O. Let us consider the total drag of a wing, which in accordance with (8.11.1). (8.11.5), and (8.8.44) can be written as c Xa
=ctcllil
-·Cx . T =
4."l<:!~t 12!~ (k) y.
fiT Ji1-
cot'x(M!..-1) 1
Having in view that the aspect ratio of the wing is AII' = 1~(Sw = 4 cot x
(8.11.7)
[2E(k)-.6. T Ji1-cot 2 x(M!.-1»)
(8.11.8)
obtain C"'a=
:~:.
Let us consider sonic flow. If M "" _ 1, then E (k) _ 1. and, sequently, the drag codticient is cx;)=c~/(nAw)
COD-
(8.11.9)
Expression (8.11.9) coincides with formula (G.4.1G) for the coefficient of the induced vortex drag of a finite-span wing (provided that 15 = 0). Hence, the physical nature of the drag force arising in a flow over a triangular wing with subson ic leadin g edges is due to induction by the vortices formed behind the wing. Accordingly, the quantity determined by (8.11.8) is called the induced drag COf'i'ficienl. If the leading edge is sonic (t.an z = V M;" - i), the second term in the brackets in (8.11.8) is zero, while E (k) = E to) = n/~; consequently, the induced drag coefficient is (8.11.10)
For a very thin wing (x _ :t/2), we may assume that. cot. x ~ 0 and E (k);::;I E (1) = 1. In this case, expro.."8ion (8.11.8) coincides with (8.11.9). In addition to triangular wings, formula (8.11.8) may be applied to wings having tho form of tetra-, penta-, and hexagonal plates with subf\onic leading edges, and supersonic trailing and side edges
~~~~sl ~::o i~i18. ~1~8t)isPf~~~~~t~~~~' (~~8.1e1t).agonal
wings, the coelti-
en.
B. A Wing in ~ Supersonic Flow
3Se
8.11. Aerodynamic Charaderistics of a Rectangular Wing
Supersonic Dow over a thin IinHe~span rectangular wing at a small angle of attack is characterized by the influence of the leading super80nic edge and the SUbSOllic tiPJ> (side edges) on the disturbed now Ilear
the surface. The simultaneous influence of the leading edge nnd one tip is present within the limits of Mach cones issuing from wing corners 0 and 0' (Fig. 8. t2.1a) if the generatrices of tllCSC cones intersect outside the wing. If these generatl'ices intersect on the surface of the wing (Fig. 8.12.1b), in addition to regions II and 1/' where the in4 nuence of one tip is obsef\·ed. zone [II appears in which both tips act Simultaneously on the disturbed flow. In region I on the wing between the leading edge and the Mach cones, the disturbed fiow is due to the influence of only the leading edge. Here the pressure cocfficiC'ol is determined br the formula for a flat plate: (8.12.1) p, = ±2«IVM!. - 1 We shall use the method or sources to calculate the disturbed flow in region II (Fig. 8.12.1a). The \'elocity at point A in this region is induced by sources distributed OVCI' wing section ACOE. The total action 01 sources BCO lind OBD 011 point A is zcro, and, consequently, integration in (8.10.3) \llust or performed OY('!' region ABDE. Aecordingl~', the velocity potential at. point A is
=
rr _d_,_
-al·...
.\~
:l.U""
,.~ l/~":'$ r~,
_d'_
r";.\-r
(8.12.2)
Point F is on the leading edge whose equation is x = O. Hence, by (8.9.4), for this edge we have r
= -(M .,/2).,
8
= (M ./2) •
(8.12.3)
Consequently. the equation of the leading edge in the coordinates r and 8 has the form r =-8 (8.12.4) Therefore, for point F. the coordinate rp = l' = -So Point B is on the Up whose equation is z = O. Hence, by (8.9.4), for this tip we have (8.12.5)
a8S
Pi.
I.
Fig.8.n.t Heclangular
Theory. Aerodynamics of an Airfoil lind II Wing
wing
ill
a
linearized :>upersollic now
We can thus write the equation or the tip in the form
(8.12.6) For point n. the coordinate Sn"""' rll' - r;... Performing intcgration in (B.12.2) with a vicw to the limits rl' =" = -s and Sa ,-. r.\, after introducing the values of r,\ and S;.. from (8.9.1'), we obtain
cpil
~- -~~roo
[V,.. %(xA-a'z.d
'~'7 t,Ill- 1 V.r.\(t~~~'zJ (8.12.7)
Let
liS
calculate the corr{'sponding pressme coefficient:
PII -. _. (2/V....,) dq;h
iJx,\,'
::a~(l.
lan-I! /
J' \
~'~~\.\
(iU 2.8)
On the houndary of Lhe Lwo rcgiou.<: 1 aut! 11 separated hy " Mach linc, the pressnrc coefficient detrrillined by (K,12.8) eql1
surfaces ;\11(1. C'ollscqllelllly, lile flow in the illdicat.ed region or thl) wing can ue cOllsidl'red conical. Let liS calc.1I1
(8.12.9)
Ch. 8, A Wing in
,\ giallce at
Fi~.
;!I
aS7
Super$onic Flow
8.12.t{/ ren'ab that the elementary [lren. is ciS
-,~
(R.12.1fl)
d'l
wlltwc y is the allgle IllcnsnrCfI rrom the Iring lip, Introducing illlo (8.·1:!.!) tlte \'alll('~ of
tit(> ndllc o[ dS from (8.12.10), we have
,~:I:~. p,,;.;"
d}'
(\Ilrl
Inn-l.V
Ph and -Pu [rolll
.~lIhSli\l!tillg-
"my
fK.tZ.8),
rUI
<:.I.r,\,
1.~~~;~::1;'1'· CI:~~'
Di\'idillg lili:-; l'.\pn':<."iull hy till' product or the wing Iwea con lined \\"illlill tile ,\Ial'll ('0111' U3 11 b~ :.!Ct') n.nd the velocity head (qx= ' ""'' p •. J'~, :2) Hlid pl'l'forming illlcgl'iltioll, we ohillin the lin coefficient for till' \ring lip!':
Intl'gl'alion
by pal'ls yic·lds
c;
:!a; l/,lf~ - 1
:20: a'
(8.12.11)
Lei liS (1I'termin(J Ihe lift cOl'ITirif'nl ('III fOJ' lilt' l'lId parts of I.he wing rt'latt!d Lo its tol,1\1 art'a S" fb:
(N.12.1I')
i."
~..:
l h. The lilt cot'fficil'nt for willg st)ctinn I ri!latl'd to llil' s('clioll is whl'l"l'
Hl"Pil
5, of this (,'\,12.t2)
Till> liFt cot'fricicllt fOI' this St'c1ioll I"t'lalvd tn
face
lot,lI \\"itl~ Slll"-
Lilt)
is
(:II·-·C,~ .~:,
"
(':,
S\\- .. ('~\,I\:
'I'll(' 1011\1 lift cuelTici('1I1 ('11,,·-
Cull
('Ill
.
}/.,:Z-I
The wave th',lg coeITicil'll1
"'"H
'3CII "
=
SII')
c.; (1-
1..)
;.,,1..
(H.12.13)
j,-;
(1-
~;"\\" I?J'~ -t)
(8.12.11,)
/IJ~,,-.I
) (8,12.J5)
i,~
(J:t
I
(1-
2;. ....
ass
Pt. I. Theory. Aerodyn"mics of an Airfoil and a Wing
FiCl···tl.l
To tbe determination of the aerodynamic moment for a rectaDgular willII'
We can calculate the cocrricienl of the moment about the z-axis according to the known data on the pressure distrihution and the lift coefficients for individual parts of the wing: Ill.,. = Mz/(q ... Swb) (8.t2.1U) The moment Jlz. of the pressure forces distributeu over the surface of tbe wing can be found as the sum of the moments of the lift forces about the z-axis acting on variolls parts of the wing (Fig. S.12.2). Taking into account that on the trhmgnlar parts of the wing the point of application of the lift force coincides with the centre of mass of the area of the relevant part. we hAve the following expression for the moment:
(8.12.17) where yep, yep, and Y n are the values of the lift force calculated for parts CDB (IlEF), EFCH, and ADB (GHE), respectively. By (S.12.ft) and (8.12.12), we ha,'e
Y'l':." c~ P"'~';;" '1'
Y1
:":CIl
(J-~';;"
SII '- -*'
,pooV!.
1
Sll
Ija;
p... v!.
.
poo~!, Su
-,-Sf:t,cn :'-: ""(i7"'"'-2-(SW-.'Js n
Yll-=c~ r ...~';;"
SI1"--'
~
)!I ..
(S 12 IS)
Introducing (8.12.7) and (S.12.1S) into (S.12.16) and taking into account the value S II = Sw/(2'J..w a'), we have
m,"~
jI';:''::'
(1- JAw
v~~-,
)
(8.12.19)
Ch. 8. A Wing in a Supersonic Flow
389
The coeffieient. of Ihe centre of presSUI"e is
e ., "
-11
:li.,,-
.T;' _ ~.'~
V:ur=T _.:!
3(2i, ... ,1.11;'.-1-1)
(R.t2.20)
If the number iff '"' for the flow oyer the wing shown in Fig. 8.12.1a is reduced. then Ilt a certAin nine of this /lumher the lip )l;lch cones int.ersect inside Ihe wing. Here regioll 1/1 appear::: (see Fig. 8.1!!.1b) in which the disturbcd flow i!< 11ff('{'ted hy Ihe )eiltlillg supersonic edge (Inti both subsonic tips. The nature of the fluw ill rcgiolls I and II (/I') is the liamc m: ill the corre!lpolldillg zolles I and II (II') of the wing ,.... hose diagralll is shtJ\\'II in Fig. 8."12.1C1. Let us consider the now ilt point ;I of region I I I (sec Fig. 8. U.lb). 'fhe region o( iufluencc of the sOIll't·e.'> 011 thi~ now coillddcs with wing s(.'('tion ABDD' IJ', The )AUer can be l'cpl'f!Senled il!< the Slim of area~ IIIJDD' ami A Hn' /I'. With this in view aud 3t'cortlillg to (8.10.3), the velodty polenti:'!.1 at point Ii is
We can (lelerllline tho integl·a!.ioll limits directly from Fig. 8.11.1b. Hence,
We c-8lcltlate the lirst two integrals by 811alogy wilh (8.12.i), and the othcr two independent I~' of each other because the regioll of integration is a pllralll'liogram (see Fig. B.12.1b). \\,ith this ill view, we have
f~ill"".o ~'t:::'"'
{-V('A .. SH)(SA
+ V2r.A(SA -'A) -(·'A·
t
I'A)
_tan-ll/8";,~A) -:-2V{SA
(tan-I
$}I)
V~
SU)('A 'D')}
(8.·12.21)
Examination of Fig. S.1:1.1b reveals that thc coordinate Slf ~-' SD' and in accordance with (B.t2.1), the quantity SlY :- -'0" Consequently, 8H -- -I'D' '"=, -I'D"
390
Pro I. Theory. Aerodynemks of en Airfoil end eWing
Poinl 8' is 011 the starboard lip whose equation is z : , l. From (8.9.4~), we can lind the equation of thb; tip in the coordinates T and $:
T-s-=-M""l
(8.12.22)
lIen('e for point 8' TD'
M...,l
'': $0' -
s,-\..;..
M",,/
(8.12.23)
Ar.cordingly, we have SI-[ .• ~
-TD' -- -SA
I- M ool
(8.12.24)
Introducing into (8,12.21) the ('oordinales TA and $,\ from (8,9.4') and TD' and SJt from (B.12.23) and (8.12,24). we find
+ Va'zA (XA
a'zA) -'
-Iall- I
XA
o:'ZA)
o:'lHl
[t1111-1 ],/
V l"A~z:'ZA
ZA)O:'
(l"A(;-a~~)a: a'i (8.12.25)
]}
Using the formllla p . -(2IV 00) Oq:i 1l 18xA' we nnd the pressure coefficient as a result of simple transformalions: PHI ,_.
±
~:,
Illn- 1
J'/ ~.,~'~\.~
l/ (;c,\(-~-cr.:~~t'.a'l
+ :
(8.12.20)
The !irst and second terms in this expression arc the pressllre coefficients PII and PI!' for the tip pal'ts of tbe wing, respectively, while Ihe lhird term is the pressure coemrient Pl fol' region I where there is no i.nfluence of the tips. The flow over a rcctanguJal' wing is of a slill more intricate natUl'e if it has a low asped ratio and the quantity A\\"VM"oo - 1 < 1. In this case (see }o'ig, 8.12.1e), new wave I'egioll~ appear that are formed as a result of intersection of tile disturbance wa'"es incident on and reflected from Ihe tips. For example, the disturbafl('es tra\'elling from section OH or the port tip reach the starboard one on sertion 0' H' and o\'crit propagatea)ongMadllinesO'O"-Il'H'" in the opposite direction, This gives rise to lie\\' Wflye zones IV and V in \vhich the naturo of the flow changes subsluntially. This now ('an be ca1culflted by using the method of sourcc..'! set out above and taking into consideration the indicated inll'kale nature of formation of the disturbance zones.
en.
8. A Wing in a Supersonic Flow
391
(a)C~c-~
FI" 8.U,] Curvl's rhnr!lct~ri7.ing thl' chungc ill the clerivativ(~ of the lift fOITl' coeflicienl (a) and Ihe cenlr("·of-pressufl' cOl'fiiricnl (b) for II I"C("tanr,rular, winl! (tIl!' aspect ratio J. w "~I!)
The known pressure (Iislriblilion and formulas (8.9.27) and (8.9.28) are used to calculale the coefficienls of the lifl force and moment, while the expressions c p = -mra/cyOl anti c.'·a .; CU: Ya are Hsed to calculate the ('oeHidents of the ("entre or preSSUi'e and drag or the wing being cOllsidCl'ed. ClIl·\·CS ('harnctel'izing thc change in the Wt nnd centre-of-pressure roeffidents of a rectangular wing with a varioHs aspect ratio havc been rOllstnlded in Fig. 8.12.3 lIccording to the I'esults or such calculations.
8.13. Reverse-Flow Method L('t liS wnsidct, one or the methods of (lC'roilYIIn the ael'oelynumir ("hararlerislics of thin wings wilh all idenlical plflllrOl'lll and locRteel in oppositely clirecte(l flows. Let liS nSSlIllle !llat Olle of such \\"IlIg~ is In 1'1 forward flow at the flngle of ntta('.k ai' und the othel' ill fill rel'n'se no\\' fll the l\J1gle of
892
Pt. I. Theory. Aerodynamic5 of an Airfoil lind" Wing
,~'d ~,~ --J~--X --~:=;==r-.0,,«,
~~·t&-ctl~!planation
Pz,«z
,I
~
9~ ilJ'O<:J
of the reverse· now mctho(l:
J-torward now: 2-fIIVl'I'Sl' Aow: 3-eomblned flow
attack az (Fig. 8.13.1.). A('col'(liug to IIII' thcorr of linearized now, the excess pressure at a point of the !'urrace is p - P.., = -p..,l'"..,u, while the pressure increment between Ihe boltom and upper sides is !lp = Pb - PI.I = pooF.., (u u -
Ub) """
poY ... !lu
(8.13.1)
The magnitude of l.he drag force produced by the pressure increment is dX. = !lp adS. A(·('ordingiy. the force for wing 1 in the forward now is (8.13.2) X. I = ilplCtI dS
jj s
and for wing 2 in the reverse flow is
X,,-
Jsj t..p,",dS
(8.13.3)
The magnitudes of the pressure increment !lPt. !lpz and of the local angles of attack a 1• a2 are measured at the same point.. In the following, we shall disregard the suction force that can develop on the subsonic leading edge. This does not change the results. Since the flows in the vicinity of the wings are nearly uniform, by superposing them we can obtain a new flow with parameters satisfying the linearized equation for the velocity potential. We shan perform the superposition so that the free-stream veloe-ities are subtracted. Now the contour of the new wing 8 (Fig. 8.13.1) coincides with the configuration of the given wings 1 and 2, but the camber of its surface is different. This camber is determined by the local angle of attack a l equal to the sum of the angles of attack: (8.13.4) III accordance with (8.13.4), the vertical components of the velocity 1,-. at the corresponding points of the surfaces are added. The horizontal velocity components upon superposition are subtrafted, therefore for the combined wing 8 we obtain !lUI = !lUI - !lua.
Ch, 8, A Wing in
<)
Supersonic Flow
393
lIenee the pressure ill('fement betweC'll the hot tOlll [llld liPPE'\' ~ide~ of this wing is llp3 = pOO\T
~
Is·i (~P.
-' ,\pJ
i'.
'J dS
A drag ill a wing is (l~ociated \\itli il di:ilurued JhJ\\ bcllillu it. The latter is an illlillitely long YOI'tex sheet l'CI~t off the trailing edge. It is clear from physical eOllsitieralions that lilt'l·c i~ 1-'1I·id 1'01'1'esponuence bet.ween the disturbed !>tal.c hehind lhc wiug mHl Ihe dl'ag force, Particul<1rly, the fo!'ce Xu and the di!lturbed sLalc behind wing 1, as well (IS the force X a:l and the disturbed state hehind wing ;2 arc in such correspondence. The flow near wing 3 ohtained as a result ai' snperposilion of the flow panel'll!! Ileal.' ,,"iug 1 aud wiug 2 has tlie propert)· that the di$w turbances ahead of it are the same as for wing 2, and behind it. the same as for wing;J, I·lellct', the dn:lg fOl'l'c of combined wing 3 e1luals the diUerellce between the drags of wing:-: 1 and 2. i.('.
X8.3=Xul-Xa~=)) (ll.Plal-:lp~a~) dS
(B.i3.i)
s Equating the right-hand sides of (8.1:.1.6) (lud (S.I:1.i), we
JJl1p,~dS= .\. J' j,p:a,dS B
ohlaill
(8.13.8)
S
This relation is the fundamental e:lpre.~sioli of the rerer,~e-!l()u; method, Going over to pr£ssure ('oefficients in (H.1J.i'» Wi' (',111 write lhb relation as (S.U.';·) ~p.a, dS ~ ) dS s s where the pressure increment coefficients are ill!; -- ~b - PIC aud Apt = Pn - Ph' Using our right to choose the angie of attack arbitrarily. I('t us adopt the same eonstant values of al and al for all points 011 1\ wing surface, I.e. a 1 = a2 :- a. We therefore find frolll (8.13.S') that
j)
J,p,a,
J) ~jil dS= ~')s ll.pzdS. or Y
aL
=
l'a2
S
lIence, the l'cYerse-f1ow method allows us to establish that the lift force of a flllt plate ill forward (Y at ) anu j'fWerSe (l"a:l) flow is tlie same' at identical angJc~ of sltRe" and frec-stream vclodties.
9 Aerodynamic Characteristics of Craft in Unsteady Motion
III the steady motion of craft, the aerodynamic forces and moments are indepencient of time and are determined for lixed rudders and ·elevators and for a gh'en altitude and \'clocity only by the orientation of the crart relative to the \'eJocity vectOI'. Ullsteildy motioll. when a cralt experiences acceleration or deeeleratioll aud vibrates for various reaSOIlS, is the most general CDSI.'. In inverted motion, this is equivalent to an ullsteady now of air 0\,('1' a craft. With such a flow, the aerodynamic properties of a craft Me due noL only to its position relative to the frce-stream velodty. bul al.so depend on the kinematic parameters characterizing motion. Le. the aerodynamic. coefficients arc a rum~tioll of lime. In unsteady now, II craft experiences udditionul forces and moments. Their magnitudes in
Ch. 9. Aerodynilmic: Ch",r",derislic:s in Unste"dy Molion
Tht' COIH"Ppl of
395
conlrollabilit~·,
i.l'. till'
9.t. General Relations for the Aerodynamic Coefficients Thl' ;Wl"Od~'Hmll ic for("('." Ill' menu'HI." a("\ ing 011 a (Taft depend 011 Ihe kinematic "ariabl('s ('hal"lll"ll'rizing ils 11lo\iol1. FOI" a giVI'1I ("ol1[igmalion and I1llilllllc H of a naft. IIwsc fOITl'S and mon]('llt!< i.H·C delcrmiliNI by Iht' night \"(.'lority v -V", (V"" is the free-strcam \"('Iorily ill it1\'l'rled now), the angu\;ll' Ol'il'ntatioll fl'llltiyp 10 the velo("ity Y('("tor V (the ang-I('." a anll ~), the angl('~ of dt'nt'("tio[] of till' fonlrol slll·farps ul"l"llllged in Ihrl'l' mutually P<'lTl'[1(iinlinr plaues (15,. f>" tllld 0" arc Ihe l"lul(I (,]" clcvator. Mid aill'I'OIl angil's. l"e~p('("ti\"C'y). and al!
(~
da.'dl.
()f (1t'\"Il'("lion
6a 11
i3
dr1·cll). alill al~f) o{ litl' l"ah' of ('IUlllgl' ill lin' llngll' of 111(' rOlll]"ol :<1ll"fal''':O
(5,
do, dl.
~"
dOt.·ell.
- doll.dt). AI"(,OI'dillgl~', Ilu' rollowillg gl'lll'l"al i"l'l;l\i()l1 holds I"or('(' or mOll1l'llt:
/. (1"""
'" "'"
II, ct, ~, 0" Oe.
1\•. D.,., n!I'
~ ..~. '0" 6", 6". ~(,. b!t" ~U
fOI"
~!:,
(!l.I.l)
The 8l'1"ollynllmk fOrl'c ami 1l10111('n\ rocHkients are funl'lions of dimcn."ionl('ss Yflriablt's rOl"I'ClSpollding to the dimcll:s ("()]'l'l'sponding to tlt<' wloeil\' I' ~ 8nd altitlldl' II. we rnll ('ol1sidel' Illp :'-.Iafh I1llmlwr M"" 1'.". a,~ allll Hl'.\·llOhl:- numher lie l'O<"fp",,"I~ (hl're ( I " . P ",. (111(1 I'''' ,11'l' 'hl' ,"pl'ed of ~Oll nd, Ihe dt'n:oi I y, ,11111 lilt' d ynami(' vbeo."ity, l'l""Pl'('tlwly, of tlte air at Ihl' flltitll(ll' /l, ;llld l i:o a typi('ai dimen:oioll of the ('["aft).
396
Pt. I. Theory. Aerodynamics of
,In Ai.roil
and a Wir:g
~~;i:~;~ion of kinematic variables charact4!rizing the motion of a craft W(' can write the other dimeu:;ioilless variables in the following form:
~,(•. ",,,.6'('.'JIV .. ~-;'I.V .. ~~~liV~ ~,).l""""~xl:Voo. C/~!J'
.
)
. V {:lI l iV"", 00:'- Q:l'I'"", cV",~'.::V,>
(9.1.2)
w~ = Q;,
Hence. for an aerodynamic coefficient in auy coordinate :;;ystem. we h8\'e
c=:/z('V"" Reo<:. a. P. b rt b,.• 611 ,
V.... ~. P. 6r , t\. 1;;.,
W X•
;\I.~, ~'/' ~.)
(})iI'
W,'
(0.1.3)
The variables M .... 'X, p, 6" 6,., ba• (}).~, wY ' and (})z detel'mine the coefficients of the aerodynamic, forces and moments due to the instantaneous distribution of the local angle!!i of attack. Here the variables M ..'" IX. p. b r• be' and fia characterize the forces and moments depending on the velocity, the angle of attack, the sideslip angle, and the coutrol surIace angles. while the \'ariables (})X' (}),1' and (})z determine the forces and moments (Iue to the change in Ule local angJes of attack caused by rotation.
~. ~. 6r- e• a• ~.~, ~II and ~z deterThe time deri\'atives mine the additional forces of an inertial natutc and appearing upon the ilccelerated Illotion of a body. Thb motioil ac,eeleratcs the air particles on the surface in the flow. When investigating the motion of craft characterized by a sIllall chauge in the dimensionless kinematic variables. the aerodynamic coefficients are written in the form of a Taylol' series in whic·h thesecond-order infinitcsimals may be retained. Assuming the nnmber Re... to be fixed, we shall present. particularly, such a series for the
V. . .
i i
Ch. 9, Aerodynemic Cherecteri\tics in Unsteady Motion
397
Ilortnnl force coefficient: c y =C"o
"uoo'-l;l/~. "('~i\'l: :('~,"'~-c~r ..\br .,c,~·-"-6~':.r~·'j.Oa
:, c;;x A(,)>: . c::'" I\(,ly . c;;': !1(.), ..
r~r a~r -;-
::e ~b"
-c
~ "" i\ -t-"
. C~
+/·)!I/\~/},(U.o:'
L1(:J"
~c;;"·"u~('J." ..\()Jy-;"
c;, . .
C,,' 1\(;'.
(B. i..)}
Hel'e Ihe coeffkielll r'l~' the derinltives dr,/oM .... ~ = = iJc y ·O'7.., cO . dC!l'rJ~ and otitcrs are cakulaled for ('erlain initial ,"alum: of the vArinhll'''. If Ihl'~e "allle1= equal zero, Ollf' 1IH1~l a!<~nllle '0-
uu. .- ,;,
thAt j.-x -:-.~ a, ('te. TIll' deriva\iv('!< call Iw eon;:i(It'I'l'd II~ til(' I'alc~ or l'iHllIgtl in II ('ocfficient or allY fOl'CP or tllOlllt'li1 dC'pellding 011 the Illllililer lJl "" Ihe angle or allnd, 01' lhe !clor~' for \,ol·\1ident being c.on:
q,
~
q'J cos PIt
(9.1.5)
where qi is the set of dimensionless kinematic variables, qi is tho amplitl!de, and Pi is the circular frequency of the oscillations. In accordance with (1J.1.5), the time derivative is
q,
= dq,ldt = -qlpj sill Pit where we hflve introduced the Stroubal number
pt
= PllIV ...
(9.1.6) (9.1.7)
398
Pt. I. TI-.eory. Aerodynamics 01 an Airfoil and a Wing
For cxample, it l/;
• a, qi .- aD (thc amplitude of the o~dllatioll~
with rc!!pect to the angle of atlilck), ,..- p;, and Pi PI1.' thCII
(dC1.;dt) iT""
~. pi
a_.
a.'''''CloCOSPq.t, -a.op~\'inpl1.t (9.1.8) Important :o;ulutiolls from a praetiral vil'wpoiut have been obtained fOl' the partkuhll' ra~e oi harmonir oscillations, They allow one to tleterminc the hasic aerotlYllnmic characteristics for the uJlSlearly 1lI0tion of willg~, boelies or revolution, and separate kinds of aircrart.. Expres!!iolls m.C{) and W,1.4) for the Ilprodynlllllic cOl!lTiciellts relate to n cmf! with an undeformcd (rigid) sllrfnce. It must be taken into IlcrollllLlwrc that in rcul condiLions such u !'Ilirfuce may deform, lor exnmple. hecausc of bending, which leads to all additionnl change ill t.he 1I0lH:tilliollllry chllracl.eristics. Com":idered bclow are problems 011 the dl'termillaliOIl or sllch chuf(lcte>risLics withouL acconnt Laken of cil'formillion, i,c. for inelastic bodies in a flow. The resulLs of iuvl'stigating Illislellrly~flow aerodynamic ('.haraeLeristics for tllc more gl'llcrlll ca~r of drforme
9.2. Analysis of Stability DerivaHves and Aerodynamic Coefficients
The' nr~l Sl'\'t'll t.erm.,; iLl Eq. (n.tA) drlt'rllline thCl static,
cr"".
e~'''', ~,e;,. l'te,), thl' d)'namic stabiliL)' derivatives. ,\!< Cllll he S('('ll f('olll (9.1.4), the lIt'I'odynamif corrr.c;i{'ub are detprmillPd by thc('ol'l'e!<ponding first am.! second order stability derivatives.
c:
ee,
The !irst ordt'r OJle'S include the va:nes or 1 ",. q,,~, c:~ and the second ol'der Olles, ~ :- a~elld)a a~, ~'~:o: _. iJ~c'l '00 aw.~, el.c. All anaJY!:lis of (9.1.4) rcyeuls thot the aerodynamic coefficients tIre d{'termilled by two groups or dt'rivllti~e!', one of which depends on the control surface yari?ble.s (ct'. ct', ... ), while the other ~Se;cOn\l~e~~~e:t!()Il~I~f~~~ ~'tl1~e' ~o;;tr~~~ ~:~~n;!I~II~e 0:0a tfl~e~;l~i~~!
eh. 9. Aerodynamic Characteristics in Unsteady Molion
399
in the angle of attack and sideslip angle upon deflcc1ioll of tile control surfaces. The values of the IInglc-,,1';t and B eorre"'[lond to the position of static equilibrium of a craft. To retain the gh'cll flight conditiolls, Ihe control fixed in p:ace. Such a flight wilh "!i.\ci!" conlrol surfaccs conlrollablc. Its conditions arc dClerrninrd COlllril'lf'ly bilily dt'l·jvaliH's lhoi {]('I1(,1H1 011 III(' illirill~i(" properlies of a crart if ("onirols Hrt' ,,/lsl'lil or an' 1"1.\('(1 ill Derinlth·cs surh as c;1, c~, c~" . . . , dl'lcl"lllil1('(1 l1S a diffcrcntiil\.ioll of 1111' iH'l"odYIl;lmic coeiTicielli!-l wiLh l'('Sp(l(~l to Ct, ~. f),. (sialic ~Iabilit~· (I('rivaljn':-;) relate III the tirst group of dcrh·alives. The partial (Irri\·alin's of the coeJTiriPlits with respect to olle of the vuriahlt's (o)~, u),/, (,l: form lIw S('{"olul gt'OIIP of whal w(' call rotary (It'riYaliy('S (rot' n:ampl(', c,;<; c,~:" ~I/.~': • . . ,), This
ci" c;;: cz,·.
acce-
grottp also indlldl'!-' dNlnlliy('s !'imilnr lo luul illso ilS Ihe:! pill"tiaJ t[CI'jYlllin'!-l of thc a~ro-
leration dericalircsilcll'rlllillt'(1
II,YII:H~tj(" ("()d~icienl.'"' \\"ilh n'.'"'llccl to onc of lhr pm'nllll!tl'fS ttl,.,
(o)lj'
or
W
i' 00,
z•
11OH-sLaliollilt·y distmll11l1ces al'C tinite, nOIl-lint·Mil;.: hCC'OHlf'S H lIoticeahle rcaltlrt'. II is r('I"I('clt'd in expilll:=ion m.J..'!) where tlH.'fe arc .">('\"1.'.·011 s('eonti-ot'llt'r Il'nll!'. Till' InllN' are:! (I('\('rminctl hy stability cI('d\'aliv('s of lite ~('l'()ull onl('f (fm ('\,Illlplt', ('~~f, nnd <;"'.) Ihn! ill'e }f 1111.'
ill lit(' fOIll,th gl"OUP, In,·('stigalioll!-l :-;ilow Ibnt 111(' illfilll'lIl"l' of 111(' dl'ri\'(1ti\'c~ Oil tht' 1)(,t"o(lynamir ('ocfficicilis is 1101 lit,' ."allll' and lilill ,1 pt·i1Clic,1I :-;igniliciluCI' i.s n f('altlre of only a parI of ,~\lcl, d(,l"i\·nli\·I'~. among \\·lJieil -"ecolld ordl'r drl'i\"ath'C'~ rOI'1lI a qllill' :-nlitll fractioll. This IlIfl1l('I1(·(' is illlillyscd illl'1lt'h spC'cilic 1:11.'"'(' dl'r(,lIdiltg Ofl 110(' <1t'l"od.\"ilnlllic \·,lliligUt"Hlioli of a {"raft alit! the comtilioll." of ils Illolioll .. \s ,,('slllt. \\1' lind the (Im'i\'nti\·('S whirl! thn 1I('I"OII,'lwmic (:Ot'I"I"I("I('I1I:-< lite COI"r{'spondillg fot·cf's and 1l101lH'nts) d~rend 011 ;tlld litl' of which of them Illay be ignored. FOI' pac-II cOl'lficil'ill. \\·e (:,111 rC',·C'al lite charactl'l'isLic lrcud ;tfi."oeiated wilh ."lIcil it j"('Lllioll. \,pL II."; {'OtlsidC'f, for eXllmple. the IOllgillldil1,d (a.\ial) force ("ocHicielll. the {'xpression rOl' \Yhich will b(~ writtell ill lile [Ol"ltl
c.-.;,'-c.\\l··
c::~'J.~
C.~()'·'J.I\.. c.;':o,:
{.~~~2., c.~br~l\r
.
c~;'fli
(!I,~.I)
\\"hC'r(~ c.'·o is Ilu' \'allle' of C.,. lIt 'J.
. ~ ..• 6" .- Or - 0, (~\ .. " (~/I(, are tliC sc(:ollfl-or(ll'r parlial (Icri\'(llin's hf UJ(' iOllgitltllinai fnt,C~' coetTicielll. wiLh rl'spct:t 10 'J.~", .. -:xfl" (for cxnmpll', <:~ :c.
()~c.ttlh!).
ill [lccortianct' witb cxprl'ssioll W,2,1). the longitudillal fOI"('(l cU('lTiis 1\ qlllllttily not (iC'P(,IUlillg Oil tlt(' rotary dl.'fjvati\"('s or the
dplll
ilC"(,I'IC'I'atiolt dcri\·alin·s. Tlti.~ qllantity is I)etl'fmillcd in the f01'1II
400
pf. I. r"eory. AerOdyn.mics of .n Airfoil .nd • Wing
·of a qnadralic df'pendence on the angle of attack and the sideslip angle, as well as thl' elevator and rudder angles. Here the components oi the coefficient c.;: depending on the derivatives c~ and c!i are of siguiflcance when control surfaces are present in the form of movable wings or empennage whose deRoctioll may lead to a substantial change in the longitudinal force. The normal force- coefficient call be writ ten ill the form of Ihe linear relation cII = cuo -:- ~ + 0/6 (9.2.2) where CliO is the ,"alue of cII at a = ~ = O. Relation (9.2.2) is suitable for small \'alues of a and 6e• The larger these angles (beginning approximately from values of 10°), the more does the non~linear nature of the dependence of the coefficient c,l on them manifest itself. The Ilon~linear nature increases with a decrease ill the aspect ratio of the lifting planes, and also with an increase in the ~fach number (M _ > 3~4). In accordance with (9.2.2), when determining the norrold force coefficient. no account is taken of the innuence of the sicleslip. rudder, ele\'ator, and aileron angle..'1, and also of the rotation of tile craft and its acceleration. At the same Lime, when a movable wing (empennage) is prosent. the quantity c~r6(' differs appreciably from that for conventional control surfaces. In the latter case, this quantity may be negligibly small. The lateral force eoefficient c~ (in body axes) can be calculated by analogy wiLb Lhe normal force coefficient cII ' i.e. by using formula (9.2.2) in which the angle of attack and the elentor angle must be replaced with the sideslip and rudder anglcs, respectively: ct. = c~~ !. c~r6r (9.2.3) According to this expression, when ~ = 6 r = 0, the lateral force coefficient equals zero, which corresponds to symmetry of a craft about the vertical plane xOy. The pitehing moment at a given number ./JI"" for a night depends mainly on parameters such as the anglo of attack a and the elevator angle Be. the rate of change in I.hese angles and lie. and also the angular velocity Qr. The influence of other parameters. in particular of the sideslip angle, the aileron angle. and the angular velocities QII: or Q~, is not great and is usually disregarded. In accordance with the above, for the pitching moment coefficient we can obtain the following linear relation suitable for small values of the indicated variables Cin a body axis coordinate system):
a
mz",...;m zo :
nl~a;-:.m~·6e+m:ZCl)z+m~-i-m:eie (9.2.4) = 6.= Cl)z= a= 6.= o.
where mzo is the moment coefficient when a
Ch. 9. Aerodynamic Characteristics in Unsteady Motion
401
A glance at the above expression reveals that the pitching moment eoeBicient is determined by two Ittatic stability derivatives m~, m~ and three rotary derivatives m~=, m!e. By analogy with the pitching moment, the yawing moment may be considered to depend mainly on the sideslip and rudder aJlglee p and 6 r. the rates of their change in time Bn and also on the angular velocity Qz. Such a relation exist.s for axisymmetric craft. If symmetry (s disturbed, however, account must also be taken of the influence or the angular velocity Ox' This can be done. for example, with the presence of a non-symmetric vertical empennage when the upper panel upon rotation about it.s longitudinal axis creates a lateral (side) force that produces an additional moment about the vertical axis. Deflection of the ailerons also produces a certain yawing moment, which is associated with the longitudinal forces differing in value on horizontal wings upon deflection of the ailerons (this is similar to the appearance of a small pitching moment upon the deflection of the ailerons on vertical wings). Investigations show that such an additional yawing moment is usually very smal1 in modern aircraft. Accordingly. the yawing moment coefficient may be expressed by the following linear relation:
mi,
P,
t
inti
=
m~p"":,,, m.: r 6r -i-- m~x(o)x
.. -
m~Y(Uy +- m~~ ':- m~rt
(9.2.5)
In accordance with this relation, the yawing moment at p = = 6, = . , . = 0 equals zero. This is always observed in aircraft symmetric about the plane :rOy. By (9.2.5), the yawing moment coefficient is determined by t.wo static derivatives m2' and four
mu". me,
me,
and m]r_ rotational deri\'atives m'il". A rolling moment (a moment aboul lhe Jongiludinal axis Ox) is due to the asymmetry of the flow caused by sideslip and control surface deflection angles or rot.ation of the craft. For a given craft design and a definite flight. speed (number M ""), this moment can be considered as a function of the angle of attack a;, the Sideslip angle p~ the elevator, rudder. and aileron angles 6e• 6,. 6•• and also of the rotational velocities Qx. Oil' Oz' Accordingly. the rolling moment coefiicien t can be written as
mx =-· m!~+m!r6r+m:a6. +m~Pa;~-:.. m:~ra;6r
+ m:6 "j}6e -[- m:xw#t
:--- m:"wy -1- ma;IIt;T.WII -:-
m~zpw"
(9.2.6)
According to this expression, the rolling moment at p = 6 r '""'" = 6. = ... = 0 equals zero, which corresponds to an ideal design of the craft produced without any technological errors. When study-
402
PI. I. Theory. AerodynamiC5 of lin Airfoil lind. Wing
ing the motion of a real cralt, I'articularly rolling motioll, account. must be taken of the presence of such errors, especially of those associated with inaccurate wing installatioll, which has the maximum influence 011 the rolling moment. Inspection of (9.2.6) reveals that the rolling moment is determined as the function of derivatives Dolt only of the first, but also of the 6r , •.• ). Therefore. such second order (mf. m£", .•. , m~j!., a function is non-linear, which o{.curs even with small angular values of Ct, ~, 6e, 6 r, 6 •• and sDlall ....elocities (I).x> (I),. 00 2 , At the same time. tbe rolling moment coefiil:ient when performing approximate calculations may be found without taking into account tbt> influence of the rates of change in the anrle of attack and the sideslip angle, and also of the acceleration. th:s influence being negligible. All the derivalives determining the aerodynamic coefficients are functions depending mainly on the number M "'" and also on the geometric configuration and dimensions of the craft. In a more general case, these derivatives depend on the arguments determining the: aerodynamic coefficients, particularly on the angle of attack and the sideslip angle, the augular '.'clocit jps and the rclevant accelerations, and also on the acceleration or the translational motion. Such a relation is found separately in '~lIch casc for a gh'en kind 01 motion of a craft. A part of the considered stability (lel'ivatives characterize the aerodynamic properties of a craH regardless of the action of the control surfaces ~c~, m~, m.~', ... ), whert!m~ the other part is associated with their deUection (c~e, m~r, m~". m~,' . ... ). EaC'h 01 these derivatives has a physical meaning (l1ld lllay be helpful in studying the motion of a cralt. The static derivatives c~ and c~ {·h(ll'i\(·tl'l'ize the pl'opcrty of a craft to change the normal and Interal fol'(,cl' depending ou the angle of attack and the sideslip angle, wh('reM the del'i\'athes c~ and t1r characterize the same property, but a:;~odated with the deflection of the corresponding control surfa('es. These del'ivatives are important for controlling the motion of the centre of mass of a <"-raft in the longitudinal and lateral directions. The static second-order derivatives ctit., c~~. c~~. and c~~ determine the additional increment of the longitudinal fOl'ce due to denection of a craft through the angles (% and ~. and also to the C·OIltrolling action of the control surfaces. The derivath'es c~6, and c~6r characterize the inOuence on the longitudinal force of the intel'action of the angle of attack and the elevator angle, while c~G, characterizes the influence of the sideslip and rudder angles lin accordanre with this, the quantities c:GeCt6e, and c~Grp6, in (9.2.1) are call('d inter-
m:
action terms 1-
mr.
The static stability derivatives m~. and m~ charact('rize an important property of a craft such as the stiffness of its longitudinal,
Ch. 9. Aerodynemic Chllracteristics in Unsteady Motion
403
sideslipping, and lateral motious;. while the deriyatiyes lII~e. mLr, m~r, and m~a rharaderize the effecti\'Cne!>s of the rfJl'rcspollding control surface. The rotational dcrivatiw!' act on the mot iOIl likp damping variables, and th£'y arc called, rpsp£'cti"cly, longitudinal damping coeffIcients (m';'z, m~, m~e), yawing damping coefficients (m~Y, m~, m~r), llllli rolling damping coeffici(,IlLS (m~:r). The componcnts m~:rwx in (O.2.5) and nt~YWy in {O.2.G) art', rl'i'pectiv('ly, the coeffit"ienls of the spiral yawing and rolling momen(.S. These moments appear Derause a craft combines translational motion with rotation about the axes OJ' and Oy, i.r. performs spiral motion. The investigation of this complicated motion is associated with the calculat.ion of the rotary (spiral) derivatives mV' and 111';11. Formula (9.2.6) for the rolling momcnt coefficient includes inLeraction terms determined by second-order stability derh·ati\'es. For example, the derivative m~B is related to the change in th£' rolling moment caused by the additional angles of attack 011 the port and starboard wing panels during flight with side-slipping. A .~iD1ilar eff('ct charact('rized by the derivath'e m~'h is; pro(lnced by deflection of the rudder. An additional rolling moment may also be caused by a change in the sideslip angle upon deflection of the elcvators. The magnitude of the lateral moment depends on the 1:iecolld derivative m~6e. Rotation of a craft about the Oy or Oz nxis may lead to an additional change in the angle of attack and the sideslip angle on opposite panels of lifting surface!'> and to t.he rel~"nnt spiral rolling moments characterized by the derivath-es 11I.~w'l and m~w;. From t.he more general expression (9.1.4) for the nerodynamic coefficient, we can ilee that it also depends 011 the other stability derivatives that were liot taken into account in the considered expressions. In separate kinds of motion, the innuence of such derivnth'es may be substantial. One of tbes;e motions is associatC'd with tIl(' rolling of a naft at a sufficiently high angular velocity n.,.. Snch rotation produc£'s the Ma,gnlls effect consisting in that with the pr('scnce of an angle of attack and a sideslip angle, forces and moments appear that are proportional to the products aO x or ~O:r' the direction of the forces coinciding with a normal to the planes containing the angles a and ~. Accordingly, the increments of the lateral force coefficients are .dcr = c?'~·'awx, and of the normal forc,e ones are /)"c/I = ce~'·~~(I)r' The increments of the yawing and pitching moment coefficients are, respectively, /)"m ll = m~"':ca(l)x and /)"m r = mr"'·~~w.~. In these expresSiODS, the quantities c?~)·~ and m~"'·t are called the Magnus stability derivatives. -' The origin of still another group of forces nnd moment!> is a!'sodated with the simultaneous rotation of a craft abont two axes and is of a gyroscopic nature. Gyroscopic forces and moments are pro-
".
404
PI. I. Theory. Aerodynamics of en Airfoil and
1I
Wing
fIg. 9.3.t
To the conversion of the aerodynamic coefficients and their stability derivatives from one reduction centre to nnother
portional to the product of two angular velocities. For example. upon the simultaneous rotation about the axes Ox and Oy. these additional forces and moments are proportional to the product Q."Qy. and the corresponding aerodynamic coefficients. to the product of the dimensionless variabJes CilxW v ' This is why the derivatives of a coeHicient with respect to Cil:.:Cil Il (c~~~jl" m~,~Wrl. etc.) are known as the gyroscopic stability derivatives. 9.3. Conversion 01 stability Derivatives upon a Change in the Position of the Force Reduction Centre
Let us consider the conversion of the pitching moment and normal force coefficients, and also of the releYantstability derivatives calculated with respect to a reduction centre at point 0 for a new position 0 1 of this centre atadistanco of x from 0 (Fig. 9.3.1). A similar problem is solved, particularly, when determining the aerodynamic eharacteristics of an empennage relative to the centre of mass that is the centre of rotation of a craft in flight and coincides, consequently, with the centre of moments. Assuming that the pitching moment (',oefficient at Ct = Q% = 0 .aquals zero (m:o = 0) and that the airspeed is constant, let us form a general expression for this coefficient relative to the lateral axis passing through poiDt 0:
mz=m~Ct-i-m.~~+ m~)~wz+ m;'z~z
(9.3.1)
in whichl thel derivatives m~, m~ • ••.• are known. We can obtain a similar expression for the new reduction centre: mzt =
+ +m:r1(j)d +m.:;l~~~
m~}al m~f~1
(9.3.2)
Here the nerivatiYes m~!, m~ll • • . • are to be found in terms of their corresponding values obtained ror the reduction centre O.
405
Ch. 9. Aerodynamic Charflcieridics in Unsteady Motion
For this purpose. let us first consider the relations between a.
~I" and
al'
~I'
OOll'
ii,
001"
;'%1' A glance at Fig. 0.3.1 reveals that
'XI = ex
+ l1ex =
ex -
(I):;;
x
where = xII, 001' = Q;lIV., and, consequent.Jy, ~I = ~ - ~:;.. The angular velocities and their derivatives do not change upon a change in the reduction c.eotres, i.e. 00:1 = 00 •• and ~:I := (.,1" Using these relations. let us reduce the formula for the moment coefficient: mz =
m~lexl + m~I~1 ~- m~%IOOIi + m~':I~'1
(0.3.3)
expressed in t.erms of the new kinematic pal'ametcl's (for point 0 1) to the form m%
= m~la ....,... m~l~ + (m;:1 - nl~lx) 00. -;. (m~'zl - m~;) ~z
(9.3.4)
By comparing (0.3.1) and (9.3..'..) we find
With this in viow, expression (9.3.3) acquires the form mz =
m~al -T m~1 -i- (m:'z 7 m~x) WZ1 -:-. (m~)z -T- m~.i) ~zt
(U.3.S)
The obtained moment coeftident can be expressed in terms of its value found for the now centre 0 1 by means of the formula m. = = mn + e,i' (Fig. 9.3.1). Here formula (9,3.2) ca.n be substituted for mu. and C,l written in the form of a relation similar to (9.3.5):
e" = c;al-:-'
e:a
l .;..
(e:" -: e~x) 00:1 ...;. (e:'" .~. e:z) W.t
(9.3.6)
By comparing tile expression obt.ained fol' In: = m%l T cux with the corresponding relation (9.3.6), we find formulas for the stability derivatives relative to point 01: m~l=m~-e::X. m:{J=m::+(m~_c;·")X-C~2. } • . •• ••• m~l= m~- ~r. nl~ft=m:~z + (m:-e;")"i-c:x~
(9.3.7)
The new stability derivatives of the normal force coefficient are obtained by comparing Eq. (9.3.6) and ao equation similar to (9.3.2): CIII
=c;lal ~.e:~~1 -j.c;pOOzt + e'~il;"1
406
PI. I. Theory. Aerodynamics of an Airfoil and • Wing
in which we assume that cY1 = cy (because upon a change in the redu('tion centre the aerodynamic force does not change). The coroparbson yields c:l=c
y.
ci,=~, c(:u = yl
8
c;ll=c: z +c:x
C;I • ..!....
yl
I
JX' Y
(9.3.8)
Similarly, we can convert the aerodynamic characteristics for rear arrangement of the new reduction centre. 9.'. Particular Cues of Motion Lon.WI.... and Latera. MotIons
To simplify our investigation of the moliOIl ofa controllable craft and farilitate the finding of the required aerodynamic characteristics. including the stability derivatives. we shall usc the method of resolving the total motion into individual components. The possibility of resolution into longitudinal and lattoral motions in principal is due to the symmetry of a craft about its longitudinal axis. The longitudinal motion (pitching motion), in turn, consists of the translational displacement of the centre of mass in the vertical flight plane (the trajectory differs only slightly from a plane one) and rotation about the lateral axis Oz. In stich motion, good stability in roll is ensured. and variables such as~. '\'. COX' and COy may be considpred negligibly small (the surfaces controlling rolling and yawing are virtually 110t deflected). Upon lateral motion in the direction of the axis Oz, the centre of mass moves. and the craft experiences rotation about the axes Ox and Og (here the control surfaces function that ensure yawing and rolling motion). When investigating longitudinal motion, we usually consider the aerodynamic coefficients in the form of the following functions: cx=cx(M •• a, 6e). c,=O.
cu=·c!I(.IJI_, a, 6e)
}
.:..
(9.4.1)
m:-==O, m,,=O. m,=mz(M ... a. 6e, a. 6,)
To study lateral motion. we use the coeBicients in the form of cu=c,,(Moo. a, 6e), m~=m:r:(Mool a. mil
=
mil (M ...
P.
P. 6n
c,=cz(M ... ,
P.
6 r)
btl br • bat CO,r, Wg , Wz> W X1
wy •
W. 01')
(9.4..2)
Ch. 9. Aerodynamic Ch3r ... cle,i$flc~ in Unsle ... dy Molion
407
It mus\. be had ill view that longiLudillal motiolL is studied ilJde~ -pendently of lateral motioll, whereas the latter i.<: ('onsidered with .account taken of the parameters of longitudinal motion. which are .determined beforehand. Pal·tkLllarly, the magnitude of the rontrol normal fotTe
ey
is found. Motion of the Centre of Man
and Rotation about
If
The division of tile total motion of a craft into these two modes is possii)le H we as:;umc that Lhe control sy~tem functions perfecti)' and during the entire Ilight ensufCS the equality of the moments M '" My, and .11 z Lo zero. Such a craft and its control system are con~ider('d as inertia/ree (falil-response). Tbe ftsslllnplion 011 the absence of incrtia signilies that wilen ('ontrol surfaces arc defiected, the angle Qf attfl.<'k Hud the sideslip angle instantaneollsly (or sufficiently rapidly) ttl.ke Oil yailles corresponding to a statically stable position ()f the craft. In these t'ondilioml, the motion of its centre of mass in tile plane of flight i,o,: ill\'C~tigated independently. When performing sneh 111\,p!>tigation. we ~hall wrile the aerodynamic coefficients ill the form c_~=c",(,Uo<, ct,~,
op,
0,). cy'='c~J(Moo. ct, 6..)
,,_,,(M~, ~,Ii,) my = y (Moe. ~. 6r), m, =
"I
1 J
lrI z
(Moe,
ct,
(9.4.3)
op)
Assuming that mil -== m z '-'---' 0, we can find the elevator and rudder angles cOfl'csponding to tile requiri'd angle of attack and sideslip angle. aud to the given traje{'tory. The possibility of Ilsing sllch a method of in vestigation of the trajectory is based on the low sensitivity of displa('ement of the centre of mass to the rotation of the craft about this "entre up \0 the in.«tant Table 9.4.1
Motion
PHelling and rolling without Pitching rolling RoBing withuut pitching Motion without pitching and roIlina
. I• + +
", + +
"
'. " "
"
+ +
+
+ +
+
408
Pi. I. Theory. Aerodynamics of an Airfoil find .. Wing
when the craft occupies a position of statically stable equilibrium. This motion due to denection of the control surfaces or random disturbances can be studied independently of the longitudinal motion of the craft. Table 9.4.1 shows the different modes of these motions and the corresponding control surface angles 6e• 6 r• 6•• the angle of attack ~. the sideslip angle ~. and also the angular velor.ities U~,
Q9+h~i;lvestigation
of a particular mode of motion can be simplified by excluding the i.nDuence of accelerations. This is justified in practice when the motion develops sufficiently slowly. When it is necessary to take the acceleration into account. one can introduce a correction to the aerodynamic coefficient in the form of a term equal to the product of the first stability derivative with respect to the acceleration and the relevant dimensionless acceleration. Ptkhlnl MotIon
Of special inlerest J:.; pitching motion, which is called the principal mode of motion. Longitudinal oscillations that are described quite well by a harmonic function surh as (9.1.6) usually appear in this motion. According to this law. the time derivatives of the angles of attack and trajectory inclination are
~= A sin Pr.t.t,
fr=Qz=Bsinpot
(9.4.4.)
where A. B are tile amplitudes, and Pr.t.. ptj are the frequencies of oscillations of the corresponding derivatives. We can consider three modes of motion each of which is described by a harmonic function. The first mode (Fig. 9.4.14) corresponds to the condition ~ = o. 6 = Uz = B sin Pot. In this mode. the axis of a body, coinciding with the dire(tion of flight (~ = 0), performs harmonic oscillations along the trajectory. The second mode of motion (Fig. 9.4.1b) is characterized by the fact that the axis of the body retains its orientatiou along the trajectory so that Uz :0_' ;. = O. Here. however, the angle of attack c.hanges to follow the harmonic change in its derivative ~. The third mode of motion (Fig. 9.4.1c) is characterized by a rectilinear trajectory with the angle between the axis of the craft and the trajectory changing sinusoidally. In this case, the angles Band ~ are equal and coincide in phase s~ that ~ = fr =. g". Important derivatives in this motion are c~ + C:" and m: + m:,:. All these three modes of motion, which are of a quite intricate nat.ure. arc encountered comparatively frequently. Two other modes
Ch. 9. Aerodynamic Characteristics in Unsteady Motion
~~;tr~~~ar
4()90
cases of motion of a craft:
o-at a zero angle 0' a\tack: to-wilh conSlanlori<'nUilion or Ihl' crart 10 = eonstl; C-il)WlIf a straight ttajeetor,.
Flt-'.U
~Iodes of motion of a craft: II-perfect loop; b-Ircc 'all
are simpler. but are encountered less frequently. }o~ach of them allows one to understand the distinction between the modes of motion 0 and ~ U. 0, = O. characterized by two conditions: ~ '--' O. 01'
'*
'*.
and accordingly to gain an idea of the derivatives m~ and m:,r. The three modes of sinusoidal motions jndieateu abo\'e ('.an be treated as combinations of these two modes. The first of them (Fig. 9-".2a) is characterized by the fact that the angle of attack between the instantaneous direction 3 of the vector
410
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
F•• 9.4.3
Banking of a craft
-V .. and body axis 1 does not change and, c.onsequenlly, the derivative ~ = O. In this ruotion, however. a varying angle (t forms between fixed direction 2 and axis 1. Therefore Qz = dO/dt ¢ O. In a particular case when OJ = canst, the trajectory in Fig. 9.4.2a characterizes the motion of a craft performing a perfect loop. Figure 9.4.2b shows n second mode of motion corresponding to the r.ollditiol1s ~ ¢ 0 and Q" == O. This maHan may occur if free fall witll a \"olocity of w = gl (where g is the acceleration of free fall) is superposed all the longitudinal motion. Here the angle {t between the fixed direction 2 and body axis 1 remains constant, while the angle ex changes (between the instantaneous flight direction 4 and axis 1). Consequently, the derivath-es .fr = Q" = 0 and ~ ¢ 0; in the gi"en case the derivative = const. In a more general c{lse (for exam~le, in sinusoidal motions), it is
a.
necessary to use botla derivatives m~ and m~:. If one of the considered modes of motion dominates, onl~' one deri"ative is significant, and it is exactly what should be used. Figure 9A.3 shows a wing for which translational motion along its longitudinal axis, at a ZOI'O ;,mgle of attack. i:; supplemcntcd with rotation abollt this axis at the angular velocity Q x = dy/dt, The nen-zero variables oW,:l' aud ~)x correspond to this mode of motion. If the rotation about the axis Ox is steadr. then 00_" ¢ 0, while the derinlh'e 'ro", = o. 9.5. Dynomlc SIoblllty Definffion
An analysis of the derivatives of the aerodynamic moments with respect to (X or ~ allows us to establish the kind of static stability. This analysis is not sufficient for estimating the flight performallce of a moving body, however, because it does not allow one to establish the nature of the body's motion after the removal of .a dieturbance and to lind the variables determining this motion.
Ch.9. Aerodynamic Characleristics in Unsteady Motion
411
Actually, if we know, for example, that the derivative m; is negative and that, ('ollsequently, thE! centre of pressure is behiud the -centre of mass, we can arrive at a conclusion only on longitudinal static. stability. But we cannot establish, for example, the amplituuc- of oscillations of the angle of attack at a certain value of the initial disturbance and how it changes in time. These and other problems are studied by the theory of the dynamic stability of a craft or of the stability of its motion. This theory makes it possible, naturally, to study not only the oscillations of a craft, but also the general case of the motion of a craft along its trajectory and the stability of this motion. The theory of dynamic stability uses the results of aerodynamic investigations obtained in conditions of unsteady flow in which, unlike static conditions, a body experiences tile 8('-1ioll of additional time-dependent aerodynamic loads. Tile eOllcept of dynamic stability is associated with two modes of motion of a craft, namely, undisturbed (basic) and disturbed. Motion is said to be undistu.-bed (basic) if it occurs along a detlnite trajectory at a \'elocity varying ac('ording to a given law at SUHldard parameters of the atmosphere and known initial parameters of this motion. This theoretical trajectory described by specUle flight equations with nominal parameters of the crait ami control system is also ."laid to be undisturbed. Owing to ranrlom disturbing factors (gusts of wind, interference in the control system, failure of the initial conditions to correspond to the preset ones. deviation of the ac-tual parameters of a craft and its control system from lhe nominal ones. deviation of the actual parameters of the atmosphere from the standard ones), and also because of disturbances due to deflection of the control surfaces, the basic motion may he violated. After removal of the random disturbances, a body moves during a certain time according to a law differing from the initial one. This motion is called distul·bed. If the forces or moments produced upon a deviat.ion from undisturbed motion are such that they return the craft to its initial trajectory, tile motion is stable, otherwise it is unstable, Figure \'.5.1a shows a stable flight in which a craft after leaving its trajectory at point M because of a disturbance continues its motion with a constantly diminishing deviation from tile initial direction of flight; Fig. !).,j.lb shows an unstable nigllt charac.terized by an increasing deviation. In the general case, the motion of a craft is determined by kinematic parameters that are functions of time such as the velocity V <» (I), the angle o[ attack a: (t), the sideslip angle r. (t), the components of the angular velocity lQ,\, (t), Q lI (t), Q: (t)1, and the pitching, yawing (course angle) and rolling angles [tl' (t), ,/1 (t), and y (/}I.
412
PI. I. Theory. Aerodynemics of an Airfoil and a Wing
I'~~, ,'/
v/
'.~
r:'L'/'7
,
M
M
,
FII. U.t
llodes or motion
K
0
K
0' a crart:
I-Slabll'; '-Unslable;
b.-amplitud.1' oj OIIClUatLons
Let us assume that the values of the same variables Po., (t), a O (t) etc. correspond to a given undisturbed motion. If at sufficiently small initial deviations (at the initial disturbances) 6.V::." 6.ao, ... (here 6.VO"" = V.;: - 1"'"". /!ao = a'o - aU • ... ) the subsequent deviations /! V00 = V 00 6.a = a - aO do not exceed certain preset values, tho motion is stable. Jf these deviatiolls increase unlimitedly, the motion is unstable. A flight may occur when the deviations neither attenuate nor increase: here neutral stability of motion is observed. Such a defmition of stability is associated with investigation of the response of a craft to disturbing actions provided tliat these act.ions impArt. initial deviations to the variables of the undisturbed flow, while the following motion is consideNd already without disturbances. [n such motion, the control surfaces remain flxed. This mode of disturbed motion due to initial disturbances of the variables is called proper or f1'oo. The proper motion of a craft can conditionally be considered as a new undisturbed motion. The stability of free motion of a craft can be iO",'estigat.ed by analysing the differential equations describing this motion. If the lateral variables and the time derivatives of the longitudinal variables in the undisturbed flow are not great, we may consider longitudinal and lateral motions independently and, consequently. study the stability of eac.h of them separately. When t.he nature of motion changes abruptly, for example. when a manoeuver is executed. such a resolution of motion into it!! component.'! is not justified. and the solution of the system of equations of longitudinal and lateral motion must be considered jointly. This solution makes it pos.'!ible to establish the nature and influence of the aerodynamic coefficients on stability in the most general form. If the intensity of action of random factors is not great. the disturbed trajectory differs only slightly· .from the undisturbed one. This allows one to use the method of' small perturbations that is based on linearized equations.
v:..
Ch. 9. Aerodynamic Characteristics in Unsteady Motion
413
SW,1I1ty ChlradlirlsHcs
Let 11$0 consider the characteristics of dynamic stability, and also the rolt' and place of the aerodynamic coefficients (stability derivative~) in investigating the dynamics of a flight using the example of translational motion of a craft also performing rotation (oscillations) abollt the axis 0: (Fig. 9.4.1c). We shall write the equation of such disturbed motion in the form (9.5.1) where 4,
= -m?;qocSlIJ t •
43
= -m~qocSl/Jf
4 Z=
-m~q ... Sl/J: }
19.5.2)
Here J: is the principal central moment of inertia of the craft about the axis 0% that is one of the principal central axes. Taking into account that m~; = mi, we obtain for the sum of the coellil'ients
19.5.3)
With a view to the value of the coefficient written as
R3
(9.5.1) can be (9.5.4)
Assuming that the coelficients 41 are constant, we find a solution of Eq. (9.5.4) in the following form: a=C,ePlt +CzeP2t (9.5.5) Here PI and P2 are the roots of the characteristic equation p'l -iT 42) P + 43 = 0 determined by the formula
+ (41
Pt,2= --O.5(4 j · I .IZ:!) ± YO.25 (ad-a:Jz
a3
(9.5.6)
We flOd the constants C1 and Cz from the conditions' that can be
determined by the deril'ati\'e ~ = ~o and the zero disturbance a: = 0 at the instant t = O. For these eonditions, the constants are
Ct =Cz=O.5~IO.25 (a, ~1Z:!)2'-aal-t/2 and the solution becomes a=O.~b-le-l.lj (i!'-e-W)
where b = YA~ - aa, A = 0.5
(41
+ 42)'
(9.5.7)
414
PI. I. Theory. Aerodynamics 01 an Airloil and a W,ng
Let us consider static stability when the derivative mr; < 0 and, consequently, a 3 > O. If we have in view here that in real conditions the damping moment is considerably smaller than the stabilizing one and that therefore I aa I > Ai, we obtain the relation (9.5.8) in whiell Ii =. V \ as \ - A~. Examination of (\).5.8) reveals that the change in the angle of attack has the natUl'e of periodic oscillations. Since the quantity ~'1 is always positive, these oscillations are damped ones; consequently, we have to do with oscilJatory stability. Let us consider the characteristics of this stability. The period of the oscillations is T=2n(\m~lqClOSlJ;1_Ab-1J2
(9.5.9)
and their frequenc)' is W = 'l.n/T= (\ m~lqcoSlJ;I_Ar)1/2
(9.5.10)
The oscillation frequency is influenced mainly by the degree of static stability, whereas the influence of damping is not great. The nature of action of these factors is different. An increa~e in the degree of static stability leads to a higher. frequency, while an increase in damping, conversely, leads to its diminishing somewhat. The logarithmic decrement is e ~ A,T (9.5.11) where (9.5.12) The larger the coefficient m'i (or m~:), the greater is the logarithmic decrement and the more rapidl~' do the oscillations damp in time. One of the important characteristics of oscillatory motion is the time to damp to half amplitude: (9.5.13) When investigating oscillatory stabilit.y, the quantity A.j Cilll be considered as an independent characteristic of this stabilit.y called the damping coefficient. A glance at (9.5.12) and (9.5.13) reyeals that its magnitude does not depend on the degree of static stability m~. Let liS consider the wavelength of oscillations (9.5.14)
Ch. 9. Aerody""mic Characteristics in Unsteody Motion
415
In prflctice, it is expedient to ensure aerodynamic properties of a mo\"ing hody !'ol1ch that the damping coefficient is ;:ufIicien1.ly great because here the time t2 is short. although the wfI\"e!ellgth grows somewhat, which is not desirable. Good prflctire dktfltes decreasing of this length. To do this. it is necessary to incff'ase the stabilizing moment. ' .... hich. in turn, leads to such a po~iti"e phenomenon as a decrease in the amplitUde of oscillation. In the Cfl.!'.e being considered, a body ha.!'. ;;tfltic stahility lhat also producf's in it longitudinal oscillatory stability. A similar analy. sis can be performed for the cafie when such static stability i!'o absent, Le. when the derivative m't > O. Solution of (9.5.4) yields a relation for C/.. lndiratillg the undamping aperiodic nature of motion. Hence, static in~t8bilitr also gi\"es rise to inst.ability of Illotion. In this case, agreement ran beobsel"Yed bet,,'eenstatic ouri dynamic stability or instability. Such agreemE'nt is not obligflt.oI'Y. ho,Ye\"er, for the g(1Tleral rase of motion of a craft. One Illny lJave a statically stable (Taft thnt, howewr, doC'!<nothavedynamic stability and in its telH..Iellry to the position of eqnilibrium will perform oscillat.ions with an incrE'm:ing amplitude. S1]ch cases are observed in practice in some airrraft at 10\\'-spce(! flight. and al~o in flying wings with a small sweep of thfl leading edge. It ron:::t be notcd that tllC overwhelming majoriLy of aircr ...ft owing to special devices have static st.ability, i.e. the ability to respond to distmbau('es so as t.o reduce their magnitude at the initial instant. This property is of Illnjor practical sigllil]rance regardless of how an aircraft behaves in the process of di::tllrbed motion. Free development of the Jisturbances ifi lIsnally attellded by denection of the control surfaces to retmn the ,,-raft to its preset flight condiliolls. The use of these control ~l1rfaces ifi a necessary condition for ensllring the preset motiOH of a statically unstnble (with fixed control surfares) aircraft. The concept of stability presumes the existence not of two separat.e kinds of stability-static and d:ynamic, but of a single stability renecting the real motion of a craft in time and its abUity to retain the initial flight c.onditions. Since the dynamic coefficients of the eqllations depend on the parameters of the undisturbed flight, tho s1lme aircraft can have different dynamic properties along a tra~ jC'clory, for example, its motion may be stable on some sections of the trfljectory and unstable Oil others. unsteady motion is described by equations whose dynamic coefficients depend on the design of the craft. Unlike craft of the aeroplane type, such coefficients for bodies without an empennage. or with all only slightly developed empennage (fin assembly), correspon<] to unstable motion characterized by slow daropiTlg of the osrillation.<:. This requires the use of means of stabilization that ('Ilsure gl"NllE'r dampillg.
9.6. Basic Relations for Unsteady Flow AerodynamIc Coefflclenb
Let us consider a flat lifting surface moving translationally at (~ = 0) and of an angular velocity Qy. The projection of the area of an aircraft or its separate elements (wings, fuselage) onto the plane xOz may be such a surface. The aerodynamic stability derivatives obtained for this surface are very close to those that would have been found for real bodies in an unsteady flow. The normal aerodynamic force, and also the pitching and rolling .moments are due to the pressure difference dp = Pb - Pu on the bottom and upper sides of the lifting surface (Fig. 9.6.1) and are .determined by the formulas
.a constant velocity in the absence of sideslip
y~.\ lllpdxd"
M.~ llllpxdxd"
(5)
(8)
M,~ - .\'jllp,dxdz
(9.6.1)
(S)
in which integration is performed over the area S of the lifting surface. Let ~lS introduce the pressure coefficient p (p - p ... )/q .. = 2 (p - p ... )/(p_lf'!_> and the dimensionless coordinates
=
£=
xIx,. and ~
=
zlx,.
=
(9.6.2)
related to the length XI!. of the Iifti ng surface. Now we obtain the following relations for the force and moment coefficients at any unsteady motion of a lifting surface with the dimensions l:
Ch. 9. AerodYnAmic Characteristics in Unste.!dy Motion
417
FI,. U.t
To the determination of the aerodynamic coefficients for a lUUng surface
Here So and ~I are the dimensionless coordinates of the leading and trailing edges of the lifting surface, respecth·ely. For a non-standard coordinate system in which the axis Ox is directed from the leading edge to the trailing one, the signs in formulas (9.6.3) should be reversed. Let us fmd relations for the aerodynamic characteristics of the lifting surface cross sections. For the normal force, longitudinal moment, and rolling moment of a section, we have, respectively dY=
r
6.pdxdz,
:l:"n
dM:T = -
dJlz = )ll.pxdxclZ,
..y
:TQ
6.pz dx dz
(9.6.4)
The corresponding coeffidents are as follows:
1
I
(9.6.5)
where x' is a chord of a lifting surface element in the section z = = const being considered. We calculate the integral in (9.6.5) over a given section from the trailing (Sl = XI/XII) to the leading (So = XO'XII) edges. W
418
Pt. I. Theory. Aerodynamics 01 IIIn Airfoil /lnd
/I
Wing
represent the difference of the pre~sure coefficients in (\).6.3) and (9.6.5) on the basis of the Taylor formula [see (Q.1.4)} in the form of a series expansion in terms of the kinem8tic variables: q,-,,(,).
q,-w.(,}-Q.(t)lIV_
)
q3=(o)z('t')=Q~(t)lJV_, gt=~=daldr, q~=~.,=d(~,)d;;
(9.6.6)
g3=~~=dw:/d'f: 't'=V...t/l (here 1 is the characteristic geometric dimension). This expansion has the following form:
.lP=~ (pqlqj i-p~jq,)=p<'«+p~~+p~),"OOx i_I (9.6.7) where p(J. = ap(J., pn ~= Ai«, etc. are the derivatives of tile difference of the pressure coefficients. By (9.6.7), the aerodynamic, coefficients call be written as cy =m z =.
l
,Ej (c~lql+ c-ilq;)
± f:"
(m1;Qt+m:"Qj)
(0.6.8)
'. I
m.,= i-;;1 (m:iq,
-r- m..~!q,)
J
In formulas (9.6.8) for the nOli-stationary aerodynamic c~effi cients expressed in the form of series, the \'alues of the stab1l1ty derivatives that are coefficients of q; and mnst correspond to Ac('ordingly, the reie"ant expansion (9.6.7) for Ule quantity derivatives take on the following value.~;
a;
C;' =
q;
c~, c~:r:, c;z;
m~l=m~, m:"; m~'~; m:' = m~, m~3:; m(.~z;
C~''',
c:.
m~'=m:, m~:>:, . . .
m:1 = m~,
m:)z .
m::E, m:z
Assuming that the craft length XII has been adopted as the characteristic geometric dimension, we obtain from (9.6.3), (9.6.7) and (9.6.8) expressions for the derivatives of the aerodynamic coefficients
in terms of the derivatives of the difference of the pressure coeffi-
Ch. 9. Aerodynamic Ch",racieristics in Unsteady Motion
419
cients: 2.x 2
1/(2.~1r.)
1=p
o
~I
C:i=-t
J
2.J2 1/(2... /1) ~o
•
J
JP~id~d{, C~'=7
0
.
JP"id~d~
~I
li(2J:/I)i o
m;i~-~ ~
jp'Il~dsd!;,
~l
ii
m~i~
-
~Z
J Jp;lsd~ds o
2.x 2
m~i=--f-
IIIZs/,) i.~
I o
19.6.9)
iG
li(2~·k)
~l
.
"x 2 1/(2.th);0
1p'/iSd~dt;,m:i.o.--;' ~l
J 0
•
JP?i~d~tl~
!1
The above expressions are linear relations between the aerodynamic deriv~tives and the dimensionless kinematic vDriables gj and go that ensure quile reliable results provided that the values of these variables are small in romparison with unity. III linearized problems 011 the IInsteady flow O\'er lifting :"lIrfares, such relations are exact for a harmonic law of the change in the kinematic variables (see 14, HI]). Let us consider some features of the slability derivaU\·cs. Sinre we arc dealing with motion in the absence of sideslip and an angular veioeity about the axis Oy (~ ,..-. 0, Q v = 0), here the deri-..'atives of the aerodYOiunic coefficients wi th respect to the parametl?rs ~ and Q y evidently eqllal zero. In addition. we have excluded the coeHiciellts of the longitudinal and lateral forces (c,., c,) and also the yawing moment coefficient m" from our analysis because inde· pendent in\'cstigations arc devoted t.o thew. It mllst be had in vjew here that when analysing the stability of a craft, these qualltitief-' are of a smaller practical significance. For a lifting surfaco that is symmetric about plane xOy and movE'S translationally in the absenr.e of rotation (Q x = 0), the prcssure is distributed symmetrically; therefore, the corresponding derivatives equal zero: m'l/ = 0, = (i = 1, 3). With rotation about the axis Ox, the pressure is distributeu asym· metrically about symmetry plane xOy, hence the aerodynamic loads produced by the angular velo("ily ul.~ are also asymmetric, and, consequently, the following stability deri-vatjy(,s eqllal zero in this case:
mil
c:·~ = 0,
°
c~ ... = o~
m~·t = 0,
m~'·'· = 0
420
Pt. t Theory. AerodynamiC5 of an Airfoil and a Wing
C.uchy-LilFilnge IntaFill
Let us consider the basic expression of the theory of unsteady now relating the parameters of the disturbed now (the velocity. pressure. density) and the potential function . With this in view. the equation of motion (3.1.22") can be given thus
a g~~d (lJ -+- grad X; =
_
+
grad p
(9.6.10)
Considering that the gas flowing over a body is a barotropic ftuid whose density depends only on the pressure. i.e. p = p (P), we can introduce a function P, assuming that P
= Jdp/p (p)
(9.6.11)
Accordingly.:
(9.6.12)
(lip) grad p = grad P We can therefore write (9.6.10) as grad (iJ
+
+ P) =
0
Hence we obtain an equation suitable for the entire space of a disturbed Dow near a lifting surface including the vortex wake: iJ
+
+
where F (t) is an arbitrary function of time determined with the aid of the undisturbed fiow variables for which 84J18t = O. Assumisg that V2 = V!. and P = P .... we fmd F (t) = V~/2 p~ c.onsequently,
+
+
+
+
(9.6.14) 8(/)/at V'12 P = V'~12 p~ This equation is known as the Cauchy.Lagrange Integral. For an incompressible medium. we have po. = I' = const. Therefore. P = = pip ... and Po> = p oo/p 00. Hence, iJ
+ V'12 + plp~ = V'~12 + p~/p~
(9.6.15)
For an isentropic flow of a compressible gas in the region of a con... linuous change in the thermodynamic properties, we have pipit. = = poo/p'!" (where k = cplc~ = const is the adiabatic exponent). A3 a result, we obtain p= k:1
of,
Poo= k~1 • ~:
Ch. 9. Aerodynamic Cn",,,cte.istics in Unsteady Motion
421
Therefore, tIle Cauchy-Lagrange integral will aC4.uire the follo\ving form: (9.G.lu)
Assuming that the now over a body is nearly nniform, linearize Eq. (9.6.16). The square of the velocity is V 2 = V~ V;, -:- 11~ = (V <Xi -;.- uf .;- [,,2 + w~
+
where u, v, and ware infmitesimal components of the disturbed velocity (u <".< V <Xi, tJ« 11 0<, and w
+
We uetermine the ratio p,'p from the I'ulillbat equation pip'! = in accordance with which pip = (p ",,,(pco) (plp ... )(h-l)!'"
= poo/p':."
Introdllcing p = poo
+ p'
into this expression, we obtain
pip = (p_ip..,) (1 --:- p'ip ",,)(11-1),11<
Expanding this expression into a binomial an accuracy within linear terms that
f07~:(1 Accordingly,
k~i
serie_~,
*)
p - p~ ~ p'.'p~ ~ (p - p.)/p"
",,'e find with
(\l.o.Ii)
Hence, a linearized expression of the Cauchy-Lagrange integral suitable for investigating nearly uniform um;leacly flows has the form ([I.6.1S)
where!f =
421
Pt. 1. Theory. Ae.odynllmics of lin Airfoil lind II Wing
Let us introduce the pressure coefficient p, the difference of its values for the bottom and upper sides I1p, the dimensionless potential q.., and also the relative coordinates and time:
p-(p-p~)lq~, t.p~;;,-p" S=x!x",
~=zlxl"
1')=ylxlt"
1"
y, z, t) - V""XIt,q) (s,
q> (x,
't-",V..,t/x"
S,
I
(9.6.19)
't)
When these dimensionless quantities are taken into account, the Cauchy-Lagrange integral becomes
p-
-2 (aq,18'
+ 8q,lihj
(9.6.20)
Having in view that from considerations of symmetry, the velocity potentials on the upper and boLtom sides are identical in magnitude, but opposite in sign, i.e. cp (x, y,!, t) = -cp (x, -y, z, t), we obtain the following formula for flp: t.P""Pb-P"~ -4 (8<j;M+ oq;I8<)
Let us represent the expressions for Cf. and dependences on qj and
qi:
3
•
q;:" ;=-1 ~ (~qlql+ip'llg,);
(9.6.21)
IIp in the form of linear
3
•
I1p=~ (p"lql-",pqlqd
(9.6.22)
;=1
We shall insert these relations into (9.6.21) with a view to the aerodynamic coefficients not depending on the time: 3
h
,
(p'l;q,+p;iq,)
i=-1
=
-4[~
.
(0:;1 q,.I. D:;' ql)
i~l
I
•
+ ~ (~'1lql +;P1i ~~l )J
(9.6.23)
;
For harmonic time dependences of the kinematic variables (ql = = qt cos Pit). the derivative iiqll{)'r=q~= _(pt)2q,
where P: = PiX,JV.., is the Strouhal number (t = 1, 2, 3). With a view to the obtained value for q'/o we find the following expressions for the derivatives of the difference of the pressure coefficients from (9.6.23): p"l = -4
[a~(lila~ T (pn~q;;IJ; P~' =
-
4(O~;'los-i-q;",) (9.6.24)
Ch. 9. Aerodynllmic Chllrllcteristics in Unstelldy Motion
423
In ac-cordanec with the above relations, the "allies of p'li and P;f are determined by the derivative.'! of titu dimensionless potential functiou with respf'ct to the rele\'(\nt kinrillatic parameters. Wave EquaHon
Let liS obtain an equation \Yhi~h the veloelty potential
or.
1
fu -7'w'
op _
1
IJp.
'* t1z. *
ay---;;Z'Tg'
ap." 1
iJp
az--Qi'·a;
= (9.6.25) These expression.'! relate to it barotropic fluid for which the density is a function of the pressure {p = p (p)l, and the square of the sp(>ed of sound a:! = dp/dp. By (9.6.25), the continuity equation becomes
+.-%_: a~ (iJ~;y _~.~+~)+~.~ +~,*+~.~-=O
(9,6.26)
Let liS exclude the dynamic variables from Ihis equation, ret.aining only the kinematic ones, To do this, we shall use the Cauchy· Lagrange integral (9.G.18') from which we shall find expressions for the pressure derivat.iv(>s with respect to the corresponding cOordinates x, y, oZ, a.nd also the time t related to the df>nsity: (lip) opfax, (lip) opioy, O/p) apia, For this purpose. with respect to x, 1/,
\\'C
Z,
sholl differentiate (9.6.18) consecutively and t:
+·*~e;rV~ ~~~-~~)
+.-%£-
=~(V= d:2:y
+'*-=Pf(V~ iJ:~t +.*--,~(Voo
-
aaY~I)
;,":t)
(1l.6.27)
u:;t - ~~~)
We shall write the relation for the square of the speed of sound at; in a linearized fiow on the basis of the Cauchy-Lagrange integral. Since P-P.
424
Pt. 1. Theory. Aerody"~mics of an Airloll "nd a Wing
then with a view to (9.6.17) and (9.6.i8'), we have aZ = a!, (k - i) (V"" acp/ax - 8cplat)
+
(9.6.28)
We can represent the velocity components in (9.6.26) in the form
V. _ V _
+ &~/&X,l
V, -
a~iay,
V, -
a~/a,
(9.6.29)
Let us introduce (9.6.27)-(9.6,29) into continuity equation (9.6.26). Disregarding second-order infmitesimals and assuming that the density ratio p,,jp ~ 1, we obtain (i--M!')
~:;
-I ::; 7
~:; + 2~:oo .~-
aL . ::~ =0
(9.6,30)
where M"" = V ""Ia"", is the Mach number for an undisturbed Dow. The obtained equation is called the wave one. It satislles the velocity potential III of an unsteady linearized flow. If in a body-axis system of coordinates, the longitudinal axis 0:& is directed from the nose to the tail, the sign of the term (2M ",,/a .... ) a'1.~/8x 8t in (9.6.30) must be reversed. With a view to the notation (9.6.19), the wave equation in the dimensionless form can be written as follo'\\'s: (1-M;,)
~~~ + ::~ + ~;% -,2M~ iJ~~'f
-M:O
~~
=0
(9.6.31)
To determine the aerodynamic characteristics of a craft. it is sufficient to solve wave equation (9.6.31), finding the potential of the disturbed velocities
V ~ &~biaX - 8q>.i&1 -
Va 8q>.!&x
(9.6.33)
Ch, 9. "'.rodynamic Characteristics in Unste
425
or in the dimen.!;ionless form iJCpb/iJ-C -
iJCPb/iJS = iJ~u/a1: -
iJ~uiiJ;
(9.6.34}
In accordance with the condition at infinity. we assume that the fluid is undisturbed, i.e. all the disturbances produced by the craft vanish. Accordingly, the potential of the disturbance and its (Ie· rivatives equal zero, i.e.
This method is Olle of the most widespread ones in the theory of Dow over craft or their isolated elements (";rings, fin assembly, empennage, body). It establishes expressions relating in the general form tile required potentiill function
r'-Vx"~ y"c z"
(9.7.1}
Here q (t) is the varying strength of the source determined at theinstant t' = t - r*la oo (where a oo is the speed of sound). lience, this strength q (t - r*la",,) is taken for the instant when a sonic disturbance wave emerges from the source and reaches point P with t.he coordinates x., y*, z* at the consifh.!l'cd instant t Wig. 9.7.1). When selecting any doubly differentiable function q; (n, expression (9.7.1) satisfies wave equation (9.6.31). Let us assume that beginning from the instant t = O. point sourcebegins to move 1rom point 0* at the velocity 'V"". Hence, at. any instant t, section 00* is filled with sources changin!l according
o
428
pt. I. Theory. Aerodynamics of an Airfoil and
o!I
Wir.g
FI.. U.t
'To the determination of the velocity potential of non-stationary sources
to the law q (t - t*), where t* is the instant a source originates at ·a point N (Fig. 9.7.1). The retarded potential at point P (x*, y*, z*) .due to these sources can be written in the form
,.
Ip'(x*, y*, z*,
t)=J
f,q(t-t*-f)dt*
(9.7.2)
where r' = V(x* + V..,t*)'l·, y*'1. + ZU is the distance between points P and N, t; and t; are the integration limits taking into .account the sources from which the disturbances arrive at point P at the instant t. Let us consider a moving coordinate system for which x = x* + + V..,t, Y = y*, and z = z*. We shall assume below that = = a ... (t - t*) - r', In this case for V.., > a.." we call hnd the following expression for the potential:
'2
·tp'(x,y, z,
t)=....!...r~...!..ql(.!!.)dr1-i-....!...I.!.q1(2)dr2. a"" J ,J , II ...
rl
II..,
rl
II..,
(9.7.3)
where rt=V(x M..,r.J2. a'1(yz-!-z2.)} M._V.la •• "'=VM~-1
(9.7.4)
The upper limit of the integrals ,; = a.., (t - tn - r' is determined by the instant t~ at which point 0 moves to point E on normal PE to the Mach Jine. The first term in (9.7.3) takes into account the innuence on point P of the sources on section AE (the leading Iront of the sound wave), and the second, on section BE the trailing front of the wave). Let us consider the case when a source is located only at moving point 0, and at the instant t point P is infiuenced by the disturban-
Ch. 9. Aerodynamic Characteristics in Unsteady Motion
427
t;
t;.
ees from the spherical waves occurring at the instants and We shall find the polential at point P by formula (9.7.3) as a result of a limiting process, assuming r ll _ O. We slu\U presume that
q(I,)_--,-lq,(-'L)a". 9(t.)---,-'Q.(-'-'-)d" a",,;~ a.., a... ~ a..,
(9.7.5)
whflre tl = t - At1; tl = t - Atl: At l , Atl are intervals during whir-h the signal is transmitted from the origin of coordinates to a point (x, y, z) with the wave front and its real' surface. With a view to (9.7.5). we have the following expression for the velocity potential: q(x, y. z, t)=q(t,)/r ..·q(t2)/r; r=Vr a'2(y2-:.· z :!) (\:1.7.6) Here q (tt)/r determines the potential due to the source at point A and q (t,J/r, at point B (Fig. 9.7.1). We consider that at the instant t -= 0 sources with a strength of q (.r;. z. t) appear simultaneously on the entire area occupied by the lifting surface. The potential produced b)' such sources at a point with the coordinates Xl> Ylt " can be written itS q.>(XI'
Ylt
Z't
t)= })
'(;T.~. ~1)d1"rJ"~.
J.\' ='-"'-""'= ,,~
IJI
wllere r = V (Xl xr'" ,,'2 Iy~ (Zl 411 is a conditioliRl elistance between the points (Xl' JIl. Zl) Rnd (:1', 0, z). The integration regions 0"1 (Xl' Yt. ztJ find 02 (Xl' Yh ZI) are the parts of plane ;1.;OZ confined within the hranch of the ~1Rch (characteristic) cone issuing from the poinl (Xl' Yh Zl)' This branch is determined by the equation a'2y i :...= (Xl - X)! - a'2 (ZI _ Z)2
As a result of dHferentiating <9.7.7) with respect to the coordinate Yl. we find an expression for the strength or a sonrce: q(x,
Z,
t)·...:_*[i.lfJ'(;T~:ll.z,.nl,-=o
(0.7.8)
The instants tl = t - Atl and t2 = t - At2 in (9.7.7) are determined according to the following values:
AtI=I.~:·~(XI-X-";... )
1 t
AI,~~(x,-x; -i;) r -'" V (XI
.x)~
a'~ {Y: . (Zt
z):!1
Relations (9.7.8) and (9.7.9) are derived in lt9).
J
(11.7.9)
428
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
When solving problems on the unsteady flow over flat lift.ing surfaces, we may assume that Yl :~ 0 because to determine the aerodynamic characteristics, we must find the values of the potential function on thi:; surface. The velocity potential cp [see (9.7.7)1, in addition to the conditions on the surface of bodies, must satisfy the conditions on the vortex sheet and at infinity, and also the Chaplygin-Zhukovsky condition in accordance with which a flow does not bend around a trafling subsonic edge, but is cast off it. In accordance with this condition, disturbed velocities near such edges change continuously, therefore the derivatives {}q;I{}x, iJcpliJy, and f}(pl{}z are continuous. The ChaplyginZhukovsky condition for the supersonic flow over a lifting sudace with suhsonic trailing edges is Ilsed in the form
.!~~~
*~.- ~;:
L-_x*
lim cp (x, D, z, t)
=
} cp (x*, 0, z, t)
(9.7.10)
:t_;;*
where x* (z) are the coordinates of points on the trailing edges, and cp (x*, 0, z, t) is the value of the potential on these edges. Vorte:.: Theory
When solving problems on the unsteady incompressible or subsoJlic compressible flows over a body, it is good practice not to seek directly the velocity potential satisfying the wave equation (as in the method of sources for supersonic velocities), but to use what is known as the vortex theory, which does not require the finding of cpo According to this theory, the disturbed motion near a lifting surface can be studied with the aid of a vortex pattern including the bound and free vortices that produce the same distribution of the velocities and pressures as the given surface in the [Jow. Let us consider a non-stationary vortex pattern using the example of a lifling sHrface with a rectangular planform (Fig. 9.7.2). The motion of such a surface is characterized by a constant velocity V 00 and small velocity increments due to additional translational or rotational modes of motion. These velocities cause a change in the local angles of attack, and also in the angle of attack as a whole, which results in corresponding changes in time of the lift force of the sections and, consequently, according to Zhukovsky's formula, of the circulation r (t) as well [see formula (6.4.6)]. This circulation is due to the rectilinear bound vortex core Ilsed to model the lifting surface with a rectangular planform. Assume that the circulation during the time M changes by the vallie ~r. In accordance with the Helmholtz theorem. in an ideal fluid the circulation of the velocity over closed Contour C (Fig. n.7.2)
Ch. 9. Aerodynamic Char<3c:terisiio in Unsteady Motion
429
fig. 9.7.1 Vortex system: l-Hfting surtac:p.. 2_.bm.md \'Qrlr·x with "arymg circulation,
or non'$!nl!onan
"orIlC~S
fig. 9.7.3 Vortex pattern of rectangular lifting surface
passing through the same particles does not depend Oil the time. Cool:iequently, when the circulation r (t) changes, a of free vortices with the circulation -l1r appears behind the surface that compensates the change in the circulation up to initial value. Accordingly, a vortex sheet forms behind the lifting surface that consists both of longitudinal vorL ices parallel to the vector VoD and moving along with the flow, and of lateral (bollnd) vortices that are stationary relative to the lifting surfact'. The intensity of vortex filament distribution in the sheet along the longitudinal axis Ox is l' = -dr/dx = - (ill' '>0) dr/dt. It follows from coupling equation (6.4.8) that the circulation is proportional to the Jift force coefficient clI . Wilh a lineClf d.epcnden~e of c li on the angle of attack (c~ = c~a). the circulatIOn IS also a linear function of '=', i.e. r = r"'a (where ro: = dr/do. = const), Therefore, the strength of the vortex sheet 'V = - (fa./V~) daldt depends OIL ~he rate of change in the angle of attack. Figure 9.7.3 shows a vortex pattern modelling a rectangular lifting surface. It consists of straight bound vortices with a vortex sheet cast oft thr.m. For Il lifting surface of an intricate planform, the vortex pattern consists of a nnmber of bound vortex filaments each of which is replaced with several discrete bound vortices from which a pail'
430
Pt. I. Theory. Aerodyn.mics of IIIn Airfoil .nd III Wing
Fl. 9.7.4
Vortex model of intricate lifting
5urface;
l-\Iiscretc obliQlW h()l'Seshoe 'V()ttex; 2'_vorlrx tflanlt'lIts C01\$tsling or d iO'Crele ()bUqu" vorl ie('/!; 3-vQI'tl'x shr~t
of free vortex filaments is shed. Such vortex patterns are called oblique horseshoe vortices (Fig. 9.7.4) Let us consider the Kutta-Zhukovsky theorem aUowing us to determine the aerodynamic loads acting on a surface element in unsteady flow. This theorem relates to circulation flow that is attended by the appearance of a trailing vortex and circulation over the contour enveloping the lifting surface. According to the Kutta-Zl)ukovsky theorem, the pressure difference on the bottom and upper sides of a surface clement in a linearized flow is (9.7.1l) AP=Pb-Pu=P ...'\'V"" where '\' = (1/V..,) iJl'Ji)t is the linear strength of the bound vortices, and V 00 is the yclocity of translational motion. Formula (9.7.H) is an application of the Kutta-Zhukovsky theorem to an arbitrary unsteady flow over a thin lifting surface and indicates the absence of the influence of free vortices on the aerodynamic loads. From this theorem, particularly, there follows the absence of a pressure difference on a vortex sheet consisting of free vortices. Let us write the strength of a vortex layer in the form of a series:
1-V.
±
(1"Q,+i'Q,)
(9.7.12)
1=1
Introducing this quantity into the formula for calculating the difference of the pressure coefficients ~p - 2 (Pb-p.)/(p.V!,) - 21/V. (9.7.13) we obtain (9.7.14)
Ch. 9. AerodYnolImic Characteristics in Unsteady Motion
Flg.9.7.S
Vortex model for non-circulatory now: l_lifting
~"rrac~:
!l11ached ,·or1iC(>s
2-,jisrFete
431
§§§§ §§§t:' \
I
d('~("d
Comparing this expression with the second formula (9.6.22), we find expressions for the derivatives of the pn'ssure coefficient difference in terms of the corresponding derivRtives of the linear· strength of the hound vortex layer: (0.7.15)
The above relations are exact if the kinematic· variables c.hange harmonically. For circulation problems, the Chaplygin-Zhukovsky coudition all the passing off of the flow from the trailing edge of a surface and on the finiteness of the velocity at this edge is satisfied. According to this condition, the strength of the bound vortices on the trailing edges is zero, i.e. "V (x, z, t) = 0
(\).7.16)
\Vhen hodies move at very low spccd:-l, or whell o."diJaljoll:-l occur in the absence of translational moLion, a scheme or non-circulalory Ilow is realized. I-Iere no wake forms behind a body, and the circulation over an arbitrary contour enveloping the lifting surface is zero. Accordingly. a vortex layer equivalent to the lifting surfRce is represented as a system of closed vortex filaments of constant strength (Fig. 9.7.5). Some results of studying non-circulatory flow, in particnlar the derivation of Zhukovsky's theorem for such [low, are· given in (1, 19]. 9.8. Numerical Method of Calculating the Stability Derivatives for a Wing in an Incompressible Flow Velocity Field 01 lin Oblique Horseshoe Vortex
The numerical method of calculating the stability derivatives for a wing in an unsteady flow is based on its replacement by a vortex lifting surface which, in turn, is represented by a system of oblique horsheshoe discrete transient vortices.
432
Pt. I. Theory. Aerodynamics of an Airfoil and • Wing
~':ii::~' h.orseshoe J,
vortex in an ullSteady flow:
t-tree vorliets
Let us consider the velocities induced by such vortices. With ,non-stationary motion of a wing, the circulation (sl.rength) of a bound vortex varies with time, i.e. r = r (til)' In accordance with the ·condition of constancy of the circulation over a closed contour, this change in the strength is attended by the casting off of free vortices carried away behind the wing together with the flow. The ·velocity of casting off the free vortices equals the free-stream velocity V""' while all the vortices are in the wing plnnc xOz. The axes
l!l~h;h~r:~~ev~~~~;~h~~ ~~:;eo~~~ec:e!:r~~~e~ff1t~:dv~nfF~~~ ~~8~li at the instant of shedding equals the strength of the bound vortex at this instant. Let us determine the velocity induced by a bound vortex of strength r (to) = rv cob (where r is the dimensionless magnitude
W·=r~~~ol(cosat-:··cos~)= _v:-rr. ~Dc:Sa~":"~~~~tK
(9,8.1)
where Go = xo/b, to = lo/b, The relevant value of the velocity produced by the free vortices is determined by the formula
W-=
_v;;r (~.~i:~+~~t.:.~:)
(9.8.2)
where 10 = lo/b. The total velocity induced by the bound and free vortices is W = W' + W-, [t will be more convenient in the following to
Ch. 9. Aerodynamic Characteristics in Unsteady Motion
Fig.
433
'.1.1
To tbe determination of the velocities induced by a "..ortex ~heet and free vortex filaments of varying strength: 1.
:-tr~e
vortlcell
use the dimensionless value of the velocity: /1/ (£0'
Co.
co ~ ');1...!-cn~a2
x) = W' (4:t/V ""f) -
u'" (~O. SOl x) = IV" (4;tIV,.,f) _ _
;Q filS Y.
(lTo:~:'
~, ~in Yo
_:. 1 ~~C~S;:!)
(U.8.3)
(9.8.4)
The corresponding value of the total dimensionless velocity is
w(So. So.
x)=WC'i:tIT'""I')'''''W't;OI~' %):....1/1"(;0,~. x) (9.8.5)
The values of the geometric parameters determining the induced velocities (the cosines of the angles aI' a z. Bl' ~z) depend on the coordinates of point .l/lxo (ao), <:0 (~o)l and the angle of inclination Y. of an oblique vortex (Fig. 9.8.1). These yailies are given in [4]. Let us consider an elernentaryvortexlayerofwidth dx at a distance of x from the bound vortex (Fig. 9.8.2). The linear strength of this layer at the instant to is determined by tile magnitude of the derivative '\'(x, to) = '_V:-.dr~!:l);
tl - to-t,
(9.8.6)
We shall write the strengths of the free vortex ftlaments fl (x, to) and f t (x, to) in the section x = canst as functions: f.(x. to)=fo(to
X-O.;:llI n Y.)} "051
f 2 (x, to)=fo(to- x . 'v~tnn:<)
(U.8.7)
The elementary velocity dV' induced at point .1I (x o, ~o) hy a vortex layer with a strength of dfo = Y (x, to) d.-r is determined 26-G17H
434
Pt. I. Theory. Aerodynamics 01 an Airfoil "nd " Wing
with the aid of the Biot-Savart formula (9.8.1):
dV' = _ d\'~.~~/Gl
(cos a'
~ cos aN) =
-
X~I(::) ~l;/~~:~' ~:':~:;j
By integrating with respect to £ from 0 to infinity anJ going over to dimensionless coordinates, we obtain (9.8.8)
Here the cosines of the angles a' and a" are determined from Fii. 9.8.1 as (l function of the dimensionless coordinates £0' S. the angle)t, and the relative spall of a vortex To :...: la/b. The velocities dV; and dV~ induced by elements of the free vortex filaments 1 and 2 are determined by the Biot-Savart formulas (see Fig. 9.8.1) as follows: dVi'-"O -
I" ~~r~·d.r
•
dV~= _ I'~~i:rd.{:
The geometric variables in thefie expressions are determined with. the aid of Fig. 0.8.2. For example, sin
W~
(0.5l, - ")/,,,
',= VT(x,--C;x)"~,"(0".5""'1,----',",)"
We find the values of sin ~~ and r z in a similar way. Integrating with respect to x from -O.5l o tan x and 0.5l o tan x to 00 and passing over to dimensionless geometric variables, we obtain
(9.8.9)
The total velocity induced by the bound vorlex, the vortex sheet. and the free vortices is (9.8.10) V~W'+V' 1-1';+1'; Let
liS
consider a harmonic change in the circulation 1'0 (ll V ""br sin pt
(9.8.11)
where p is the angular frequency and r is a dimensionless constant. In accordance with this Jaw, we ShAll write the circulation (R8.11) in the form y (x, lol -' -bfp cos pl1 = - V..,p*f cos (ptl) - p*il (n.R.12)
Ch. 9. Aerodynamic Characteristic~ in Un$teady Motio"
435
W(' shall reprl's('nt formulas W.H.7) for the circul:tttOllS a:; follows:
fd.r,
to)
V .. bl' sill
[JI (/0-
.r_
".~I~L'Ilr.
LI
rd.r. (0 ), l'"brsin[p(t o-·' ')_:~I,:'I""X)] lulroducillg the StrouhaJ number p* '- up V,,,
(9.8.13) wllere the l11illliS :;igll correspollils to til(' (!U8ntity 1'1' and the plus sign 10 j'". After illscrlillg the obtained \",I/u{'s of the strength'\' (,t., to) and of til£' circulations ]'1(~1 Cr, 1'1) ililo (9.8.8) nn(1 W.8.9), respectively. \\"{' lind cxpressiolls for the indllced velocities W', V', V;. and V; ill the integral form for (\ h;lrmonic (sinllsoidal) change In the circulatioll. It is not difficult to see that the total induced velocity l' IV' -- V' : r;· .. r~ call he represented ill a general form in terms of the dimen.<:ionless velocity fUlIction v (So. ~o. x, p., to):
r--, '~~r
V
(S!I,
~!I'
X,
(9.8.1-1)
P*. to)
wher!', in tnrn,
v'" Vl1)(~O' ~o. x, p*)sinpto-!·vl:.')(~tI, ~o' x, l'*)c()spl~ } t;(]):....; w' (~o. to, x)-+- p*v;' (~o' ~o, x . .fI*) :, [,."; (So· So' x, p*) (U.S.1:'!) v12 ) "~P*V2 (~, 1;0' x, p., : v'2 (~. ~, x. p*) We determiue the dimensiolllc::,s velocity w' W,oS.J), and we find I,he func~ions v;, t-';, t'~. and u~ from an of illtegrals W.8.8) and W.B.9) after introducing inlo I.hem, respectivelv, tile harmonicalJy clullIging quantities y (x, to) and rIO) (I, to). The above relations describe a sinusoidal change ill the strength of a bound vortex. If this change is cosilillsoidal, i.e.
W.8.16)
1'0 (I) -' V""bf cos pI t!ten we hilve y (,t., to)
r I (2) (x,
10) =
= bJ'p
sin pI".., 1/ ,p. sill (pt -
p*~)
F",ur CO!' Ipttl- p* (S=FO.5Io tllli x)1
(9.8.17)
Wilh H view to the above, instead of the lirsl of expressions (9.8.15) we have the general relation t;
= ~.(I) (So,
;0.
x, p*) cos pto -
[;(2)
(So. ;u, x, p*) sin /Ito (!J.S.iS)
436
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
One of the important problems in unsteRdy aerodynamics is the one on the harmonic oscillations of craft with small Strouha) numbers (p* _ 0). For such numbers u(1)
(So,
~, x,
O)=w'
°a(~:) Ip*_Q = vi! (~, ?t. x,
TV, Ip*_O + :;! 1,>*-0
0)
(9.8.19) (9.8.20)
The last derivative is used directly for calculation of unsteady now parameters. The rmding of this derivative, and also of other functions determining the dimensionless induced velocities is described in (4), vortex Model 01 • Wing
A vortex model of a wing in the form of a nat lifting surface in an unsteady circulatory flow is shown in Fig. 9.8.3. If a wing has curved edges, they are replaced approximately with a contour formed of segments of straight lines. Hence, in a most general case, a lifting surface has breaks. Let us divide the entire area of a wing into zones by drawing sections parallel to the axis Ox through the breaks in the contour. Let us further divide each segment l6 between the breaks into strips and number their boundaries from right to left (the boundary having the number p = 0 coincides with the tip of the wing, and having p = N with the longitudinal axis). Usually the values of the width of a strip lp,"_l differ only slightly from one another. Let us consider the strip between the sections p and p - 1 in any of the zone.'i 6. We shall denote the dimensionless coordinate of the leading edge in the secHon p by So. p, Ilnd of the trailing one by The relative chord of the section is
No
G.,.
s.p -
bp = bplbo = So l' Lel us assume that a strip consists of n cells. For this purpose, we shall divide each chord in the relevant section into n parts by means of points with the coordinates ' •• P '~".p~0.5b. (1-<0'7) } G. . ,p-t =So'P-I";- O.Sbp _ l ( 1-cos7)
(9.8.21)
v=O, 1, 2, ... , n The cel1s arranged along the line v separating them along the span of the wing form panels whose number is the same as that of the cells in a strip (n). We shall designate the boundaries of the panels
Ch. 9. AerodYr"lamic Charilcterislics in Unsteildy Motion
fig. U.l Vortex model or J-wlng: 2-obli(Jur
il
437
wing in rill Ilnsteady circula.tory no\\": h",.,w~h"~
'-IlIJl'ls;
hy \I {for the leadillg edge 'I' """" 0, for the tl'uiling one v = "J. We shall designate the coordinates of points at till' intersections of the lines v and p by the subscripts v and p. Hydrodynamically, the plane of the wing is eqlli\'alent to a Yortex surface that is approximately depicted by a system of bound discrete vortices. Each of sl1ch vortices evidently consists of oblillu('" horseshoe vortices adjoining one alJother, while the total number of these vortices coincides with the number of cells in whicll they are accommorlatcd. We shall characterize the bound vortex filaments by a serial number fL counted from the lIose. Let liS introdll!:e (or the sections dividing the plane of the wing along its span, ill additiOIl to the numbers p, the serial number k conn ted from a tip where \\'e a~sume that k = O. Accorrlillgly, O:::;;;;k ~ ,y, We shall denote the points of intersection of the lines fl and k by the SIII1IO subscripts, \\.'e consider here tllllt the coordinates of points on the leadillg edge ill St"!ction k and the magnitude of the chord in this section bk (pl of each lOne 6 arc known. We shall determine the posit.ion of discreLe YorLices a:o follo\\"s. We separate the chords in the sections ilnd ~"_I with points haYing the (:oordinates
;h
~p." . So.I:-:· O.5b" (l-cos 2!~n·I:1)
~Il'''-I ~c ~o.h-I -- O.;)b"_l ( 1·- cos 2~~:- I j.I ~--'
I, 2, ... ,
It
:1)
(9.8.22)
488
Pt. I. Theory. Aerodynamics of an Airfoil and 011 Wing
Poinls with identical !J.'s coincide with the ends of discrete VOl'· tic.~s. WI' hal'e the followillgrelatiolls for the dimensionless coordinates of the middles of the oblique bound vortices, tlu'!ir spall lA'''-l' and sweep illlgl('s tan x~: L,:
~::tl
05(~h
S)II, •• )-=O.5(£oh..j~h_l)
o 2;) (b" ; b,,_11(1-cos t:,~_,=-O.5(l:"
1,.J..t,I.It_I)'
2"'2:;1
1
n)
i~'h_'- I~b:-l =tJl,"-I-~)1 I,
I
(9.8.23)
tan x~: ~_I' - (~. "_I-~" ,,)l(bll, "-1- bll,") jJ.~··1. 2, ... , Il; k,...·l, 2, ... , N The coordinates of the centre of another vortex symmetriC about plane xOy 011 the port half of the craft and also the sweep angle arc as follows:
o~: t.1
=s:: ~-Io crb~: L. =. -t:: L'}
crt h . h-I "".. l h . 1,_1. crx:::t_."""-'X~:tl
(9.8.24)
where the l'ariables for the port side of the wing al'e designated by the faclor o. For a wing with rectangular edges, the sweep angles of the bound vortices arc constant (x ... : const.) , and each of t.hem transforms into an ordinary straight horseshoe vorlex. H a wing plane has breal(s in its contour, then the corresponding breaks are present in the vorLex filaments too. The circulation of obliq\le and conventional (straight) horseshoe vortices along tile spall is constant, while Iree filaments parallel to the axis Ox are shed off the ends of the bound vortices. The free filaments propagRte downstream, and when the flow is circulaton' they pass away to iurmity. In addition, when the circulation of the bOllnd vortices changes with lime, free vortices of the relevant strength are shed off tltem. The vortex model being considered is very convenient for computerized calculations of a Oow. This is due, flrst, to the suffkiently simple relations describing the disturbed Oow near a wing. and, second. to a number of important properties of the system of algebraic equations which the solution of the problem is reduced to. One of these properties is that the diagonal terms in the matrix of the equAtion coefficients play a dominating role; the solutions themselves hllve a great stability relative to the initial conditions. A significant Ieature of computerized calculations is also the fact that the usc of oblique horsesl1oe \'orUces instead of the cOllventional ones leads to suhst.antiall'limpliflcation of the calculations and to more accurate r('slilts.
Ch. 9. Aerodynamic CharacieristiC$ in Unsteady Motion
439
Calculation of Circulatory Flow
System of Equations. Let u:; JT'p['esl'lIl the dimensionless cil'clIlation of the oblique hOl'scshoe \"Orte:\: rl"~' 11_\ = r~. I;, /,_11 (1'"",bo) in the following form:
1'".,.. A-I ('1')",,"
~t [1':1, h. II-I qi (Tl ; r~i.~. II-I ql ('tl]}
q, ....... ct.
(1)."(",
(9.8.25)
(Il,,; T=l'",I-'bo,
or in the expanded form f ... , II. II-I
(T).~'~ r~, Ii. II-I a..!. r~. h. h- I ~+. r:~ II. II-I (O).'(}
+ r:~II. 11_\ wx-:-l':~ II. h-I
(I).
:..
(9.8.26)
r;~ II, II_I ~z
where r:~ Ii. A-I arc dimensionless functions not. depending on the time; the only Hme-dependent \'ariables are qi and q~. We shall assume that tbe kinematic variables change according to harmonic rl'lalions (9.1.5) and (9.1.6) which CIUI be represented in terms of t.he dimensionless Lime T as follows: qi :..: cos prT, -q'{pr sin ptT (9.8.27) wllcre qT are the amplilllfle \·a.llIes o[ the yuriables not depcnding on lhe time, pi '-' P1bo:l l "" is the StroulHd nllmber (Pi is the angular frequencr), and T '--' tV",lb o' With a view to (0.8.27). the circulation (9.8.25) is expressed as follows:
q;
qr
r ll . h. 1,_1 (-t) =
f q1 [r~i. k.
k-I
1-=-\
cos ptT
-r~~ II, Ir-l pt sin pr"t] (9.8.28)
Ily (9.8.6) 3
'\'(s~: ~_I, "t)= V"'" ~... f/Tpi {r~i. II, II-I sin (pt (T-~~: :_1) 1 j'l
(9.8.29) where S~:hlt_l is the longit.udillul coordinate or the middle of nn oblique bound vortex. The expressions fol' the circuh\lion of the rree \'orlic.es (9.8.7) have the rorm
,
r 1 (:!) (6~::- t."t) <..-= V ,,-bo.~ qi fr~: It. II-I cos {pi [T- (stt: :-1 i_I =FO.51". h-I lun
xt:: n_I))}-r~l. h, II-I pi ~in {pi /T- (st::!-1 +0.51/1> /,_\ UIIl )(~: Z-t}HI
(9.8.30)
440
Pt. I. Theory. Aerodynllmics of lin Airfoil lind II Wing
Introducing (9.8.28). (9.8.29). and (9.8.30) into (9.8.1). (9.8.8). and (9.8.9), respectively, we obtain for a certain control point
!;;;~~e~: :r~\e~~ltsh~f'~ie~7v~~~ v~nsd~,c~~ev:l:'c~:!~~o~"W~'~h~ii :;s~:: that the control points are at the centre of the lines drawn through points with identical values of '\I (except for the forward and the rear points), i.e. at an identical distance between the free vortices. Hence the coordinates of the control points (indicated by crosses in Fig. 9.8.3) are determined by the relations ~~: ~- 1 = 0.5 (fv. p + sv. P-I)'--= 0.5 (£0. I' :.. So. 1'-1)
I 1
+0.25 (bp---:-bp_il (1-cos2f!-)
(9.8.31) 0.5 G,,·; ~P_I) '\'=0,1,2, ... , 1l-1; p=O, 1,2 •.. , N The relative value of the total induced ,'cJocity at this point v = (W' + V' .-!. V; + l';)/V «> is determined by the dimensionless velocities vtl) and ",(2) that are evaluated in accordance with general relations similar to (9.8.14.) and (0.8.15). Let us designate by octl) and OV(2) the additional velocities at the control point being considered prociuce{1 by an oblique vortex located on the port half of the wing symmetrically about the plane :zOy. The coordinates of the middle of such a vortex are £~::-l and -~~::-l" The values of out I) and OI~2) are determined with the aid of (9.8.1), (9.8.8) and (9.8.9), account being taken of the change in the direction of the coordinate axes and the transfer of their origin. We shall add these values of the corresponding dimensionless velocities V(I) and ut 2) for symmetric motion (for qi = ex, w z, and ~~:
f.-I =
even functions r~~II.II-1t rt~It.II-1) and subtract them for asymmetric motion (for ql = W", and odd derivatives of the dimensionless circulation). We shall determine the total velocity at a control point with account taken of the influence of all the other vortices, i.e. calculate it by double summation of the induced velocities over the number of strips N and panels n. This velocity is determined by the values of. the derivatives of the dimensionless circulation r:~k.lI~l and
r:~II,II_1'
To find them, we must use the condition of flow without separation in accorrlance with which the total dimensionless velocity at a control point induced by the entire vortex pattern of the wing equals the undisturbed component determined by Eq. (9.6.32'): vv, I'. P_I = Vv.;~ 1'-1 = (W' +- ,"....L. V;': V;) V· =
-W",S~::_l-WzS~::_I-Ct
(9.8.32)
Ch. 9. Aerodynamic Characteristics in Unsteady Molion
441
The kinematic variables in this equation can be represented in the form of a harmonic relation: (9.8.33) By satisfying Eq. (9.8.32) with a view to (!).8.:~3), we obtain t.he following three systems of linCllr alg~brlli('~C]lInlionsfol'lh() required derivatives or the dimensionless circulation .v
-it ~
r~~".11-1
and
r~I~II.II-l:
"
~ (V~I.) ~t: I~:}- l±oV[.I.)~: I~':' t- I) r;/. /.. h_1
1I=1J.l..-1
..
.v
+ £t ~
"
r
~ (v;.~) t':~.:. t, - I =crv~.2.) ~': ~':. -I)
11-\ J.l.. I
(0.8.34).
+p~ ~ ~ (l)~l,):::~}-I ±crv;.I'):::I~~~-I)r;:i,It,)'_1 --0 1I=1J.l=1
p=O, 1, 2, ... , .IV; v=O, 1, 2, ... , n- J; k=l, 2, 3, ... , X
The quantity v~~P.P_l is dcterminecl in accordnnce with Eq. (9.8.32)
as follows:
We find the dimensionless variables in relations
(!).8.3·~)
with n vie\\' to the
V~I.)~:';;~II=V(Ll(£~:~:~-:::L ~~:~:~::' 11,.k_1> %t::t._j,pt) (21J.l. h, 11-1_
vv.
1>. 1'-1
_1I
(2)
u, II. /I-I ,.1'. I,. h-I )L. h p, p-h '-v.I'. 1'-1> h.},_t,x, •. 1,-1,
(G'v.
II) )1. II, It-I __
ov".p.P_l
(I) ("!:IL, h. II-I -1<)I,h_l.
Oll~·~)'~'::':'lh-1
,..1'. h. ii-I
"~:::.I;-t:)o.,,\.. I)'I'_I'
_1I
,-.
/"
".1,_1'
1/
'"
p,)
1(9.8.36)0
Pi
v (2 ) (i~~: :~:
;:1,
LII.,,_I> -x~: ~- I, pi)
O~~~'. ~,:
:::1, I
442
Pt. I. Theory. Aerodyn/lmin of an Airfoil and
011
Wing
Tbe dimensionless geomet.ric parameters in these relations are determined in the form
s~::::~:l
;'-t·-S~:~-t' ~~:;::~:I=C~:~-l-I;~:~-I h-I -...."h. 1,_t1bo·= !;It. /.-1 -~".I<; IIlI<
(9.8.37)
;{i::~-I=: t::=:-~::~; CJ~t::~:;::: ~~,~: :;-1 :.,~: t-t
El;Ich of the tltree systems of equations obtained includes (n - 1) N linear atgebraic eqll
d:lt..k-l
atives r~~It.Ir_' and is nN. To close the system
r!;.~.h_II,,=J.I.::":O' r!:~,h_IIJ.I_J.l.=O' q,
'" (,)." wz (D.8.38)
Proceeding from (9.8.:11.) illHI W.H.:IS), W~ !ihall compile tllree syst.ems of equations for determining the derivativt's of dimensionless circulation, taking into accouJll the ~pecilh: values of ql and The first sy~t[)m is lIl'led rOt' calcllJRting the deriviillves
q;,
r:.
lI •
It_I, r:.I,.h_l:
J.. ~ 4:l LJ
~
~~
(V(2)J.l.It,k-l. OT(2) II. I"h- I ) -I V,I'. p-I
L Io=JJ.l'--i
p.* -4i""LJ L.J h~lll
((1)).1,11.1,-1 l-\.,p"p_!
l_av(I) II.I,.k-l)r"
\·.1'.P-1
J.I,h.I,-1
v,p.p-I
1
-1\
r;,-,.I,.h-I-
_I
"" "
,y ,:~,~,(v\',):;:,~:t-'+av\'.':;::;';'c')r:"'.H p*
L
_~ (L·V.)/~:I:'-'lh-I+(JV~,I');"'·I:·~f' .1)r~,h.k_I=O
"=Iw-I
r~, h. /<- t I,( ~II. -- 0, r~. I" p
h-IIJ.l=". -= 0
',0,1, 2, "'. N; v=O, '1, 2, ._., n.-l: k-t, 2, 3, ... , N
(9.8.39) II
I
Ch. 9. Aerodynamic Characteristics in Unsteady Motion
\\'(, HIIII Lht' {I('rivatives r:~~I/.I'_1 alll' r;:~1<.h the :
"
.\"
-+,- ~ I,
~ (l:\.I,)r,:I~.:.f'-1 I"
---f ~
as a rcsult of ~olving
ot:V,)::::::~'-I) r;;~
I,. 1<_1
1
"
.\"
-I
~ (v~.~) :,'''I~':' :,-1
Ot';.2.) ::: :,'_: I' - I) 1';;',' II. II-I
11.- I
I,~
,\
/I
.Z~ '~(v~:'):":I~.:.i'-I, (:w~.2.)::"I~:.t I)r;~~"
~,_d
S
...!...p*
443
I'
I
II
1<-1 \
(9.8.40)
,
L ~: ({;~.I'):":I~'_:t"-I.i-ot:V,):.':/'·II'-I)r~,':"." .- () k __ II'=1 p=O, 1,2 .... , /I;; ,·,---,0. 1.2. ... ,11-1; 1.'--=_1, 2, :l, .... ,y
I
III £qs. m.8.:3\J) <11111 (\l.HAO). we IHlYf' cho~f'1I tIll! pill:'; sign in tllC parenthesC5 hCCI1U5C Wfl nre considerillg symmetric rnoliOlls in wllic-II the distrihution or the circulAtion O\·£ll' tllf' span is I\J~O symmetric. The thil'd syslt'm of, efjlHltions aIJo\\"~ lIS to determinn the dedva-
r;::",..
ti\'('s I, 1 1111(1 I";:~II.I'-l chnrncterizing the 11~~·mmetric Ilature -of motioll of n crllrt. Choosing t.he millll5= sign in (\J.H.34). we call wl"it(' thi~ system as s "
-b- ~
~
(0\.1,)
i:: ::: !,-I _(JC~"I.)I~" /'.:. i' -I) 1';':.' 1:.1:_ I
,,=11'=-1 .\'
--:G- L
"
~ (d"2.):,:/:~t'-I-OO~"::,I:::i:~!' I) r;~:'I'.
~-I [.1
.\"
1 /I
I;~' 1'\ (U\.2.) ,',': I~:" t' .\'
4- p* k~1
1_ clt·F) :,': ::.
.
(1J.8.41)
f' -I) r~~ .
1';;1 (~·V.)~,: ~,... ~.- 1-1Jl·V.1f': /~.:.I' I)r;':~ I,. I, _1,,"0.0 r~:;
/,,/,-1
I"
I' •
.c.
0,
r;i: II. II-I
11"-"" ·~tJ
"._0,1. 2, .... .Y; \' 0,1,2, ... , I.: 1, 2. J, '.', .y
11~1;
444
Pt. I. Theory. Aerodym'mics of an Airfoil lind a Wing
We find the dimensionless geometric variables in Eq~. (9.8.39)(9.8.41) from (9.8.37). The solution of these equations allows us to determine the derivatives of dimensionless circulation for an unsteady flow O\'er II; wing at llrbitrary Strouhnl numbers. Let us consider the problem of motion at low Strouhul /lumbers (p! -+ 0), which is of major theoretical and practical signifIcance. In accordance \vitll (9.8.19) and (9.8.20), we have V~_I.)~: l~~ 1"'_1 = V~: ~: ~::; V~~);:;~ I' - J = p'tiJv~?) t:~:. II, -l/iJP't
Introducing these expressions into lowing systems of equations:
*'
.'1
"
~ ~ (v~:~: ~:l±av~:~: ~:l)
(9.8.:~4),
r:,', I., "_I =v~~ P. p_1
II~I"-"I
r~j, I,. 11_11"""" .. =0, \'=
1, 2, ... , n-1
(V.8.42)
we obtain the fo]-
(9.8.43) p=-""' 1,
2, ... , N;
k=--,l, 2,
±a()v~~.)::: ~.: I"-I/apt) r ,,'~' /'. 1,-1
.. ,.V
II
} (9.8.44)
r~i, II.k-l 1),=".=0; p-=" 1, 2•... , N; v-I. 2. .... n-l k~I.2 ..... N Consequently, the pl'oblem on the oscillations of a wing with low Strouhal numllCrs reduces to solving six (instead of three) systems of equations but simpler ones than for arbitrary values of p~. The first system of equations (9.8.4;~) contains no derivatives
r~:k,k-l and is solved iudppendently of system (9.8.44) for the derivative r~l.k.k-l' Its value is used to calculate the right-hand side of the equations of t~e second system (9.8.44), the solution yielding the derivative r~~k."--l' The dimensionlE'ss coefficients of equation (9.8.43) and (9.8.44) are determined from the relations
(9.8.45)
Ch. 9. Aerodynamic Characierisfics in Unsteady Motion
-·,(E~::::~::' ;~:::~-::. ih-~:!.):::
!:..t-
111 .1.-19
I
illl:!)
~
up!
.
(S~:!:~-.:. a;t~:~:~,~l.
445
y.~:LI)
-x~:Ld
It,ll-I.
Let liS write Eqs. (9.8.43) and (9.S.44) with account taken of the specific yalues of qj and Q-h and also of the nature of motion. i.e. symmetric or asymmetric. The system of equations for determining the derivati . . .es r:,l.It-l has the following form: ....
11
::t ~
1,·=111-1
p-:J,2, ...• ;.\";,·=1,2 . ...• k=.1, 2 •...• X
We find the derivative r~."'"_1 equations: "
or
Z Z (tfv":;: ;:l·:-O"vr.:;: ;:1) rff, .\"
_
(9.8.46)
n-1;1
solving a second system of
.:.
k. k-I
k=II-I=1 =..;
I I
l~,I<,k_IIII-=II.--'O;
.v
)
Z (v~:;:~:l·:·a~:~:~>:b r'~.It.h_I=-1
11 (011(2)11-, A, A-I
~ ~
v,
p~Pt-l
.a
c.lJ.2)1\, h,
v,
It-I)
P8pr- 1
(9.8.47)
k=IIo1=1
;.;
r'~,
h, 1:-1
p=1. 2, .. " N; k,..-=l. 2, ... , iY
r:,II,:k-ll"'=II.=O;
,\,.=1.2, ...• n-1;
We use a third system to find the derivative r:~Jt;.Jt;-I: 1
N
4i" ~
,11,1:-1 ",,11.,11-1 0; v,p \ hn (~.JI.JI_l+(JVV.p.p_l)rll.II •• _l=-SV.P_l I
'~I .~I
r~~
II,II-dll
~II- •..:: 0;
p=1,2• .•• ,1'f; v=1. 2• ...• n-l; k=1.2, ...•
I
.vJ
(9.8.48)
446
Pt. I. Theory, AerodYnllmics or an Airfoil "nd a Wing
A fonrth system of equations is intended for cair.ulClting
r~~II.II-1:
(9.8.49)
r~: 1<, 1<_11..._,..=0;
1'.--1,2, .. " N; . . . =1, 2, .'" n-l;
k_1. 2..... N We compile a tifth system for oetermining the dedvatives r:~ k.
k
\:
(9.8.00)
A sixth system of equation!:', given below, allows r~~ Il,
II_I:
Z
.
to find
.
~ (tIv;~:~,:l-ovt:;:~:~)r~·~k,/r-1
k _I II_I j\.
=-
"(
k~1 11~1
", :~!rl
ov<2111,1<,1<-1
r:,,,,h_,I"~'I1.=O;
ilt,(2)J.I,k,It-I)
~"'j:r-I
-0
p=l,
2, ._.,
k=l,
2,
r~~It,h_1
(9.S.01)
N; v=1,2, _._, n-1;
.. , N
AerodynaMic Chllr.cterlstlcs
The val lies of the circulation derivatives r~~Jt,h _I anti r~~u-I call he llsed to calculate the di~tribution of the pressure coefficiellts, and according to them, the wing slability derivatives. Here we calcula.te the difference of the pressure coefficients for the bottom and upper sides in the form of the series (9,(j.~2):
en, 9. AerodynamiC CnaracterisliCl in Unste<'ldy Motion
447
In uccortlance \\·ith Zhukovsky's formula (!J.1.15), we Ilan'
JP:....·2y, ,;,.\. '''.. 2"Y~.";
~
,21';";
{JUl .•
pfJI: .. 2yfJI;,
,2"\",".
p,~: .., 2~'
(11.8.33)
Let liS consider fI cell containing II point with the coordinates £i.:t-l' ~:Ll Jor which we slUlll calcldate the pressure. We determine the strength of fI bound vOl'tex at Llli~ point by the appropriate value of the circulation related to the interval Letween tlw control points ~~1l,1<,1<-1 ~~::~;-1 - ~~:-::~"I in the s(,(,tioll ~:::~-l:
Let us wrile the strength of the vortex layer and the circulation of a discrete vortex in tim (orm of the series W.i.'12) ltnd Hl.8.25):
Introducing these exprcssioli!> into (0.8 ...)1), we oblaill relations for the uerivatives
'--- r~i, II, I, _ J' :-"s~"
h, k-l
(\1.8.56)
With account taken of these deri\"atiw:::, the expressions for the streng~h of the vortices arc determined by the quantities P~:h.""l and P~:II,II-1 (9.8.53). Let us introduce them into formulas (9.H.5) in which is replaccd wilh series (0.8';)2), the vltluc of XII is taken equal to bo and x' to the local chord b h . h -l:
dp
(!l.8.;')7}
448
Pt. J. Theory. Aerodynamics of en Airfoil and a Wing
Let us replace the coefficients on the left-hand side of (3.8.57) with their expressions in the form of series similar to (9.6.8): ,
Cy
=
.
~ (C;'1iql-f c;'liq,) 1-' 3
•
3
•
~ = i~1 (m~qlq,+m;q'ql)
I l
I
(9.8.58)
m~= ~ (m~qlql'"l"'m~llql) i"",1
A3 an example, let us write the right-hand side of (9.8.58) for the normal force coefficient in the expanded form:
Two other coefficients, m; and m~, are expressed similarly. As a result of the indicat.ed replacement of the coefficients cY' m~. and m~ in (9.8.57) with series and of a transition from integrals to sums, we obtain the derivatives of the aerodynamic coefficients of the sections in the following form: 11;o1l"
Cy~lh.JI-1=~ ~ k,1I.-1
r!I,JI.k_1
11=J.14
C;~IIt.'Il_1=~Il~" r!I,Il,k_1 h,ll-t
(9.8.60)
11=110
(9.8.61)
(9.8.62)
We obtain the stability derivatives fot a wing as a whole by using the following relations for the total aerodynamic characteri&-
Ch. 9. Aerodyni!lmic Chi!lri!lcteristic;s in Unstei!ldy Motion
449
ties of a wing: 1.J(2b o)
C.,= ~~; lTI:
~ c~b(C)dC
t./(-:b.l
'J
J
~-; s:
mx= ~.~bo
m;b:!(~)d~
lol(Zb g )
~.
(9.8.631
c;b(gd~
\\'e can express the coefficienls cu' m" and m" for a wing in tlte form of series (9.6.8), and their correspond ing values c~, m;, and m.~ for the sections in the form of en.S.S8), After using the deriv;..tives (9.8.60)-(0.8.62), we obtain the stability deriv
ci
mil
N
c: 1.: L l
•
=
c: = 1
11,,-1 X
1l=1l. ill. 1I-1
~ r~i. II,
JJ,=Il. l1~il • •
;'_1
(9.8.65)
~~! ~ 1/1, II_I ~ r~J, II, 1,=\
I'~jlo
(9.8.66)
For asymmetric motioos (qj=w,,):
C;i=C~I=O
(9.8.67)
m~l=m~'=O
(9.8.68)
(9.8.69)
450
Pt. L
Th~ory.
Aerodynamics of lin Airfoil and. Wing
Let us write these expressions in the expanded form with account and also of the numtaken of the corresponding values of q, and bers of the bound vortices ).1.0 = 1 and Il. = n closest to the leading and trailing edges. For symmetric motions (q/ = a, CI):):
q"
.
~
m.~ =m~ .=m~~ =m:~=O
(9.8.70)
(9.8.71)
(9.8.72)
•
N
th,II_I
m~: _ ~b! ~ /,,-1
n.
~ r~, It. A-l~: t
w,
4b3 ~ =-s;;; LJ
.
h;1
"~I
11=-1
",=1
171:'=
ih
, 1<-1
..;,
t,""
I
(9.8.73)
J
",-1
mz
-
.10-1
h
It, 11-1'0,"" It-I
1 t
.
(9.8.74)
~~ ~ III. h-l ~ r~: It, II-IS~: Z, h-l J
For asymmetric motions (gj = 6):<) (9.8.75) n!
N
m~~= ~b! ~ lit,
11_1
A=I
~£
m",
4b~
N
=-s;:;- L: Ii-'I
L:
r~~ II. II_I~~: ~_I
).1=1
_
h.h-j
n~.1:
(9.8.76) k
h r).l,II.II-lt).l.It-t ).I.
" ... :1
The to\.al values of the aerodynamic coefiicients can be determined according to the found stability derivatives, using (9.6.8).
Cn. 9. Aerodynamic CnllrillcierisliC$ in Un51eilldy Motion
4~t
Delorm.tlon of • Wing Surface
The above numerical method of aerodynamic calculations related to a wing with an undeformed (rigid) surface. In real conditions, however, such a surface may deform because of bending or deflection of the control surfaces. The flow over a wing with account taken of its deformation is considered in 14, 19]. The magnitude of the deformation can be expressed in t.he form ~
(" C, .)
= 10
(., C) 6 (.)
(9.8.77)
where 10 (E, ~) is a time-independent function determining the changing form of the surface, and 6 h) is a time-dependent kinematic variable characterizing the scale of the deformation. In addition to 91 (q1 = a, 92 = w;e, 93 = Wz ), we can also adopt the dimensionless time function q4 = 6 h) as a variable determining the unsteady flow of a deformable wing. The right-hand side of expression (9.8.26) determining the circulation will
additional~y
include
the sum
r~.It.It_16 + d.It.It-16
Iwhere the derivative 6 = (d6Idt) boIV",,]. This will lead to a corresponding change in the aerodynamic coefficients, the general expression for which can be written in the form of a series
c
=
c«a
+ J~ + cW.<w", + c";.< ~'" + c'·"w z + c;':~z + c06 + cig; c = I1p,
cv' m",. mz (9.8.78) The deformation of a wing changes the boundary condition of flow which instead of <9.6.32') acquires the form
~= -a('t)=w:t:(-t)~.-wz('t)~
+ ~,' 6 (.)+10 (;, 0 i(.)
(9.8.7!l)
where (8/0/8"£) 6 ('t) + h (6, ~) ~ h) = uJV"" is tbe dimensionless vertical component of the velocity of an undisturbed fiow due to deformation of the wing and damped in unsteady [Jow by the disturbed stream.
WiLh a view t.o (9.8.79), we shall write a system of algebraic equations for determining the circulation derivatives ~i,It.It_l and
r7,~It.1t -1 which are then used to lind the derivatives of the aerodynamic coefficients, and their local and total .... alues with a view to the deformation of the aircraft. The results of solving this problem are given in detail in 14, 19].
pt. I. Theory. Aerodynamic5 01 an Airfoil and a Wing
4ei2
Influence of Compressibility (the Humber .tI 00 ) on HOon-Statlonary Flow
In deftnilc conditions, the cvaluqi = bfpl8q, and CPq, = O'f'/aq, in th£' following form (see 14, 19»:
q; ~ ~,-o.
±
'-I
l
(9.8.80)
where ~o is the velocity potential of stationary flow over a wing with a finite airfoil thickness at a zero angle of attack. Let us introduce the functions 'P. 1jlQI, and 1jlQ'/ related to the velocity potential and its derivatives by the following expressions:
If:; ='¢qic~SWG~.P:k-21fq;sinWG
} (9.8.81)
",q, =k-2"'''lcosw~-p1-1'1j)ql.sinw; Cfo='tJ,lo, w=M:"k- 2pt
where k = Vi - M:l,. After the corresponding substitution for the derivatives in (9.8.80) of their values from (9.8.8t) and then introducing cp into (9.6.31), we obtain the following three equations: kZ
a;i~Q t- a;~~Q
=0
a2w~; _I. a~2 .
821l'Q;
k2
a7;:i
a~~:i + a~!:i +(pr)2M:'k-~ql = 0
_
-1-
"TI,
+~, a~2 T
+ a~o
k'l
. .
(p')2M ' k-2.""1 i
...
't'
=0
I
(9.8.82)
(9.8.83)
Ch. 9. Aerodvnamic Characteristics in Unsteady Motion
4fl3
Considering one of the ba::;ic ca.o.;(>s or Do\\' that is characterized by low Strouhal numbers (pi ~ 0), inste;'1I1 of Eq (0.8.83) we obtain (').8.84)
Let us consider a rigid lifting sHriace. Instead of the flow of a compressible fluid over it, we sha11 study the flow of an incompressible fluid over the transforme(\ surface. The coordinates of points in the incompressible flow are related to the coordinates of the space of the compressible gas hy the expressions XIC ,...,.
xlk,
YIC
= y,
Zic
=
(9.8.85)
Z
in the dimensionless form tiC =
Ile/bo.le =
£.
Tllc
~IC -= Zie/bo· Ic
= '=
Ylcfbo,lC
= 11k,
(9.8.86)
i;Je
Let us change the pl;Ulform of the given lifting surface. nsing the relations between the coordinates in the form of (9.S.S6). In accordance with these relations. the dimenSionless geomotric variables of the transformed surface can be written as follows' (U.S.87)
where edge;
£0
and ~a are the dimensionless coordinates of the leading
blc = bIClbo.lc. b = blb o, blc = b/k For the centre and running chords in (9.8.87), we have introduced the symbols bo and b, respectively. For wings with straight edges, the planform is usually set by dimenSionless parameters, namely. the aspect ratio Aw = lo:'bm, the taper ratio 11,,· bolb l • and the sweep angle of the leadillg edge la. The corresponding dimensionless geometric vari(lbles of the transformed wing in an incompressible Oow are: =0
Alc•w = /cAw. 11IC,"'" = 11"" Ale ....- tan Xo,le = Aw tan
xo
<0.8.88)
where Ale,lV = l!cibm,lc, Aw = lJbm, 11lc,w = bO,IC/bl.ICI and Ilw = = bolb l • Figure 9.8.4 shows the given and transformed wing planes. The transformed plane retains its lateral dimensions and is extended in a longitudinal direction in accordance with the relation $Ie = ~
xi]ll - Me..
4154
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
(a)
FI.9.U
Geometric dimensions of a wing: a-given \\"Ing In a CQmpresslble now; b-tranMormpd wing In an IncompreSSible now
Let us go over in Eqs. (9.8.82) and the first equation (9.8.84) to the new variables Sic. 1"]lc. and Sic related to S, 1"], and S by expressions (9.8.86). As a result, we obtain
(9.8.90) Let us now consider the motion of the transformed lifting surface in an incompressible fluid. Let us write the disturbing potential of the flow over the surface as follows:
,
cDlc=k
.
[Illo + t~l (/lI~~' IC q;, Ic+
(9.8.91)
where $,c is a dimensionless quantity equal to the potential related to the product bo, Ie V "". Everywhere in the disturbed region of the flow O\'er the surface, the potential must satisfy the continuity equation (9.8.92) Introducing (9.8.91) into this equation, we obtain equations which the derivatives of the potential of the non-stationary incom-
Ch. 9. Aerodynillmic Chillfillcter;s!io in Unste~d)' Mot;on
4ee
pressible flow over the transformed wing mllst satisfy: 8~o;tlrq. it
,j2¢'I"
ie
,j~f~ T~ Q2;it _~ q2:~t + Ie
ie
~o;
•
i)"::~L Ie
1
II,
(9.8.9:1)
= 0
By comparing Eqs. (9.8.8{1) and (Y.8.Y2), and also (9.8.90) and (9.8.93), we find that the functions "1'0. "1''11. and ""ril determining the wlocity potential for a wing in a compressible Dow, and the functions ¢Ih $·t~·ic. and
(~, 11. ~)=¢~~,te(Ste, 11t·;. ~Icl
1jJ'ii, Ie (S. 1),
6) =
(1)~i. Ie (~Ic. lll e • ;,c)
}
(9.8.94)
where ~=~Ie.
t)=lllcfk,
bO-=b!Jk
Consequently, for finding the potential function and its derivatives for a compressible flow over a given lifting surface, we must solve the equivalent problem on the unsteady incompressible flow over a transformed surface with the corresponding boundary condi· lions, The formulas obtained allow us to directly calculate the relevant values for the given surface in a compressible flui(\ accord· ing to the found aerodynamic characteristics in an incompressible flow for the transformed surface. Particularly, the formulas relating the differences of the prcssllfe coefficients ~md their derivatives have the following form: tlPo=tlPO.lc1k.
pqi
,-,""p~ci.
IC/"}
p'li = p;ld' lelk =SlcM!,p~1.IClk3
We use these data to fmd the diffcrcllcc of the pressure coefrlcients at the point of the surface being considered in a compressible flow:
(9.8.96) Taking (9.8.96) into account, we can obtain the aerodynamic coefficients of a wing in a compressible flow. while by nsing (9.8.84)
4fi6
Pt.
r.
Theory. Aerodynllmics 01 en Airfoil end II Wing
we can establish the relation between the stability derivatives of this wing and also of the transformed wing in an incompressible fluid:
C~i=C~i.IClk Y Y. ic • mq'=mqi,lelk2 x x, Ie ' m:' =
C;'-=(c,ii.,C-.U2mQi.,e)/k'l V y, Ie "" ~,Ie 2 I )/k' m;' =- (m~i,le ,:....M x· x • "".:Ii:
m~:'l~elk. m~; =~ (m::'I~e
where [.:Ii: =.
~~, IcIe S~.,\ ~ic p;~' ic~le;le d~ie d~tc w,
,
J = 2110 ,ie z·
S .... Ie
,
j' jr p'li,IC£2 de. dr ie Ic .Ic
~Ic
s\\", Ic
j
(9.8.97)
-M!,!z)lk 3
I
(9.8.98)
I
I
The following formulas for the coeIfLcients of the normal force and pitching moment for ql = ell = 0 correspond to the relation I1p~ = I1Po"e1k between the values of the difference of the pressure coefficients in a compressible and incompressible flows: (9.8.99) With a view to the stability derivatives (9.8.97) and the values (9.8.99), the total normal force and pitching moment coefficients or a wing in a compressible flow have the following form: (9.8.100)
Welcan write the corresponding rolling moment coefficient m. in the form of the third of expressions (9.6.8). 9.9. Unsteady Supersonic Flow over a Wing
Let us llnd the solution of the problem of an unsteady supersonic flow over a wing in the form of the velocity potential q:l (Xl' Yl' Zl> t) of a transient (pulsating) source (9.7.7) satisfying the wave equation (9.6.30). Let us consider the expression for such a potential for harmonic oscillations of a wing when the kinematic parameters vary in accord-
Ch. 9. Aerodynamic Characteristics in Unsteady Motion
4'57
ance with the general relation q,
=
q; exp Opt)
(j
=.0
cr, l'J.~,
fJ)1)
W.n.!}
It follows from (9.6.32) that the h011ndary conriiliOlls 011 a wing
also vary harmonically. The derivative of the velocity potential on a wing (9.7.8) (let liS call it the now Jown\\"ash) om he expressed by the harmonic function (0.\J.2)
The required disturballcc potf'ntial is related to the downwashes.
~~rJ~~~~ul~i~~'~~~~~~: (~.~~~~ti:: f~l'~~~~: t~
are determined in ac-
: =t/21 =/~\~l~ ~ ~/:,~::!'~i ~jJ~~'(aa]'2)}
~~ ~
(9.9.3)
After introducing (9,9.2) and (g,9.3) into (9,7.7) for a wing (assuming that y -'.co 0), we obtain Ip(XJ, Yl' ZjI
,I
rr JJ
t)=, _ _ Hee iP1 ~
eI(%"
[.:;;t(;t.,.1/. Z)] pO vy
to)
Xe- jP(.~'~~:'::"'" (eiP ":C!.""! e-;P .,,,,-:~,,.1 dJ/= Taking into account that the sum in paTell theses on the righthand side equals 2 cos (pr,a",a'Z), aIHI the (]lIantilies p* = pbo/V""
(I)
= p*Jl;,'-x"1
(9.9.4)
we obtain the foHowing expression for tile velocity potentia!: fP (xt. X
Yll %1'
t) = -
rr ['. au (x,
JJ a(x"
-k He q$e ]
y, z) y=l
iN
- ,....:.'cos -,) ('"') d, d, "oJlx> - , -
(8.Y.5)
:,)
whel'e r = V (Xl x)Z a'Z [y~ -;- (ZI Z)2J. With a view to the harmonic uature of llloLion of the wing, wecan write the potential function as follo'..-~: fP(x. y, Z,
t)=V""bO.~(Ql"jqJ ;=,1
= V""bo He eil'!
.
L
j~l
l-rp1j9J)
.
q; lip"} (x, y. z) -;-- ip*Ql"j (x, y, %)1
(9.9.6;
4ea
PI. !. Theory. Aerodynemic5 01 en Airfoil and ... Wing
where qJ -= qje iP !,
1j '""" dqJldr:= (dqj/dt) b,/V"" =
ipjqje iP !
(9.9.7)
In accordance with (9.9.6). the partial derivative of the potential function with respect to y is a'l'
Ty (I, y,
Z,
. t) = V""ho Re e'p!
h3
_[ flIp'll
qj --ay(x, y, z)
;=1
+ip*
&:!I~j
(x, y, z)J
(9.9.8)
Inserting this ucrivatiye into the left~hand siue of (9.9.2), we .obtain the following relation for [OIP (x, y. z)/oyly=o: [ ;: (x, y.
+ip* 8:U'" (x, y,
Z)]
(9.9.9)
Introducing this expression into (9.9.5). substituting (9.9.6) for the velocity potential on the left~hand side, and taking into account that e - ;"'~,-l;) = cos CIl (:t"l -x) _ i sin W (Xl-X) b~
bo
obtain relations for the derivatives of the potential function:
XCOSW(.1'I;:.1'),p*[ uti (x.
y. z)Jv=o sinw (:t";,;.1')}
x Cos (
y, z)
= -
~
j J {[~ (x, G(..-,. z,)
ay
, W(:el-:e) t [Off"l ( XCOS-b-.--p; ~ x, y, z
)J
b:~""
) d.1'rd Y
y. z) JII=O
.
IIl(II-.1')} JI_oSlO-b-.-
X cos b:;.. d.1'rd, .
(9.9.10)
Cn. 9. Aerodynamic Cnaraderi~tic~ in Unsleady Motion
4~9
For ullsteady motion wit.h small Stroulwl numbers (p. -- 0, w _ 0). Eqs. (9.9.10) acq\lire the form :t
'P qj (x, y, z)=
!I
I
{.I', y, %)111
j (!J.n.10')
_..!... ~ ~ [iJ~'lj
cpqJ(;r, y, z)"'"
~ (/.
-*}) {[ _ M; (:, -Ii ¢'"b o
ii::)
(.c, y,
[}!
dX,d'
)] y=u 'rr d, }
I , y, %
I I
Let liS introduce the rlimcn~ionless coordinates (9.6.19), adopt.ing the centre chord (XII = bo) as the charact.eristic geometric dimension: S = x'bo, TJ = yJb o• ; = zlbn (9.9.H) Considering tbat to determine the aerod:\tnamic loads it is sufficient to find the value of the potential function on the wing (y = 0), let us express (!J.fl.W') in terms of the dimensionless coordinates as follows: q>
'1(,
"
0 t) '_1
~
--'-jjcr [£] (il]
:1
ll"{'
d,_d; r
(H.U.12)
.
We can use the found derivathes q.qj and \foi, to determine the dimensional valuc of thc velocity potential: fP(Sl' 0,
~d = VOI>bo.-~ IIfQj(SI' O. ~I)
.
qj..!-q,q)
1--1
Equations (9.9.12) allow
liS
(s\, (), ~tlqjl
(H.U.13)
to find the velocity potential at
fi:i~~~r~~~:n~~~10;\I~)e \:.~ '~~t~~~l t~~~e n~~irdo~~~~~~!sSd~~~~~~~~dl~; the derivatives [Drp'/j /811],1_0 tlnd [acp'jIDTjI ..=o. The values of these dowllwashes must satisf)' the boundarr condition (9.6.32) on the wing {l} = 0) in accordance with which we have
a~ala'l ~ -1, a.p./8tj - 0 18.., = -~1' aq.-=181l'= 0 a«lw:1811 = -SI' 8~I/DTj = 0 {l(pW_<
(j ~ I)
(9.9.14) (9.9.15) (9.9.16)
480
Pt. I. Theory. Aerodynamics 01 lin Airfoil lind eWing
In addition, the values of the downwashes being sought must corrcspond to the value of the potcntial function on a vortex sheet (9.6.34). In theoretical investigations, (9.6.34) is replac.e
.
Jl ['1"1(0, 0, Cl q, (')'eq>"'(s, . 0, "q,(.)[ =
±
['1'"
i""l
(s',
0,
"q'(")e-cpo, (s',
0,
"q,(.,)]
(9.9.18)
where by (0.9.1), we have gJ (t) = qje iPt , qj ("C"') ,,-, qje;P{l-(X*-X)}VooJ;
qj ('t) = iqjpe iPt 9} ("C*) = igj p*eiPlt-(Z.-Z)/Vool
After introducing the values of qj and their derivatives into (9.9.18), we obtain (without the summation sign)
"·Hp'q>" (s, 0, "
=e-jp·(~·'-~lCfQj(~"', D, ~)+ip*e-tP.(i·-~lcp;J(~*, O,~) (9.9.19) Since e-tp*(~*-b=,cos {p'" (s*-E») - i sin (p" (,*-t)J or at low Strouhal numbers (p* _ 0) e-ip*U*-U=
1-ip* (~*-"
after introduction into (9.9.19) and separating the real and imaginary parts, we lind the following expressions for the derivatives on the vortex sheet (excluding the term with p.!):
'1'" (s, 0, ,) =
(9.9.20)
Beyond the limits of the disturballce cones (a Mach cone or wave), the supersonic flow remains undisturbed, hence the potential function is constant, Therefore, the additional disturbance potontial
Ch. 9. Aerodynemic Characteristics in Unsteady Motion
461
!1~1~:J designations of the coordinate systelll$: Q-connonlional U. O.
~);
b-lr3Mrormrd
(1.1'
o.
tt'; '. ~-C'hlraC'terjslic coortllnal"-B
in this region, including t.he Mach wave, equals zero, Le. ~ (.. ~, " t) - 0 (9.9.21) Simultaneously el"erywhere in plane xO, except Cor t.he wing and the wake vortex behind it, the condition is sat.isfled that q> (<. 0, ., t) - 0 (9.9.22) according to which the function lP is continuous and odd relative to the coordinate Y ('11). Let us consider Eqs. (9.9.12). Their usc for numerical calculations is hindered by that if the <,quality ;1 - ; = ±a' (~l - ~) holds, the integrands have a singularity of the order of ,-1/2. It is therefore necessary to transform Eqs. (9.9.12) to a form convenient for snch calculations. To do this, we shall introduce a new system of coordinates (Fig. 9.9.1.): 6t - 'Ia', ~t -~, ., - • (9.9.23) The wing in the new system of coordinates is shown in Fig. 9.9.1. from which it can be seen that the lifting surface has been made shorter, while tho transverse dimensions have been kept unchanged. 'It. and ~I Equations (9.9.12) transformed to the coordinates have the following form:
et.
Cl'QJ(SI.t.
O. tl.d=--i-~ \" [ ••.~qJ] _, d;~dtt ••atJ "I 111Tt (9.9.24)
462
Pt. I. Theory. Aerodynamic$ of an Airfoil lind II Wing
To simplify the solution of Eqs. (9.9.24), we shall introduce the following relations for the derivatives of the function q> used for small Strouhal numbers (p* -+ 0):
~'I(s" 0, ,,) - F'I (s" 0, ,,)
q>~J(£1>
0,
S!)=~J (Sh
0. t;,,)-
M;~t ~J(Sh
} 0, SI)
(9.9.25)
After inserting the value of CPqj (9.9.25) into the first of Eqs.(9.9.24). we obtain an integral equation for determining the derivative pqj:
F"i(t
_t, I,
0, )__ -'-rr[.!!:.!...] JJ ,
1, t
8'1t
"
11t=O
d',dC, rt
(9.9.26)
" Let us introduce (9,9.25) into the second of Eqs. (9.9.24): rJ (SI, h 0,
~t. 1)-
M!,;,t.
t
pqJ(£I,
It
0,
~I,,)
M!,(St',t- St )
[.!:L!...]
Ct
81)1
} dS!dtt Tlt-O
rt
Substituting for the derivative Fl J on the left-hand side of the equation its value from (9.9.26) and having in view that by (9.9.25)
""] "I-O=-= [""] [~ ~ T1t=o we obtain an expression for finding the derivative Fqj:
F~J (t_1,
to
0') = • .",1.
t
_..!.. f r [~] d~!r,d~t IE J J 8Tl' Tlt~O
(9 9 27)
..
"
We perform our further transformations in characteristic coordinates (Fig. 9.9.1). namely,
+"
r - Is, - s",,') - ,,, s - (S, - s"",) (9.9.28) selected so thal only positive values of the variables will be used in the calculations. The value of ~o.o.t representing the displacement of the characteristic system of coordinates is determined as shown in Fig. 9.9.1 (the coordinate lines from the displaced apex 0' pass through breaks of the wing at the tips).
Ch. 9. Aerodynamic Characteristic$ in Unsteady Motion
463
Fig. 9.9.1
To the Dumerical calculation of a wing with combined leading and supersonic trailiDg edges in an unsteady Row: I . JI-resions 101 fmdlnG: HlP down.
wa~ h
Equations (9.9.26) and (9.9.27) acquire the following form in thecharacteristic coordinates:
;, F) (rjt
&F~) i i~'[.~
1 "..
0, $j) =-
-z;:
(9.9.29) ] 11,=0
drtls V(r1-r) ($I-S)
For further transformations allowing us to eliminate the singularity in Eqs. (9.9.29), we shall introduce the variables
v = Vr
1 -
T,
U
=
VSl -
S
(0.9.30)
Equations (9.9.29) acquire the following form in the variables (9.9.30)
(0.Y.31)
where (9.0.32) To perform numerical calculations, we shall divide the region. occupied by the wing into separate cells as shown in Fig. 9.9.2.
464
Pt. l. Theory. Aerodyn
II
Wing
Let h stand for the width of a cell in characteristic axes. This quantit~' is the dimensionless distance along the r tor s) axis between the edges of a cell for points n, n + 1 (or i, i + 1), respectively. It is equal to hi ~ 1I(2Nb,) (9.9.33) where N is the number of parts into which the wing half-span l/2 is divided. Letrandsbe the coordinates of a fixed point (point P1 in Fig. 9.9.2) at which we are finding the velocity potential (or its derivatives), and m and i be the coordinates of the running points used to perform Dumerical integration. Accordingly,
= rh,
rl
'I
=
$h, r
= mh, ,
= ih
(9.9.34)
Assuming that the functions D J and E J [see (9.9.32)1 within a cell are variable, let us write Eqs. (9.9.31) in terms of the sum of the integrals over all the cells:
f'l(r" 0, ")~ -~ •
1"
(r" 0,
"')~
;:-1._1 'Vtm..-I)II'V(i+l)1I.
2}:B
)
J
m..,Oi-O
'VUiii
'Viii
D,dvdu (9.9.35)
';-t;-I'V<m+I)iI.'V(i+t)h
-f :B:3 J
1II=01-0}f"iiii
J JIiii
E,dvdu
The integration regions including the projection of the wing onto the plane T) = 0, the vortex sheet, and also the sections of the disturbed now outside the wing and vortex sheet are replaced with a sufficiently large number of whole cells producing a conditional wing with toothed edges (Fig. 9.9.2). A cell should be considered to belong to a section of the integration region if its centre is on this section. Inside a cell with a sufficiently amaH area, the derivatives Dl and E J can be assumed to be constant and equal to their values at the centre of a cell. Accordingly, Eqs. (9.9.35) can be written in a simpler form: Jill
(rh $h) = t
-~ ~
±
Dj • nt. iB;,
m,
i, i
m-=li=1
;, (rh, 'h) ~
-~
±±
E;. ,•. ,8;. m. i.;
m=I{=ot
(9.9.36)
Ch, 9, Aerodynlmic Chlrlciefistics in Unsteady Motion
where D; .•. ,=D,I(m-1/2)".
(i-1/2)hJ
E; .•".,=E,((~-1/2)h. (i::-1I2)"J __- - !~ B; .•. ;., =(V r-m-i- I-Vr-m)(V ,-;-1·-1 s- I)
I
488
(!I.!l.:17)
Formulas (9,9,36) allow us to determine the values of F"J and
pqj and, cOllsequently, the derivatives of the velocity potential (see expressions (9,9.25)1 if within the entire region of source infiuence (including the wing plane, vortex sheet, and the disturbed flow outside of them) we know the quantities D},m .• and E J•m•1 (the flow downwashes), and also the values oC B;,m,";.i (the kernel functions). The dowilwashcs arc determined differently for the following three regions: (1) the wing, (2) the disturbed region confined by the leading edges. tips. and surface of the Mach cones issuing from the corresponding points of the wing (cn(l regions). and (3) the vortex sheet, Let us consider the dowllwashes in each of thl'sc regions, having in v iew the case of small Stroullal numbers (p* _ 0). For a wing with supersonic (or combined) Jeading and supersonic trailing edges. it is sufficient to know the downwashes on the lifting surface (section I in Fig. 9,9.2). They are determined by boundary conditions (9.9.14)-(9,9.16). in accordance with which
d,'.:,=-I(j=I);
I
Ij,'.:,=h(m-i)/2 (j=2)} '.' (j=3)
d,'.~,= -~'h (m;.i-I)/2-~""
EWlli=M~(h(m+i-1)/2-i-~o.o,d/a.'
(j=1)
E\'~i={M~lh(m-i-i -1)/2·+-£,. '. ,]/(2<%')) "h (m-I) (j = 2)
il)'.l..= -M:" Ih (m+i-I)/2·l-!•. '. d'
(9.9.38)
(9.9.39)
(j=3)
Section II (Fig. 9.9.2) between the characteristic axes, the leading edge and the Mach lines issuing from the vertex of the wing is occupied br an undisturbed flow region; hence, all the downwashes equal zero and, consequently.
(9.9.40) Let us determine thE! now downwashes in disturbed region III between the leading edge, the tip, and the Mach cone issning from the points of a break in the wing contour (Fig. 9,9,3). The velocity potential in t.his region (outside of the wing and vortex wake behind
466
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
fig.....J
To the numerical calculation ot
a wing with subsonic edges in an unsteady now: III,
IV-~,Ions
down~.. n
for
IIndlng
thf
h
it) in accordance with condition (9.9.22) is zero. Therefore, from (9.9.29). we obtain the conditions for region Ill:
0)
(9.9.41)
01 These integrals can be written in a somewhat diDerent form:
[ ', J -" - '- r J D,--' -~O· I"""" . / - 1E, .~ ~O • o V't- r VII-I
0
0
j'-'-' - rD,-'-'-~O· Y'l-' 0
o
YSI- a
•
,,11 - 8
yr1-r
0
r./' r (,
r '1-'
'0
E, ./' r
St-I
~O
(9.9.42)
(9.9.42')
The obtained equalities are Abel equations whose right·hand sides identically equal zero. Hence it follo\...·s that the inner integrals for $ = $1 IEqs. (9.9.42)] and for r = Tl IEqs. (9.9.42')1 equal zero. Accordingly. we obtain the following equations for determining the down washes D~lIl) in region III (Fig. 9.9.3): "
I D~IlI)
.,-h ~,
\'
~,:~
D~I1I)
ds
=_
"-~
~
VSt- 1
ii
dr
V't- r
= _
1 0
! I
DJ
ds
VII-I
r,-h
Dj
ds V'I-'
I
(9.9.43)
Ch. 9. Aerodynemic Characteristics in Unsteady Motion
467
The
Assuming the values of the down washes in the cells to be constant and going over from integrals to sums, we find in accordance with the variables (9.9.30) the following relations for determining the downwashes in a cell (point) with the coordinates r 1 = rh and $1 =
=sh:
D~lIJ) (rh, sh) = ~lll) (rh, sh):-:- -
:;:-1 ~I D j I(m-O.5) h,
(i-O.5)")B;.ml
;::-1
L;
E j [(m-O.il) h,
m.,...\
(for points to the left of the axis
(i-0.5) "IB;.
m
sd;
.1-1
D~IlI)(rh, sh)= -i~1 D J [(m-O.5)h,
.-1 E
E}lIl) (rh, -;;h) ~ - i~1
j
[(m-0.5) h,
('-0.5)hIB; .• (i-0.5)"IB, .•
s,).
I
I
I
(\1.0.45)
(9.9.46)
(for points to the right of the axis In Eqs. (9.9,45) and (9.9.46), we have introduced the notation B;::,
"1=
Vr-m-i-1- V;: -m,
B;, {
=
Vs--m -;-..1- V:;- i (\}.9.47)
According to the above relations, the flow downwash at a point is determined by summation of the downwash values in all the cells in the corresponding strip m = const (or i = const). The calculations are performed consecutively beginning from the cell at the vertex of the wing where the down wash is determined by formulas (9.9.38) and (9.9.39). Let us now calculate the downwashes on the vortex sheet behind the wing (region IV in Fig. 9.9.3). We shall lind them from condition (9.9.20), which we shall write in the transformed coordinates q>qj(St, 0, sd=-"rp"J(st, 0, ~n ' . rpqj (St, 0, ~t)='P~j 0, ~D-(l'(St-·5tl'P~J
m',
} (0.9.1,8)
Pt, I. Theory. Aerodyn~mics of ~n Airfoil and II Wing
468
where ;: and ~r. = ~ are the transformed coordinates of points on the trailing edge. Relations (9.9.48) can be represented with a view to (9.9.25) and the characteristic coordinates (9.9.28) as follows:
~';(r"
0, s,)
~F'I(
~
-
~"
";", ";-") r*·!-s* $*-r* 1(-,-2-' -2-)
(r,f-$,) f"1 ( ";-,, , ";-"
.of ex' [ r\-:,sl-;(r*-:-s*)
1
q
]F l( q
r*~-~*,
)
I
(9,9.49)
$*;r.)
Thestl relations hold for the conditions (9.9.50)
where $* and r* are characteristic coordinates of points on the trailing edge. In addition to (9.9.49), we can use Eqs. (9.9.25) for the vortex sheet, which with account taken of (9.9.26) and (9.9.27) have the form
(9.9.51)
Replacing the derivatiYes 'l'qj and CP1J in (9.9.5f) with their relevant values from (9.9.49). we obtain integral equations for determining downwashes on a vortex sheet (region IV in Fig. 9.9.3): -
-
2~
rr
1~
j'",I
o 0
(l
DJ -:r~d~';;;d'~= l/(rl-r) (sl-S)
(Ej...,.,,~d~';;;d'~=c
~
l"(rl- r )
(SI-' S)
(9.9.52)
Cn. 9. Aerodynillmic ChiliTilicteristiC5 in Unsteilldy Motion
469
Both equations are solved in the same way. Let U5 consider as an example the first of Eqs. (9.9}i2). We shall isolate on section IV a cell with the sides h and the characteristic coordinates $1 = sh and r 1 = rh. We shall write the left~hand integral of the eqnation as the sum of four integrals:
',I'D, '0 U
+
d,d,
_.''('
7]-h
'l-h
Il
i 1"::__ i
d,
~)
Ylrl-r)(sl-s)
'I
Y~I-~
'l-h
V,
v<'-, + i 0
'1
V:t
J
T
S
'l-h
~
~l
Dj
d,
rl - r
'I
1":"_, i
Dj
7]-/1
"1 _$
''I'D, Y
v:"-,
l/~r_r
(9.9.53)
q-h
As we have already indicated, the entire integration region should be divided into cells and the tlownwashes assumed to be constant in each of them. We determine the coordinates of the cells by means of the integers m, i, rand (Fig. 9.D.3). Csing (0.9.30), we go over to the variables v and u: V Z = (, _ m) h, u~ = ($" - i) It !n.9.54) The first of the integrals (9.9.53) has the following form: 'l-h 'l-h Vii Vi;
s
y~s_$ ) ~-I
DJ
Y(i-rn)h
~,,~ 1
y::-r
=..
o
.1
;:_1
du
Y(;-m+lJh
~ 1
)_
2du
j
.\
D, dv
1""trir_rn)h ~
2D J dv
Y
'::--1 ;-1
(9.9.55) ;=1
"'~I
where B r.m .•.• is an influence function determined by the third formula (9.9.37). For the second and third integrals on the right~hand side of (9.9.53)~ we obtain the relations:
(9.9.56)
(9.9.57)
470
Pt. I. Theory, Aerodynamics of an Airfoil lind
whel'e
Br,m.;,.=~~~::~~~;~~:
II
Wing
B;,;,;,
i
(9.9.58)
For the fourth integral (9.9.53). we have the formula
r
d.
't
d,
-
sl-h l~ )-h Dj~=4hD/(rh,
-
(9.9.59)
sh)
By summing all the above values of the integrals, from (9.9.52) we obtain relations for numerical calculations of the down washes in a cell \vilh the coordinates rand ;(Fig. 9.9.3): ;-1 ;_1
If,I\') (rh,
sh)=
-ih Fqj {r*h, s*h)- ~
~ DJ(mh, ih)Br..1I • 7, i (9.9.60)
E~!\') (rh,
+ cr.;h
sh) =
--iT,- [iT, (~*h. _s*h)
(r-i s_r*,_s*)F rt , (r*h. s*h)]-
~
±EJ (mh,ih)Br.1II .... t
111=1 "=1
(9.9.61) We perform numerical integration by double summation over all the cells within the confines of an inverted Mach cone issuing from the point being considered (rh, sh) excluding the initial cell (at the vertex) at which the downwash is determined. Its magnitUde is found consecutively beginning from the cell on the trailing edge (for the starboard side) and i* (for with the smallest coordinate the port Side). The coordinatea of points on the trailing edge are related to the coordinates by the expreS5ioD9
r'"
r. s
(9.9.02)
The calculations are performed as follows. The smallest values of the coordinates r* and s* are used to determine the nerivatives FIJj (r*h, s*h). pi) (r*h, s*h) for the first cell, next the above rela· tions are used to calculate the down washes and innllence funetions for the entire region within the Mach cone issuing from the adjacent cell (with the coordinates rand 8). After this, double summation of the products, of the down washes and the innuence functions is performed, and the required vaJues of the downwashes at a point with the coordinates 'r and"8 are found by formulas (9.9.60) and (9.9.61). Passing along a strip = ronst (S = con.sl) to the following cell with the coordinates ((8 f) h.,ri.
r
+
Ch. 9. Aerodynamic Characteristics in Unsteady Motion
471
we repeat the calculations and find the corresponding value of the dowm.... ash in this cell. By performing similar calculations, the dowilwashes can be found on the entire vortex sheet behind the wing. The values of the downwashes are used to determine the derivatives f"li and pij at each point of the lifting surface. Xext q:.'Ij, CPqj, and the corresponding values of the velocity potential cp are found (9.9.13). This quantity directly determines the pressure coefficient and the difference between its values on the bottom and upper sides of the wing [sec formulas (9.6.20) and (9.6.21)1. The difference of the pressure coeHicients (9.6.21) can be written as the series (9.6.7). The derivatives p'l} and pqJ in it arc determined in terms of the values of F'II and Fqi; by using (9.6.24) and (9.6.25), we obtain the following relations:
~) =
_
p~i (~, ~) =
-
p'l I (1;,
+M!,a'
+-
~~~' ~tl
'3.\ {-a.'FQj(~tt ~t)
[~ iJF(;l~(;t, M";,
Q
af i
(/;a
;11 _ C
(9.9.63)
aF'Ij~;t, ~t)]} a.;t
-I
Hence, to find the derivatives of the difference of the pressure coeHicients pljj and pqj determining the value of /lp (see (9.6.2t)l, it is necessary to calculate the derivatives OF'll/iJ'St and fJFqjlfJ~t. There are different ways of calculating these derivatives. In aerodynamic calculations, particularly, the method o[ differences is used, in accordance witll which
":::i ~ '':;~J +
[p'lj
(r+ 1, s,
i)-i'i (r. 5>1
(9.9.64)
The derivative fJFqj/fJ;, is calculated in a similar \....al'. As a rewe can write the following relations for the values of p'lj and
s~lt,
pqj:
p(IJ(~, ~)~-~[F'I}(r-:-1, $-;' l)·-P'lj (r, $)1
pqj(~, ~)= :~a {_a.'Fflj(r, s)
+":;;- lp7 (r-i-l, j
;~-1)-Fqj(~,
(9.9.65) s)]
PI. l. Theory. Aerodyn~mics of !In Air/oil ~nd
472
!II
Wing
To improve the accuracy of calculating !he derivatives (9.9.65), it is necessary that the values of
pill
and J7lj be found hI more cells
than the number in which p'j) and pqj are found. Hence, in this case the interval 6.~t can be increased m times (where m is an integer). Accordingly, the derivative
iJ:S:'
~h
=
IF'll (r+m, s+m)_pllj
(r,
s)]
(9.9.68)
We determine the derivative arJ/ost i.n a similar way. With a view to the new "alues of ap1j/oSt and aF'j/OSt, we change formulas (9.9.65) for the derivatives p1) and p';j. We use these derivatives to find by meanf' of (9.6.9) the derivatives of the lift force, rolling moment, and pitching moment coefficients. As a result of substitutions of (9.9.6::1) into (9.6.9), where we assume that Xli = boo we obtain relations for these derivatives that can be written as follows:
,•
'. ~
•
i. +M'_C~2.t ~ }
aCg=cV.ha~y=c~l.t.
(9.D.67)
a'c~:=c:.:h a'3c;Z = c:l, t+~_c:l.t (9.9.68)
(9.9.69) The following symbols have been used in these formulas 11<2bo)
~t
-1/{2bo)
'D. t
c~. t = 4b~').'"
.~
•
) ~Q.
c~:!. I = 4b5~~2A\\' [:~ '"
J
~t
J
-1I(20Q) lo. t
'tl
ds i • dbt
it
) -//(21)0)
•
;J~;t.
iJF
11(2&0)
C;l,t= - 4b~~l).".
1/(2bo)
Q•
j
/{(200)
j
r(St. bt)dstdbt t
it
\
.:. iJF(1,
(9.9.70)
;i:' ~t) d~t d~t
-1/(200) fu. t
iJF"-~t ~t) ~i dS i dbt]
Ch. 9. Aerodynllmic Chllrllcteristics in Unstelldy Motion
C;~t= 4b~~:A".
't
If(2bn)
~
OFWZ,!t,
)
47$
~!l d~t d~t
-1.'(2b,,) ~o, t
•
c;l. t =
-
4b~~~ZAw
"
1/(2bo)~!
J
{o,
~
F Z (~l' ;d dS t d~t
. 1/(2b,,) ~u. t
c;i.
4b~~~2A\\" [;~
t=
(9.9.71) ~i
I
I!(:lb u)
.~
iJF(":~tl' ~t) ~d~1 d~t
-1/(2b,,) ~o
1/(2bul~: )
",
)'
(IF
t
za~~' ~) Sd~dCJ
-1/(2boY.to,t
m;, t = 4b~~:2A\\"
;t
1,'(2b o)
j
.~
aF'l ~t ~t) ~t dSI dS t
-1/(2b o)t o.t •
mft. t =
--
4b~~~3Aw
ti
1/(2b ol
i J Fa (~t. sd ~t dS dst l
-1/{2b,,) ~o. t •
m~2, t =
-
1{(2bo)
j
4b~~~'A". [;~
tt
.:.
J
oF
a
(9.9.72)
~~:. ~t) ~I d s! d~l
-l!( 2b o) to, t 1/{2bo)
,i
J J
iJFabt·
t:t)
1/(2bu)
't
(0'
)
)
s~d;t d~t]
-1/( 2b o) to, t
4bS~:ZA'"
m;'\ =
of z~i:' 'I) £1 dS t dSt
-11(2b o) ~o, t •
m~'{. t =
-
4b~~~8A'"
it
1/(2bo)
J F Wz (Sh stl St d~t dSt
j
-If(2b o) ~o, t •
m;'~z.t=- 4b~~'Aw [;~
l/(2b o)
.
~t
J .J
(lFWzii:'
(9.9.73)
~t)Sld5tdSt
-1/(2bo) '0.1 1/(2bu)
tt
til
) J of Zi)~:' ~t)
-1/( 2bo) i o, t
s~dStdSt]
474
Pi. I. Theorv. Aerodvnamics or an Air/oil and a Wing 1I(2bo)
~t
-1I(2b o)
i o. t
m;~1 = _ 4bii~~2}.w
rn:t
t
=
,~
et
1/(2b~)
4bij~:aA."
~
~
-1I(2b,,) •
m;{ = _ 4b3~:3A.".
(0)
aF
)
1:.
1/(2bn)
~t
i o• t
~:~. St) \;t ds t dS t (9.9.74)
(St. 1;d St dS l ds,
t
1I(2b o);t
)
j
-1!(2bo)
i o. t
[.~~
-1I(2bo)
FWx
x
:Q
fJF xa~':'
tt>
St
d£t dt t
~,
1 J" x.~:. ~.) t.'. d•• dt.]
(9.9.75)
It must be taken into consideration that the velocity potential cOn the leading edge is zero. Consequently. (9.9.76)
Accordingly,
it )
q
aF
Ja~tlo 'I)
ds, = Ft , (st. ~t)
~~. t
i~
J
" of
J~i:·~t) StdSt=stF'Ij(st,
St)
!io.t
.t
-J
F"J (St. i;;d
dS t
lo.t
(9.9.77)
'J~
ilF'I J (st>
~t)
,.,.. A: _,. t;-F" (t.
--.;-,-'ot5tu'et-~tSt
lo.t
., -
\ F'J (S" ~o: t
t.) t. dS.
.)
'=" ...t
Ch. 9. Aerodyni!lmic Chi!lri!lcteristics in Unstei!ldy Motion
..
) of'' i ~~:'
471i
~I) sf d~t .= ~t~F'I j (st, ttl
,:
lo.t
-
) Stp"J (s" ~t) d61
to.t
In a similar way, we shall writ.e the relevant. expressions related to the derivatives pili. Let us go over in (9.9.70H9.9.75) to the characteristic coordinates sand r (9.9.28). Assuming that the velocity potential (or the [unction D) is const.ant in the cells and replacing the integrals with sums, with a view to (9.9.77) we obtain relations for numerical calculations of the derivatives of the aerodynamic coefficients. These relations, used for finding by (9.9'(i7) t.he correspondin.g derivatives of the aerodynamic coefficients of t.he lift. force ~. ~, C~I, and also for determining from (9.9.68) the derivatives of
c::,
the pitching moment coefficients sented by the relations: C:. t = 4bg~'"",,. h
h
m:;, mi, m~·;. m~;:
FrJ. Cr*,
can be repre-
s*)
;. C:t, t = -
4bg~~~;.w
liZ ~
~ pi" (T, $)
I
s
~
4I>!a"),.. h Cy,2t=--,,-
{a' ",.,i -, M!, f.C'. ~r,
- ~ [(;:'+S*)hI2+s.", .• i F"f',
r.
-') S
(9.9.78)
[
s') +h ~ ~ F"(r, s)} s ,
C;,'t = 4b3~';"w h ~ p"'z (r*.
s*)
;.
c;t, c:i.
t
= -
t
=
4bB~~~l.w
h'1.
~ hPj,), (r, "$) s
4bB~~11...
h {;:
~
"" ;:-
i'z (r*. $*)
t
(9.9,79)
476
PI. I. Theory. Aerodynsmics of an Airfoil and sWing
,. +h~~F"'(;:, SI}
,
m~. 1 = 4b~~~%}'w h {h «r* +8*) h/2 -:-~. o. tJ Fo. cr*, $*)
"-h~,~r(;, S)} m:I,t= - 4b~~:'AW h'l ~,~ [(r~s)hJ2-:-£o.o. t1 r(r, "8) ;
_
~hJo.'JAw h { +
m,2.t- - - , , -
a'
M:7.,
'"
1-* +s-"') h/2
~ (r
(9.9.80)
"
- ~ ((r* + 5*) h/2+~. o. t]2 F~(r", s*)}
,.
The aerodynamic derivatives of the rolling moment coefficient are calculated by the following formulas:
m::t= m~f.
t
=
m~:Il = x2, t
4b~;S).w h
h rls.-r*)h!2j F'"x(r*.
;*)
" 4b~~:3)...... h h h ({s-r) hJ21 F\i):Il (r, S) Z
_
4.f;~o.'3"W I~
,
h {~ 'V [(s*-r*) h:2J M~ LJ .'
,.
F~X(r* • :;*)
- h [($*- r-*) 1I/2J [(r* + ;-*) h/2 +~. o. tJ F"'x (T*, 'i-h ~ ~ s
l(s-,)h!2J FO, (r',
,O)}
s*)
(9.9.81)
Ch. 9. Aerodynamic Ch~racteristic5 in Unsteady Motion
477
If we take the wing span l as the eharacteristic dimension when calculating the rolling moment coefrlcient, and the half-span lI2 when determining the kinematic parameter {J)z, then the new values of the rolling moment coefficient and the corresponding derlvatives are determined by the relations
mz1 = Inzboll. m~fl = 2m~:t:: ; (boll)s (9.9.82) By analogy with (9.9.69). we shall give the following working formulas for the deriva~i"('s:
..
.
a.'m~l' = rn~;:lh a.'zm;~·1 = (m~et)l ,-- M!.. (rn:f.lth m~et....,...· - 8a.'Aw (boll)~ II ~
(9.9.83)
,(s·- r*) h/2)l"·-.: (r*, s*)
(m;r,lth = -16a.'Aw (boll)~ h {:~ ~ Us* -r·) 1112J
(9.9.84)
"
,.
+!o .•. tI F"'(i",
"h
h ~~ 1(,-r)"'2J F"'(;:, ')}
s In the above relations, the symbol the cells along the trailing edge, and
~
signifies summation o\'er
th;~symboL S~. double sum-
s malion o\'er all the cells on the lifting surface. When performing calculations of the total aerodynamic characteristics. one must have in view that these calculations are carried out in the system of coordinate axes depicted in Fig. 9.0.1 (with the rule of signs indicated in this figure). All the geometric quantities are measured and the pressure coenicients are found in the system of coordinates whose axis 0% passes through the vertex of the wing (see Fig. 9.11. t). When the mean aerodynamic chord bo is taken as the characteristic dimension, the aerodynamic coefficients and derh'atives are found by converting the value.~ of c", mz, mz, and their derivatives by the following formulas:
Cfl.A~-'C!l' C~.A=C:t c~.A=c;bolbA c:.z'AA. = c:'zbolb A• C;.zAA = c;z (bofbA.)2
}
(9.9.85)
478
Pt.
r.
Theory. Aerodynamics of lin Airfoil lind II Wing
mz,A=mzbolbA•
m~'A=. mzbolb m~,\=m~(bolbA)2 A.,
}
(9.9.86)
m;'zA,A =m:z (bolbA}Z·. m;'zAA = m;z (bolbA) m:t. A =m:tbolbA. m:;AA=m:t(bolbA)'l', m::AA=m::t (br/bA)3 (9.9.87)
In the above expressions, it is assumed that aA=(daldt)bAIV"",
wz.A=QzbAIVco
~z, A = (dQzldt) b).IV!.. ~:t,
A=
(dQ.)dt) b~/V!'
} (9.9.88)
9.tO. Properfles of Aerodynamk Derivatives
Let us consider the general properties of the aerodynamic derivatives as applied to finite-span wings with a constant leading and trailing-edge sweep. Investigations show that at low Strouhal numbers (p* -+ 0) the value of /I derivative is a function of three arguments: kAw = AW V 1 - Mi." tan Xo, and 111l" These arguments, called similarity criteria, are determined, as can be seen, by the aspect ratio All'. the leading-edge sweep angle Xo, the taper ratio 11",. and the Mach number M co. For an incompressible fluid, there are two such arguments: Aw tan :>:'0 and 11,," If a rectangular wing is being considered, it is necessary to assume that Aw tan XO = 0 and fill' = 1; for a triangular wing. flw = 00. Diagrams characterizing the dependence of the stability derivatives on the similarity criteria for wings with a taper ratio of 11w = 2 in a subsonic flow are shown in Figs. 9.10.1-9.10.10 as illustrations. The mean aerodynamic chord bA has been taken as the characteristic dimension when_ calculating the data in these figures (the kinematic variables ;"A, W:.A, c:,z.A, the coefficient mt.A)' This makes the dependence of the aerodynamic derh'atives on the geometric parameters of the wing more stable, the results approaching the relevant quantities for rectangular wings. When CAlculating the rolling moment m:tl, we chose the wing span l as the characteristic geometric dimension, while for the kinematic variables W.xl and ~Xl we took the half-span l/2. The origin of coordinates is on the axis of symmetry, while the axis Oz passes through the beginning of the mean aerodynamic chord bA • When nece5sary. w(' can convert the obtained (lerivatives to another characteristic dimension and to a new position of the axis Oz.
Ch. 9. Aerodynllmic Chllrllcterisiics in Unstelldy Motion
Fig. Uo.t
~~ea:~~: i~al~buela~~~n~t: s~t'it ity derivative c~ for a wing in a subsonic Dow
Fig. '.to.l Change in the quantity kC:,z'AA detennining the derivative
~~;i~~t~~~
Fig. '.to.l
a wing at subsonic
.
.
valuesofyc:~At and k Sc:6.z ~g:rd~:r;~~i~~ o~h:he c~o~~~~ie~~ 'with respect to oX..%. !for a lifting surface at M"" < 1.
clI
479
-480
Pt. I. Theory. Aerodynamics of .an Airfoil "nd " Wing
'flg.9.1G.A
k't:'
k't;,t.
ValUE>9 of ,z:..1 and A determining the derivative of Cy, A with respect to (l)t, A for a !!Ubsooic now over a wing
'flg.9.to.S -Change in the variable kmx~3.:1 determining the derivative of the rolling moment coefficient with respect to (l)xl for a wing at M",,
fig. 9.tO.•
Values
of
k2(m~fl)2 change in coefficient for It wing
k'(m;f1h
and
characterizing the the rolling moment with re.!lpect to ~l:l in a subsonic flow
Ch. 9. Aerodynamic Characteristics in Unsteady Motion
481
r-'--'-- ------~?y
FI'II.9.tU Function km~ A determining the derivative m~ A for a wing in a subsonic flow
-1.2
f--;---"-
/(m~~,'-----'-----'------'------'---""
Flt· 9.tU
valuesofk3m~AAl d~termining
the
and k'm~AA2 dC!rivative
m~AA for a wing at subsonic veiocities
FI'II.""o.t Values of km;.zAA determining the derivative m~~:...A {or a wing atMoo <1.
I
'---'-~
Pt. J. Theory. Aerodynllmics of lin Airfoil lind
482
II
Wir"l9
0.',--,--",_
" Fig. '.tUG
Variables
k3m~lA~
k3m~~~:\'}
anc!
characterizing
the
change in the derivative m~ZA'\ for a wing at J/", < 1
-{J.2
~~F::/p"'~
-o.'1I--f""~-=.J, -O.6L---'-"'--'--~_-'---_-"
According to the data shown in Figs. 9.10.1-9.10.10, the aerodynamic deri"fltives are determined as follows:
..
.
(kc~)/k, C~,AA = [(k3C:;'-AI) + M~ (k3c:;'-A!!}JIk3
(9.10.1)
c:::...'\ = (kc~.A A)lk, c;~.Z:...A = f(k3c;'.~t)+ M~ (k3c~~Al)J/k3
(9.10.2)
m~ln = (km~I·)/k,
m:r =
f(k::m~fl)l + ~'JI!. (k!m~~t}Zl!k2
(9.10.3)
m~.A = (km~. A)/k.
m:.AA =
[(k3m~1.1) ~
(9.1u.4)
m~:A..A = (km~·AA)/k. m~ZA A =
[(k3m;'ZA1)
c; =
.
.
.
M!,
(k3m~,AA2)l/kJ
+ .U:'" (k3m~ZA~)J/k3
(9.10.5)
We can nse the graphs (Figs. 9.10,1-0.10.10) to find the stability derivatives for the entire range of suhsonic numbers M """ including an unsteady incompre~sible flow (ld "" = 0, k = 1). The data of these graphs point to the increa.se in the lifting capacity of the wings when going over from an incompressihle (iJl = 0) to a compressible fluid (Moo < 1). Here the stability derivatives at M"" < 1 can be calculated sufiiciently acclIr(\tcly according to the PrandtlGlauert compressibility rule c = Clc (1 - M;,,)-1/2, where c and CIC are aerodynamic yariables in a compressible and incompressible fluid, re~per.ti\'ely. The lower the aspect ratio, the smalJer is the action of the compressihility, i.e. the smaller is the influence of the number M"", < 1. This is riue to the fact that the disturbances introduced into a flo\\" attenuate with a decrease in the cross-sectional dimensions of a wing. i.e. \\'ith a (lC!crcase in the aspect ratio. Xear the tips of a wing in a slIhsonic flow, the air flows from the bottom side where the pressure i!'l- higher to the upper one. This leads 00
Ch. 9. Aerodynamic Characteristics in Unsteady Motion
483
to lowering of the lifting capacity of the wing. The intensity of such ov('rnow grows with a decrea~e ill the aspeet ratio of a wing; consequently, an increase in this aspect ratio improyes the lilting capacity. An' analysis of the illfillence of tile leading-edge ~w,·eep flngh.' "1.0 shows that it.s growth leads to an appreciflble riimillislling of the d('rivative c~ for a wing wit-h a high aspect rfllio. At small aspcct ratios.. the influence of the angl(' Zo is virtually absent hecause in this case nen an illsignilicant change in "1..0 leayes the planform of t.ilt' wing almost ullchanged. A change ill thc lapC'r ratio II,," at ~iven Yalues of AIV and 1.0 has H slight effect Oll the lifting I':aparities, especially for low aspect ratio.!; of a wing. Tlw {'xplana\ion is that. fit Moo < 1 most of the lift force is protillc('cI by the lifting surface near the leading edges, which is I!~pcciall,.. noticeable with low aspect ratios. A lar~e "aIne of the tap('!" ratio II\\" (at 'i." 1':01l.<:t) corre:.<ponds, as it w('re, to cutting off a p~H·t of the poorly liflillg taU surface of the wing. All anlllysis of tht, fOHlld v
484
Pt. I. Theory. Aerodynamics of.,n Airfoil end ., Wir.g
FIg. ••to.U Value.!! of a;'c~ determining the
c:
stability derivative for a wing in ~a 8Uper.sOniC now
~9."O.U
Change in the quantity Cf.'c:,!"AA determining the derivative c~~\ A for a wing at su· personic velocities
Flt,9.tO.U Relations
a:flc;~A2
for
a:'3 c:'AA1
and
charac~ri~ing
the change in the derivative of t,he coefficient ell with ~spect to fora wing atM", > 1
aA
_ _ _ _---=C"'h.-'-'."A:::'",o:::!dynamic Chllraeleri$tics in Unsteociy Motion
485
",---,-/r""
'.'f--7i"--";-f----r'+----P-e..-
0.6
Fig. 9.10.14
Values
a'3 ..<:':.-ii\
of
nnd
a'3c;.Z:\.~ determining the deriv. ~.tive of
cy
",:ith respect to ~l"\
In:1 s.upefSOD1C
now
O\'er
<1
winlf
..
Fig. 9.10.15
functiun (L''''.~';''' rilarad(Orizlug till' rlLangt· in the dcrhative, of the rolling- momt'nt \'oe)Jlci~nt with re~p('('l 10 w.Y\ for I
a.'m:?'
Flg.9.f0.t6
Values of ',/.'2
a'2(m~'I\]h
and
(m;jqh characterizing the
change in the ucrivathc of the rolling moment coefficient with
~~~~~~nl~ 81;~ ror 11 wing in a
-O.1J'
486
Pt. I. Thllory. Aerodynamic5 of an Airfoil and a Wing
Flg.9.tO.t7 Variable a'"t~.A determining the derivative m~A for a wing atMoo > 1
-J r---t---'<::6"""'--t----j
a'mtA,
., ., Fig. 9. to. t8
Valu~s of a/3m~~AI
11
m.~t.
for
It
a
5
wing in a
'Ilf
.5'V"_hlrl.X:
°l'l..-:~' ' _ ....
\
r_
'II
and
a'3m ~\2determining the deriv-
ative
L"1. ,
a'h;~
.0
I,w"
l.........
J.
,l",km4~0 4 IJ I
-
I
I
•
-0.5
"
SUpeI'llonic flow
10
,I~
, .s:-. ,•
Fig.9.to.t9
6
~~~g d::l~a~I';:.1 m:.~AA
1
for a
.
""~
IS
a'A""
a:.\ w
'1",~2
,If--
J
FUnctio~ a'~~zAA (j)dewrmining
~,
a,Jml,t ... 10
Aw/aJlX.-O
•
'\ '\
./
Ch. 9. Aerodyn&mic CharecterisliC5 in Unsteady MoHon
487
fig. 9. to.10
Varia~1(!5 ':\: 3m;.'~~ and
.
a'3m~:;;:...'.t characterizing change in the derivative in a supersonic !low
the
m~:zA"\ -1.~
mined with the aid of formulas (9.10.1H~UO.5) in \...-hich the quantity k = V 1 - M;" is replaced with ct' = VJroo - 1. The graphs allow us to arrive at the conclusion that the lifting capacities of wings at MOD> 1 grow with an increase in the aspect ratio. although to a smaller extent than at subsonic velocities. Particularly, for a rectangular wing (XO = 0), the increase in ~ wben M">O > 1 occurs only up to Aw = 3-4, and then the derivative ~ remains virtually c·ollstant. For triangular wings. the constancy of t1, is. observed beginning from ct'}.", ;:e, 4, Le. when the leading edgi-s transform from subsonic into supersonic ones. The derivatives of longitudinal and lateral damping (m~.·AA and m~"\ 1. But they grow in magnitude at supersonic flow near values of Moo> 1. For these velocitics, as when Moo < 1, a sharp change in the magnitude of the non-stationary derivatiycs is also attended by a change in their signs. At large numbers Moo (Moo;:e, 2.5-3.0), the aerodynamic derivatives with dots decrease, tending to zero at Moo _ cx>. The influence of the planform on their values is not so great at the usual velocities (sllb~ or supersonic), but becomes considerably greater in the transonic region where the number M;>o equals unity or is somewhat largel" than it.
.f.B8
Pt. I. Theory. Aerodynamics of an Airfoil and a Wing
9.11. Approzimate Methocls for Determining the Non-Stationary Aerodynamic CharaderlsHes
~~~::::a:::" Harmonicity
The harmonicity of oscillations, which is the basis of the met.hod of calculating non-stationary aerodynamic characteristics set out above, reflects only a partial, idealized scheme of unsteady flow over a craft.. In the general form, such flow may be characterized by other time dependences of the kinematic parameters. At present, methods have been developed that can be used for unsteady flows described by any functions. But owing to their great intricacy, it is not always possible to use them in practice. It is therefore necessary to have Jess strict and sufficiently simple methods of calculating the total and local nOll-stationary aerodynamic characteristics. One such method is based on what is called the hypothesis of harmonicity (sec (191). By this hypotheSis, an aerodynamic coefficient for unsteady flow is represented by series (9.6.8~ in which the stability derivatives c'lJ, rfii, and others are functions of the mean angle of attack "0' the Mach, Reynolds, and Btrouhal numbers for a given wing. In the Hnear theory of flow of an ideal fluid, such a relation is limited to the Mach and Stronhal numbers. According to this theory, the aerodynamic derivatives depend very slightly on the Strouhal number for low-aspect-ratio wings of any planform at all numbers Moo when the latter aro large. We can assume that in a first approximation this is also true for any other time dependence of the kinematic variables. But no\\" with any function describing wing motion, the stability derivativ('s can be takfln from the results of calculations or experiments obtained for a barmonic change in the kinematic variables, and the values o[ these variables can be determined on the basis of their true time dependences for the given mode of motion. This is the essence of the harmonicity hypothesis. It allows us to obtain more accurate results when the stability derivatives depend less on the Strouhal number. When solving problems on the flight stability of cralt, t.he Strouhal numbers p* quite rarely exceed values of 0.05-0.07. Consequently, ill practice, we may use the non-stationary characteristics for p* -+ 0 with sufficient accuracy and, therefore, proceed in our calculations from the harmonicity hypothesis. The harmonicity hypothesis allows us to obtain more accurate results if the change in the kinematic variables can be represented by a smaller number of terms in the Fourier series corresponding to a narrower spectrum of frequencies characterizing this change.
Ch. 9. Aerodynamic Characteristics In Unsteedy Motion
489-
When studying non-stationary now oyer craft, researchers use rather widely what is known as the steadiness hypothesis. The content of this hypothesis consists in that all the aerodynamic characteristics are determined only by instantaneolls values of the angle of attack. The local values of the pressure coefficients nre the same as in stationary Dow at an angle of attack equal to its true value at the given instant. According to this hypothesis used in the linear theory, the aerodynamic coefficient is
c,
(t) ~
Cll"
(t)
(9.11.1)
where ex (t) is the true angle of attack at the giwm instant. anll c; is the stability derh'ative ill steady now. If a craft rotates about its lateral axis at the angular velocit)· ilz, the expanded hypothesis of steadiness is used according to which c,(t) ~
<':"
(t)
+ C:'w, (t)
(0.11.2)
Experimental investigations show that the hypothesis of harmonicUy yields more correct results than that of steadineS5. This is especially noticeable for wings with low aspect ratios. An incre-ase in the aspect ratios is attended by Jllrger errors because at large values of Aw the dependence of I.he aerodynamic derivatives on the Strouhal number becomcs more appreciable. This is not taken into account in the approximate methods.
'.......w_ . . . .
According to this method, in a stationary supersonic Iin(>al"ized Dow over a thin sharp-nosed airfoil, the pressure coefficient at a certain point is determined according to the local angle of inclination of a tangent to the airfoil contour ex - ~=-- (for the upper ~ide). and a. - ~J. (for the bottom side), Le. the corresponding yalue of this coefficient is the same as that of a local tangent surface of a wedge. By formulas (7.5.20) and (7.5.20'), in which we assume that ~N = ~ on the bottom and ~1. = -~ on the upper side of tht' uirfoil, at the corresponding points we have ;;b-2("+~)Il'M~-I. p.~-2("-~)!VD1~-·1
(~.11.3)
where ~ is the local angle of inclination of the contour calculated with a view to the sign fol' the upper side of the airfoil. With account taken of these data, the differencl' of the pressure coefficients on the bottom an(1 upper sides is
!ip = where ex' = VM!. - 1.
Pb -
Pu =
4a./ex'
(9.11.4)
490
Pt. I. Theory.....1I,ocIyn.. mics of lin .... irfoil lind II Wing
tJ.p can also be applied to non-stationary fiow if instead of a we adopt the value of the local summary angle of attack determined from boundary conditioll (9.6.32) in the form It is presumed that formula (9.11.4) for
as = -v/V ... = a + 00••:;; - oo;X (9.11.5) where; = x/b o, "i = 2zll, and 00:1:1 = O:l,U(2V ... ) [with a view to the chosen coordinate system, the sign of OO~X has been changed in -comparison with (9.6.32)1. Let us write tJ.p in the form of a series:
tJ.p =
+ p«lXlOO:l:l + p«l~ooz
pGa.
(9.11.6)
Substituting for tJ.p in (9.11.6) its value from (9.11.4), but using from (9.11.5) instead of a, we find the aerodynamic derivatives:
al:
po. = 4/a',
=
p«lXI
4Z1a', pf>lz
= -tU/a'
(9.11.7)
According to the tangent-wedge method. for all wings we have (9.11.8)
Knowing the quantity (9.11.7), we can find the stability derivatives:
c~=
1~ j j tpd'iii;
Awlb o
o
c;~~
rI?-
'.v.;bo J l~'dxdz
x~
m A t ij.
mlCfJ =
__ -
--f- j j pmJ,'lzdzdz
(9.11.9)
Oi"o
A6
m~='~
ri"r pGrdzdz.; -- - '"
J
).6