Aerodynamics

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. 0O

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 ! 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 ~ PI~l (1-\'o)d. < < 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 Ph .=-: /JOC.b on the bottom side- is (".~pouding \',ducs of !he }-,'(q",,/,) :1ud cx" . Xa'(q",).)·

r" .-

Intro("\ioll of the shock and the characteristic. Downstream. cHrving of the shock is due to its intt'l"

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,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.< = 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
(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

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 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


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.'-~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;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('. 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 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

(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~ 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~~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 overlween 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
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 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 AI= 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 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;, but also on the t (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. 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

tal.c hehind lhc wiug mHl Ihe dl'ag force, Particulclor~' for \,ol·\1ident being c.on:
(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

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.tigation. we ~hall wrile the aerodynamic coefficients ill the form c_~=c",(,Uo 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 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:, . . .

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 (iJed 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 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 *-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-
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) :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~: 1s 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

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 ( "'(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) ( 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

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:: :'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",, 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---' 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~"\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

mz~=~

ll·i'l~ p"'--zxdxdz. o~o

OiQ

We locate the origin of the coordinates at the nose of the centre chord of a wing (Fig. 9.11.1). We adopt the centre chord bo as the characteristic dimension for m~ and OOz. and the span land halfspan u2, respectively. for m X1 and 00:.: 1, For such a wing. the equations of the leading and trailing edges in the dimensionless form are all follows: x_·_~ (t)u.+t)71",otan'l.. :Z }

__

0 - 110 -

(9.11.10)

4'1U'

XI=1!-= -1 + (l1;~t _

IJ~;:-t '-.,ot:nX·)i

cn. 9. Aerodynllmk Ct-lIrllcleri$tics in Un$leady Motion

491

fig. '.H.t Diagram of a wing ill the calculation or stability rlerintiVl's accord in\.! to the mcthorl 0[' i(lcul w~'dgcs

Let us write the ratio of the ccntrp chord to the span and the aspect rat io of a wing: b,ll~

2'lw'['w (~w -i- 1)1,

Aw ~ 1',8 - 2~wl1l(~w+ 1) b,l (9.11.11)

The tungl'nt of the sweep angle along the trailing edge is tan Xl = tan 'l.o -

(2b o") (1 - 1hl",)

Introducing (9.11.10) and (9.11.7) Into m.11.9). stability derivatives of a wing:

+

tl:....+,~~"'~3 (

i.\\

t~1I '/ n ) ~J

(9.11.12) W('

obtain the

(9.11.13)

III the part.icular C(lH' of rectangu lar wings for which tan Xo = 0 and 1'jw = 1, we ha .... e

(9.11.16) m~f' = -2/(3a'}

(9.11.17)

m~~= -2la', m~·= -4/(3et'}

(9.11.18)

492

PI. I. Theory. Aerodynamics of an Airfoil method makes it possible to determine only the aerorlynnmic derivatives without dots. The results obtained coincide with the accurate solutions according to the linear theory for inl'lniLc-span rectangnlar wings, and also for triangular wings with supersonic leading edges, at small Strouhal numbers (p* _ 0). For linitc-aspect-ratio wings, the tangent-wedge method yields more accurate solutions when the numbers M.",. and the aspect ratios ).w are larger.

1. Se.d,?v, L:1. Sirn~lari!y and JJtmeM,tJllal .1Iet/jod.< !II Jlechal!!(~, tran;:. by Klsln, \ltr Publtsbers, \Ioscow (1982). 2. Sedov, L.1. A Course Of ConUJII./Um .1fecllanics, ,"ols. 1-1\-. Croningen, Wolters-NoorthoH (1971-1972). 3. Kbrislianovkh, 5.A. Ob/ekanie tel guzolll lirl baLldkh skorustyakh (HighSpeed Flow of a Gas over a Bodr), !'\auch. trudy IsAGI. vyp. 481 (194U). 4. Bclotserkovsky, 5.:\1. and Skripach. A.K. AerodLnamiclll.'s1rit protZIJOd/!!le

~!~~~~~t,~:s :f~aC:!ft ~n~{~/\Vr;~ a~"s~~Ub~~IJ~!~'v~~~i:~::t\!~~~~~i:~:~

(1975). 5. Loitsyansky, L.G_ Mtkhallika ~hidk('s1i i ga:a U'luid \lC{'hanics), Xauka, Mosco",' (1970). 6. Pred\'oditelev, A.S., Stupochenko, E.\"., 10110\", V.P., Pil'!Shanov, A.5., Rozhdestvensky, LB., and SamuIJov, E.\'. Termodir/llmid~l'skfl' funk/sit vozdukha dli/a temperatur ot 1001) do It 0')1) K i datl"flil nt O.U(}l dft 1000 atmgratiA-i luakt.ii (Thermodynamic Functions of .\ir lor Tcmperatures from lova lu 72 01)0 K and Pres~ures irom 0.001 to tOOO atm -Graphs of Functionsl, Izd. AN SSSR, Moscow (1960). 7. Prerlvoditelev, A,S., Stupocbenko, E.V., Samu"1lov, E_V_. Stakhanov, J.P .• Pleshanov, A.S., and Rozhdestvensky, LB. 1'llv/ilsy fl'rmJdinamlc!teskikh fu.nktsii vozdukha (dlya temperatur 01 61)(}() do 121/1)'1 K r dav/tali ot 0.001 do IIJ(hI atm) [Tables of Thermodynamic Functions of Air (for Temperatures from 6000 to 12 000 K and Pressurcs from U.001 to 1000 atmll. lzd, .'\N 5SSR, Moscow (1957). 8. Kibardin, Yu.A., Kuznetsov, S.I., Lyubimov, :\. :'01"., andShumyatsky, 11. Ya. Atlas ga:odinamiclltsklJ.-'/ funktsit pri hol'slti!.:11 skor08tyakll i vysokikh temperdturakh lJosd/Uhnogo potokll (A tlas of Gas-Dynamic Functions at High Velocities and High Temperatures of an Air Stream), Goscnergoizdat,

ltloscow (t961). 9. Kochin:N.E" KibeI, LA" and Roze, ~.V. Teordi(/reskdya gidromekhanika (Theoretical Hydromechanics), Parts I, II, Fizmatgiz, Mo!lCOW (1963). 10. Fabrikant, I.Yn. Aerodinamika (Aerodynamics), Nauka, \losco\\' (t964). 11. Arzhanikov, N.S. and 5adekova, G.S. Aerodinamlka hol'shikh skorQstel (High-Speen Aerodynamics), Vysshaya shkola, Moscow (1965). 12. Irov, Yu.D., Keil, E.V., Pavlukhin, B.N., Porodenko, V.V., and Stepanov, E.A. Gazodiaamicheskie Junktsii (Gas-Dynamic Functions), lI-Iashinostroenie, Moscow (1965). 13. :,!khitaryan, A.:'>1. Aerodinamika (Aerodynamics), \lasninostroenie, :\Ioscow (1!"!jfi).

494

References

14. Krasnov, :'S!; variabll's, 143 two-dimensional plane motiOD, 127 EulC')', II:~, 131i l10w ratl', 89 !la~

dYllamic~,

fundamental, 12!J, 201, 3119 syH"nl, 1291(, ilI4 Grollll'ka's, UIII, 3G JlOdognlph. 180 J1ugolliol. 172{ itltl'l!rn-difTt'rctltial, 25/1 kinpm;ltifs, fundanlC'nlal, 201 La1!ran!{I', 135 Lapla'orce{s), aerodynamic, airfoil, 293f body, 27 comrlex. 246f conversion to another coordinate sY,'ltem, ItOf dissipative, t27 drag, 38, 42, see also Drag gyroscopic, 403f

l'orctI{s). lateral. :~,:) lift. 38 . .'.::!. ::!;iOr. :.!54 ((at pl;.ltc. 248, 3113 muimum. 47 Dearly uniform Dow. 236 . ,386

(;a5, equa\.i()ll~, l2\lif. l·'~') [low from res(!l'\'oil·, 1SllT heating, 58ff ideal, 134, 149, see 1Il.,;(> Gas, perfect interaction with body, 15r, 18 ioniz.ation, tWf

ri;c~i~f~go~rff\~~1~,~,!)~g3!

mass, :.!7 extern;.ll, work, t2ii OD rnOViu!! body, 25ff normal, 38, 416[ producl!d by pressure, 293 ponderomotive, 27 side, 38, 42 suction, 24~. 382fI correction factor, 383[ sideslipping wing. 305fi triangular winq, 362 surface, 25f. iOil work. 12-41 viscous. Viti ForJ!lula. see also Equation(s) Blot-5avart. 05 conversion, coordinate sy~tem9, 82ij

Euler. 117 Karman-T$ien, 268 Prandtl-Glauert, 26B. 302, 306 Reynold~ generalized, 33r Sutherland's. 63 Zhukovsky. 248 Zhukovsky-Chaplygin, 247 Frequency, oscillations, 414 Friction, $I!t a/so Viscosity in turbulent flow, 3211 Function, conformal, 2'.Of

~~~tY:iidf~trt:Jtion. 354, 358, 360f

potontial. 79, 243, 354f, 367{1, 457, see also Velocity potential derivatives. 458 doublet, 3541 gradient, 80 stream, 89f, 9B, 202 Gas, see; also

Air,

Flow(s), Fluid

::l~~r~~ili:r' el!~tr~~ity~8f1945211

diatomic, mean molar rna9S, 70 diffusion, 121 dissociation, 60, tB9H dynamics, Hi, see alsQ Aerodynamics, high-speed

mixture, 67 mean molar ru;J.~, 70 parameters, at stagnation point. 191 perfect, see olso Gas, ideal calorically, 65 equO-tion of state, 65 thermally. GS recombination, 60 stream, configUration, 149f viscous. (low in boundary layer,

'34

Gradient,

~~~::r 've1!~ity,

32

potential function, 80 pressure, 107 velocity, 34

Half-wing,. inllnile triangular, rn Heat. speclrlC, 61£ Heating, acrodyn:;l1oic, 15, 18, .-':\ Hodograpb, t 7\)fi

H~b~~~~!iS~f

reverse influence, 17 continuum, 16, 20 harmonicity, rm~, intl'raction, 402

ir!~~~~:l.an s:~;:~f;o~tc

supel's