1、REPC)RI 970THEORETICAL LIFT AND DJiMPING IN ROLL AT SUPERSONIC SPEEDS OF THIIN SWEPTBACKTAPERED WINGS WITH STREAIM.WWE TIPS, SUBSONIC LEADIING EDGES,KND SUPERSONIC TRAILING EDGES -By FEASK S. MALVESTUTO, Jr.,KENNETH MABGOLIS, andEEEWIEETS. RIBSERSUMMARYQn the basis of linearized supersonic-jiow theo
2、ry, generalizedeguations were derirecl and calculations made for the lijl anddamping in roll of a limited series of thin sweptback taperedwings. Results are applicable to wings m“ih streamwnketipsand for a range of supersonic speeds for which the uzt vO), A., and u:c.=pV2SbThe evaluation of equation
3、 (15) is simpltied by empIoying the integration procedure used forthe rolling-moment coefficient of each of the elemental triangukr areas (see fig. 5) is given bydCl,C= dCm-lCza. The contribution to(16)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
4、THEORETICJSL LIFT AX() DAMPING IX ROLL OFwhere WOis the latera.Idistance of the center of pressure ofan elemental triangIe from the X-axis of the wing and dCLCis t-he lift coefficient of an elemental t.rimgnIar area. Theanalysis of reference 3 shows that for a pressure distributionof the form z(r),
5、which is the form of the pressure of equa-tion (14), the lift of a.nincremental triangle can be expressedas(17)where xl is the height and AS,= t-hearea of an eIemental tri-angle. The quantities xl and yOgiven in termsof v and otherparameters may be expressed a.sfollows (see Q. 5):3 c, eOvO=ZInvRegio
6、n OGEb(lm)l=2eo(l + mv)I(18)3b(l+m) vyo=8 l+mv JUse of equations (16), (17), and (18) leads to the followingintegral expression for Cl,=:b3(l + m)I(m)p 1sPdv1617 (I+na)b-zeo+ (1+ mvy.J=yFn (1-hn)b-i-27nOOCr(19)SWEPTBACK WINGS AT SUPERSONIC SPEEDS 401Region of wing within wing-tip Mach cones.For ther
7、egion of t-he wing within the wing-tip Mac-h cones, therolling-moment c.oeftlcient is ealuated by employing theapproxin.mte surface velocity potential given in equation (6);that is,The veIocity potential given by equation (6) can be expressedso as to apply to the rolling wing by considering the IOCS
8、I “elope of the airfoil surface with respect to the flow direction.For the thin wings of zero camber considered herein, theslope at any lateral station y is the Iota.1angle of attack ofthe wing and is equal to p/V. Substituting pq/V for a inequation (6 gives the approximate linearized surface veloc-
9、ity potent,iaIfor the rolbg wing; that is,The evaluation of equation (20) carried out inyields(20)append(l+m)b3Y. Tr3(l+m)f(j, expressed in oblique coordinates. (See fig. ofappend- Approximate solufion1.2Lo.8 $u9.Bu .6.4 -,Exacf Soluflh for tip regim.2(2) 10 .2 .4.6 p=”?-:!.8 Loz -1-. -_Y_w!(a) Sect
10、ion A-?J y= Crmatimt.) Seotfon S-S; z= Constant.FIGURE7.-C!hordwfss and sprmwfse pressure dfstributiom- forlfft in eectional planes throughthe wing-tip region.on crossing the Mach line emanating from the break.In particular, when a portion of the edge is para.Jlelto thestream direction (as is the ca
11、se for the streamtise wing tip)Czu;(vm) .“the due of dl,w IS unity. Hence, the term containing_du:(vnr)dow is zero and the lift is found to drop to a smallmagnitude on crossirg the tip Mach line in the wirg-tipregion.Provided by IHSNot for ResaleNo reproduction or networking permitted without licens
12、e from IHS-,-,-404 REPORT 970NATIOIWIJ ADVISORY COMMITTEE FOR AERONAUTICSFigure 8 shows the chordwise and spmnvise pressure dis-tributions for rolling. A similar situation of large fhitedrop in pressure across the wing-tip Mach line exists forrolling ardogous to the lifting case. (See reference 8.)T
13、he interesting result obtained is that the rolling pressure isnegative in the wing-tip region. This behavior is due to thefact that the sign of the pressure in the wing-tip region isaffected only by the angIe of attack of the leading edge ofthe plan form on the opposite side of the rolI axis which f
14、orpositive roll is negative. (See reference 8.)3.0 1 1 1 Exact sdufkm- Approximate soiufion2.0 3N 1.5b b4/o “NQ. . t)1I.5 880t1L. - .-,. -.5 a0 .2 .4 .6X-I .8 Lo(a) Section A-* #= Constant.(b) Section S-S; z= Constit.GURE S.Chordwise and spmmise pressure distributions for rolling in sectiensl pIanes
15、through the wing-tip region.The exact soIution for the rolling-pressure distribution inLhewing-tip region has not at preeent been determined, Inview of the good agreement between the appro-Clp / y.3 “ .3 . L,22 ,.245 5055 60 .2 0 “:2 .4 .6 .8. .A, deg .?.-FIctuRE 12.Someillustrative variations of da
16、mping-in-roll derivative Cl= with Mach nnmber, espect mtio, eweepback, and taper ratio.1 CONCLUDING REMARKSBased upon the .concepts of linearised supersonic-flowtheory, the lift-curve slope CL. and damping-in-roll deriva-tive Clphave been evaluated for a limited seriesof sweptbt-ickwings (with strea
17、mwise tips) of arbitrary taper and sweep.The investigation was limited to a range of Lh!la,chnumbersfor which the wing is completely enclosed between the Machcones springing from the wing apex and from the trailingedge of the root chord of the wing. An added restriction isthat the lhfach lines from
18、the wing tips may not intersect onthe wing.The r.wxdtsof the analysis are presented in the form ofgeneralized design curves for rapid estimation of t-hedcriva,-tives. Some illustrative variations of the derivatives withaspect ratio, taper ratio, “Ma.chnumber, and leading-edgesweep are aIso presented
19、.*LANGLEY AERONAUTICAL LABORATORY,NTATIONAT. ADVISORY CONIIVIITTEE FOR AERONAUTICS,LANGLEY FIELD, VA., February 15, 1949. IProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-409APPEI)IX AEVALUATIONOF POTENTIALFUNCTIONFOR LIFTThe integral expression for
20、the potential.=_h JJ d dq%-,o - w-m)1=E=gCD+-41q=Q (uu)(Az)When the ppropliate substitutions and simplification areperformed, equation (Al) becomes(tP)Q=l.(qa71/u/ / / /x,y or uw,vwFIGGB.E13.SlcetcLtof wing showing region of integration (area SW.0) for the ewhmtion oft-hesurface veIocity poterdia fo
21、r IKt and rollThe limits of integration are readily obtained from figure 13and the pot entiaI function may then be expressed asThe integration yields(o).:; &-&% )( b.lf (#Auwzfw+ )Ti%en equation (A6) is expressed in terms of r,y coordinates,the formuIa obtained isand is given as equation (10) in the
22、 text. -:Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-EVALUATIONOFThe intograI expression for the potentialAPPENDti BPOTENTIALFUNCTIONFOR ROLLmay be evaluated by the same method used in appendix A.upon substitution of the new variables and simplif
23、ication, equation (Bl) becomeswhere the limits of integration are obtained from figure 13. .When the integrations indicated in equation (B2) are performed, the following expression is obtained:( 1+3m bM )J(4)?=*, Vwuw +.y 1WC )( bM ).vw+w l+?nuw wEquation (B3) transformed into z,g coordinates become
24、sp 23(2?n+ 1)+ t(m+ 1)-2?iaz /2(B+ TM) (b 2y)(d)?=(1+ ?n)B/(Bl)(B2)(B3)4) and is given as equation (21) in the text.REFERENCES I 4. Cohen,Doris:The TheoreticalLift of Flat Swept-BackWings atISupersonic Speeds. NACATN 1555,194S. -1, Brown,ClintonE.: TheoreticalLift and Dragof Thm Triangular5. Evvard,
25、 John C.: Distributionof Wave Drag and Lift in thoWings at SupersonicSpeeds. NACA Rep. S39,1946. Vicinityof WingTipsat SupersonicSpeeds.- NACA TN 1382,2. Brown,CIintonE., and Adams,lilac C.: Dampingin Pitch and 1947.Roll of TriangularWingsat SupersonicSpeeds. NACA Rep. 6. Evvard,John C.: The Effects
26、 of Yawing Thin Pointed Wings at892, 1948. Supersonic Speeds. hACA TN 1429, 1947,$ Malvestuto, Frank S., Jr., and Margolis, Kenneth: Theoretical 7. Evvard, John C.: Theoretical Distribution of Lift on Thin WingsStabiIity Derivatives of Thin Sweptback Wings Tapered to aat Supersonic Speeds (An Extens
27、ion). AACA TN 1585, 1!348.8. Moeckel. W. E and Evvard, J. C.: Load Distributions Due toPoint with Sweptback or Sweptforward Trailing Edges for a I Steady Roll and Pitch for -Thin Wings at Supersonic Speeds.Limited Range of Supersonic Speeds. NACA Rep. 971, 1950. NACA TN 1689, 1948.410Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
