Biot, M. a., Loads on Supersonic Wing, 1949

8/6/2019 Biot, M. a., Loads on Supersonic Wing, 1949

1/5

Reprinted from the JOURNAL OF THE AERONAUTICAL SCIENCESCocyright, 1949, by the Institu te of the Aeronaut ical Sciences and reprint ed by permission of the copyright ownerMAY, 1949 VOLUME 16,No. 5

Loads on a Supersonic Wing Striking aSharp-Edged Gust u

M. A. BIOT*Cornell Aeronauiical Laboratory

ABSTRACTThis is a calculation of th e chordwise lift distr ibution, total lift,

and moment on a two-dimensional wing striking a shar p-edgedgust at supersonic speed. A direct solution is established byconsidering a distribution of sources in a fluid at rest. An alter-na te meth od usin g Busem an ns conical flow is also shown t o beapplicable. The time history of the total lift and mid-chordmoment is discussed. It is shown that the total lift increaseswith time and reaches a maximum that corr esponds to the steady-state phase of the flow. The mid-chord moment goes thr ough amaximum independent of the Mach Num ber if the latter valu e islarger than 4/r, while this maximu m can become infinite for aran ge of Mach Nu mbers between 4/r and 1.

(1) INTRODUCTION

C ONSIDERABLE ATTENTION ha s been given lat ely t ononst at iona ry flow problems of wings flying atsup ersonic speeds. Most of th e work, however, ha sbeen concern ed with th e aer odyna mic forces on anoscillating airfoil from the standpoint of flutter analysis.The pr oblem of th e wing hitting a sha rp-edged gust is ofa different nat ure a nd turn s out to be actua lly muchsimpler than the oscillating airfoil problems.

It is shown in section 2 tha t it may be tr eated by adistr ibution of sources of a simple type along th e chordand that the pressure distribution may be derived byelementa ry methods. The procedure does not intr o-duce a moving fluid but consider s a fluid at rest inwhich n onsta tionary sources are distribut ed in a layer ofvariable extent. This point of view, wh ich is closer t oacoustics t han t o aerodynamics, is somewhat novel andseems to pres ent ad vant ages of simplicity an d closenessto physical r eality in cert ain categories of problems.The pressur e distribution derived by this meth od isapplied to the calculation of th e time hist ory of lift a ndmoment on the wing in section 3. Par ticular att entionis given t o th e value of th e mid-chord moment , whichsta rts from zero, rises to a maximum, and goes back tozero. The value of this maximum and related da ta isevaluated in section 4. These results are of part icularinterest to the designer.

The der ivation of th e pres sur e as given in section 1is only one of the m ethods tha t may be us ed in thisproblem. As an independent check an d as an illustra -tion of th e application of Busem an ns met hod of fonicalflow-to a nonstat ionar y problem, an altern ate derivationis given for th e pres sur e distr ibution in section 4.

Received August 15, 1948.* Member of Cornell Aeronau tical Laborat ory consu lting sta ff.

Also Pr ofessor of Applied Ph ysical Sciences, Brown U nivers ity.

In a paper by Schwarzl procedures used in oscillatingairfoil th eory a re extended to th e problem of a wingstriking a sha rp-edged gust at supersonic speed. Re-sult s for an oscillat ing down-wash lead t o th e gust prob-lem by a Fourier int egral repr esentat ion. This methodconstit utes a considerable detour a nd introduces int er-mediate r esults of a tra nscendental nat ure which ar eactually not needed and are more complicated tha n theresult. It may be verified that the expression derivedin the present paper for th e pressur e distribut ion isequivalent to that derived by Schwarz. He does not,however, discuss the physical aspects of the problem orderive expressions for lift and moment.

(2) DERIVATION OF THE PRESSURE DISTRIBUTIONThe wing of chord 1 enters a uniform gust of upward

velocity v. at the super sonic velocity I (Fig. 1). Th evelocity componen t norma l to th e wing mu st remainzero, an d this condition is equivalent to the genera tionof a velocity norma l to the wing which can cels-the gustvelocity (Fig. 2). This m ay a lso be considered as areflection of the gust on the wing. Because thevelocity is super sonic, th e pres sur e distr ibution on oneside does not influence the pressure on the other, andth erefore we need only consider t he bottom side. Th epres sur e distr ibution on top will be th e sam e except for areversal of sign. For the same reason the pressur e dis-tr ibution is not influenced by t he tr ailing edge, and

e RurtVAir at rest

lpigure I*ing eZltrrlng Vertic al Gust

Figure 7%flecttanf he Gust an the Wlnp

296


2/5

LOADS ON A SUP ERSONIC WING 1 29 7

sr

Fi~re 3Limits of Intepatlon for Integral (2.6)

everything is th e sam e as if th e wing were of infinitechord. The pr oblem is then r educed t o finding thepressur e in a fluid, originally at r est, due to th e pr esenceof a uniform distr ibution of norma l velocity vo along t hechord from a point 0, corr esponding to the edge of th egust , to th e leading edge A. If th e wing ent ers thegust at time zero the length OA = Vi.

Now, such a distr ibution of norma l velocity ma y berepr esent ed by a distr ibution of source singular itiesgenerated continu ously at the leading edge and remain-ing constant ther eafter. The velocity poten tia l of su cha source appearing suddenly with a constant intensityat t ime tl an d location xl, isC#+ogr-~log c(t-jtl)+41 cyt - t# - t21(2.1)with

r2= (x--CQ+y2This expression satisfies the wave equation

wher e c = velocity of soun d. Moreover, for su fficient lysma ll values of r with resp ect to c(t - ti), it repr esent s avelocity field

9 = @o/s) log r (2.3)identical with th at of a steady sour ce in t he incompres-sible flow. Hen ce, by an alogy, it ma y be conclud edt ha tth e un iform d istribu tion of such sources will produce aun iform norma l velocity componen t vo.

The pressure generated by this source along the x axis(y = 0) is

a+p = -p z = n4c2(t _ jy_ ($$ _ xl) (2.4)

where p = fluid density. Note that the source locatedat xi suddenly appears when the leading edge reachesth at point, i.e., at a time tl = - (xi/V).

The local lift 29 at a point x and time t, due to theun iform distr ibution of such sources from xi = - Vt tox1 = 0, is given by2p = ; Pcvo1 dxl-rQ &[t + (x1/V)12 - (x - Xl)

(2.5)or with the change of variables

x/et = 4, x1lct = h, c/V = sin p1M=----sin P = Mach Number

2p = 5 pcvo1-l/sinr d(l + .$isin dji2 - (.$ - &)2 (2.6)In integrat ing this expression special att ention must begiven to the limits of integration. The function un derthe integral sign must be tak en a s vanishing when-ever the r adical is imaginar y. The range of integrationis th erefore limited bet ween th e two roots of th e equa -tion,

(1 + EIsin j2 - (5 - [Jz = 0These roots are

p = E-1 ,5(2) = E+ll+sinr 1 - sin p (2.7)These quan tities, plotted as functions of E, are repre-sented by two straight lines (Fig. 3) which intersect at apoint of abscissa

and ordinate4 = - (l/sin p)51 = - (l/sin /*)

The interval of integration is th us limited t o the shadedar ea bordered by the two str aight lines [Eq. (2.7) ] andth e axis (& = 0). We mu st th erefore d istinguish be-tween two ranges of values of [: for - (l/sin ) < 5 < - 12p = z pcvlj1 xEl()

2/


3/5

a!% JdfJ%ii\jiiL OF THE AERONAUTICAL SCIENCES-MAY, 1949

Leading E4as-A4 -I 0 +I

Lift Dlltrihtlon he to the curt

I I U-/ 0 +I )-I, ct

Fipure 5Li*t Mstributfon For Vwlous kch Numbers

2p = ; pcvos 41w d( 1 + .$I in EL)~ (5 - &) I (2.9)= 2pczr,os-l (;;F;;Jlr cos p

)

In this region the lift distribution depends on the time.Substitut ing ,$ = x/ d ,

2 pcvo2p = - cos _-l?r cos /A (x + ct sin Kct + x sin p (2.10)

The cos-l bra nch is between zero an d ?r. It will beobserved th at th e lift distribut ion depen ds only onx/et-i.e., th ere is a similarity law an d we may dr aw aone par am eter fam ily of lift distribut ion curves with th evariable x/d and the Mach Number as a parameter.The a ppear an ce of th e lift dist ribut ion is shown in Fig.4. We see th at t he production of lift is not limited tothe region of the gust. The lift propagates ahead of theedge of th e g&t with t he velocity of soun d. The liftdistribution for various Mach Numbers within th e re-gion a ffected by the gust edge is shown qu alitat ively inFig. 5.

More often we are inter ested in the stalling momentMl,2 about the mid-chord. We find, after integration,I

(3) LIFT AND MOMENTIt is useful for practical pur poses t o obta in th e values

of total lift a nd moment as functions of time for a gustacting on a wing of chord 1. Becau se of similarityproperties, such quantities may be expressed by meansof nondimen siona l functions of V t/ l only. In com-

put ing th e time history of lift a nd moment , we mu stdistinguish between three phases.

Phase 1 .-Where th e tr ailing edge is still outside th eregion where lift is produced, ( V + c)t < 1.

Phase Z.-Wher e th e tr ailing edge is in th e regioninfluenced by the gust edge, (V + c)t > I > (V - c)t.

Phase 3.-Where th e entir e wing is outside th e regioninfluen ced by t he gust edge, 1 < (V - c)t.P h a s e 1

We integra te th e lift distr ibution 2~ over t he chord 1an d split th e integra tion into two inter vals, one inwhich t he lift distr ibution is un iform an d th e oth er inwhich it is not. Thu s the total lift is-lL=2 s dx+2--Vt s

ctP dx (3.1)- c l

or-i = 2cts~,l;np~+ 24Elf: -

2ctsl dp_-l E d4.4 (3.2)This form is rea dily integra ble. We find

L = 2pcvoZ( t/q (3.3)Similarly, the pitching.moment about th e leading edgeML is

--cf +crML=2 s (Vt+x)pdx+2-vt s (Vt + x)pdx- Cl (3.4)orML = 2(#

M,,, = ; IL - ML = p~o12 ; ( )1 - 7 (3.6)

Phase 2Similarly, t he lift a nd moment during Pha se 2 are

s--cl

s1 VIL=2 pdx+2 P dx (3.7)-W --d

--d 2 - vtML=2 s -vt (Vt+x)pdx+2 _-cl (Vt+x)pdxs (3.8)or


4/5

LOADS ON A SUPE RSONIC WING 29 9s1L = 2ct P & + 2ct[fiE l_, - M l/ 2-I/ sin II -=7(1-7)i[s in -l(%)+;]+Pm PThe second ph ase originates at T = l/(1 + sin p) an d

term inat es at T = l/(1 - sin p). It is foun d t ha t inth is ra nge the moment Ml12 may go th rough an othermaximum and tha t this maximum may be larger thanits value in th e first pha se. Value of th is maximum an dth e value of T at wh ich it occurs ar e given in Table 1 forvarious Mach Numbers.

with z = (l/sin p) [(Z/ Vt) - 11. We findTABLE

M a c h N u m b e r [M1/2!,oz.pcvol~&I0.281

with L, = 2pcv01/cos p. The sin - br an ch is ta ken be-tween - (r/2) an d + (s/2). The sta lling momen tabout the mid-chord Ml,2 is given by

%=$(I - y){sin-l[($j - 1)-&--l+:}+

i (T)2dw (3.12)

At th e end of Pha se 2,l = (V - c)t, and we may verifythat the above formulas yield L = L, = 2pcv~l/cos ,uand Ml,2 = 0.

1.25 1.10 0.255VT 1.00 /4

The ma ximum of Ml,2 in the second ph ase is greaterth an the maximum M112 = (1/4)pcQ in th e firstph as e, if (l/ sin p) < (4/7r) = 1.27. The value of th eabsolute maximum of th e mid-chord moment is plott edagainst the Mach Num ber in Fig. 6. This ma ximumis independen t of th e Mach Number if th is MachNumber is greater tha n 1.27.

(5) ALTERNATEDERIVATIONO F T H E P R E S S UR EDISTRIBUTIONBY THE ME THOD OF CONICAL FLOW

P h a s e 3 The above results may be derived by an entirelyIn this phase the lift and moment remain independent differen t pr ocedu re. We ma y compu te t he velocity

of time. We find Ml,2 = 0 an d L = L, = 2pcv~/ cos p, field due to the gust reflection on the wing. Becausewhich is the lift in st ead y-stat e flight of a wing of an gleof attack v ,/ V . The lift increases all th rough Ph ases 1and 2 and reaches its maximum in Phase 3.- The timehistory of th e mid-chord moment requires special at ten-tion, as shown in the next section. &S O(4) MAXIMUM VALUE OF THE MID-CHORD MOMENT

Wi t h a n o n d i m e n s i o n a l t i m e -v a r i a b l e T = V t/ l, t h em id-chord m om ent du r in g Ph as e 1 i s [c f . Eq . (3.6 ) ]

M l/ z = pcvoZ27(1 T) (4. l) 0.25T h i s c u r v e i s a p a r a b o l a w i t h a m a x i m u m a t T = l /z .T h e v a l u e o f t h e m a x i m u m i s

i

[M1/2l,m. = (I/~)PG Q (4.2)I t i s i n t e r e s t i n g t o n o t e t h a t d u r i n g t h e f ir s t p h a s et h e m i d -c h o r d m o m e n t i s i n d e p e n d e n t o f t h e M a c hN u m b e r .

D u r i n g P h a s e 2 t h e m i d -c h o r d m o m e n t i s a s g i ve n b yEq. (3.12)


5/5

300 JOURNAL OF THE AERONAUTICAL SCIENCES-MAY, 1949

we deal with a super sonic wing velocity, th e effect of th egust is th e same as if th e chord were infinite. There-fore, the principle of similarity applies, and the velocityfield is similar at a ll insta nt s except for a scale factor pro-portiona l with time. In other word s, th e velocity fielddepends only on the variables A x

5 = x/d, 9 = ylct (5.1)The velocity field is dist ur bed by th e reflection on th ewing in a region bounded by the str aight lines AFF an dth e circle FEF center ed at th e gust edge 0 (Fig. 7).In th e sha ded ar ea AFDF th e field is un iform an dcorr esponds to the steady-sta te motion of a wing withconst an t angle of at tack. The tr an sient field where th eeffect of the gust edge is being felt is inside a circleof radius ct cent ered a t 0. The field inside thiscircle ma y be compu ted by Busema nn s conical flowmethod.

By the transformationf = x cos 8, 11= X sin 8, x = 2s-/l + s)x = s cos 8, Y = s sin % >

(5.2)

The wave equa tion [Eq. (2.2)] is tr an sform ed to La-places equa tion in th e X, Y, plan e

(@#+X2) + (W#J /dY2) = 0 (5.3)Consider now the componen ts of th e velocity field

u = b+/bx, v = @J /byThey also sat isfy Eq. (5.3). It is conven ient to int ro-duce th e complex var iable 2 = X + iY. Let us in-vestigat e th e velocity field on th e bottom side of th ewing. The v componen t of th e velocity field is

v = Re : [log (2 - Zi) + log (2 - Zz) - log Z](5.4)

where Re = rea l pa rt of; 21 = ie; Zz = ---iem; sinp = c/V. This exIjression sat isfies Eq. (5.3) a nd th ebounda ry value of v on the circle an d the wing-i.e.,v = -uo on ODF an d 21= 0 on FEO. In term s of rea lquantities, I

v= -erotan-1{

[s - (l/s)] sin %> (5.5)?r [s+(l/s)]cos%+2sinp

Now we ar e inter ested in th e pressu re distr ibution onth e wing. This pressu re distribut ion for v = 0 may bederived from th e above expression for v by ma king u seof the equations of motion and continuity

p (dupt) = - (h#/bx)(5.6)

By elimina tion of u an d tr an sforma tion of variables,we find th at , on th e x axis, cos 8 = + 1. This imp lies

This may be written

By integration* = pm cos-l (,:+;?J?r cos ,u (5.9)

which coincides with expression (2.9) above.CONCLUSIONS

It ha s been shown th at the pressure distribution on asuper sonic wing striking a sharp-edged gust may be ex-pressed by a simple formu la. This pressu re distribu-tion is obta ined by direct integrat ion of a variable dis-tr ibution of sources in a fluid at rest. It is also shownth at th e sam e result is obta inable from Busema nn smet hod of conical flow. Fr om th e time hist ory of tota llift an d moment it is concluded th at th e largest value ofth e total lift is reached in the last phase-i.e., when astea dy flow ha s been established-while for the mid-chord momen t a ma ximum valu e [J &1/2]_ = (1/4)pcv,J2is rea ched if th e Mach Number is larger th an 4/?r.This maximum moment is independen t of th e MachNum ber. However, for Mach Numbers between 4/7ran d 1, th e maximum mid-chord moment varies with theMach Nu mber an d becomes infinite at Mach Nu mber 1.These conclusions are, of course, subject to the usuallimitations of th e linearized small pertu rbat ion th eory.

REFERENCE:1 Schwarz, L., Ebene lnstationaere TJzeorie des Trap&e&e bei

cos %(bv/b%) = [(A2 - l)lPCl(~PlWUeberschallgeschurin digkeit A VA Goettingen, B43/J/17, July,

(5.7) 1943, tr an slat ion by AAF No. F-TS-934-Re.

Documents

Biot, M. a., Loads on Supersonic Wing, 1949