1、(JJ:)q iFOR AERONAUTICSTECHNICAL NOTE 3071FORCE AND MOMENT COEFFICIENTSTHEORETICALON A SIDE SLIPPING VERTICAL- AND HORUONTAL-TAILCOMBINATION WITH SUBSONIC LEADING EDGESAND SUI?ERSONIC TRAILING EDGESBy Frank S.Malvestuto, Jr.Langley Aeronautical LaboratoryLangley Field,V%WashingtonMarch 1954 -. .-. 7
2、. -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-IT NATIONAL ADVISORY COMMITTEETECENICAL NOTETECHLIBRARY IC/WB, NMIllllllulllulllllllllllllFOR AERONAUTICS 00bb21b3071.THEORETICAL SUPERSONIC FORCE AND MOMENT COEFFICIENTSON A SIDESLD?PING VERTICAL- A
3、ND HORIZONTAL-TAILCOMBINATIONM1ITlSUBSONIC LEADING EDGESAND SUPERSONIC TRAILING EDGESBy llcankS. Malvestuto, Jr.SUMMARYTheoretical expressions have been derived by medns of linearizedsupersonic-flow theory for the lateral force due to sideslip P, theyawing moment due to sideslip % , and the rolling
4、moment due to side-Slip Cz$ for tail arrangements consisting of a vertical triangularsurface attached to a symmetrical triangular horizontal surface. Theresults are valid, in general, for a range of Mach nuuiberfor which theleading edges of the tail surfaces are swept behind the Wch cone fromthe ape
5、x of the arrangement and the trailing edges of the tail surfacessre ahead of the Mach lines from the tips.A series of design charts are presented which permit rapid estimatesto be made of the force and moment derivatives. A discussion is alsoincluded on the application of the expressions for the pre
6、ssure distri-butions determined herein to other plan-form shapes of the tail surfacesand possible wing-vertical-tail arrangements. A solution to a two-dimensions.1“mixed type” boundary-value problem which is needed in thepresent analysis but which may also be of interest in other “conicalflow” malys
7、es is presented in an appendix.INTRODUCTIONThe prediction of the stability of complete airplane and missileconfigurationsrequires a knowledge of the aerodynamic forces and mmentsacting on all the component surfaces of the afiframe and the rates ofchange of these forces ad moments with the attitude,
8、velocity, andacceleration of the associated surfaces. The rates of change of theaerodynamic forces and mmnents when linearly related to the attitudes,velocities, and accelerations are comnonly called stability derivatives. . - .- - . Provided by IHSNot for ResaleNo reproduction or networking permitt
9、ed without license from IHS-,-,-2 NACA TM 3071Theoretical estimates of stability derivatives for a variety ofwing plan forms with flat-plate cross sections are now available. Infor-mation, however, relating to the stabili derivatives contributedbyvarious nonplsmar tail systems is stiU meager. IMost
10、of the availablederivatives sre for configurations composed of low-aspect-ratiosurfacesrefs. 1 to 3). In reference 4, however, sideslip derivatives have beenpresented for tail arrangements for which all the plan-form edges are.supersonic. In references 1 and 5 appro-te estimates of the damping-in-ro
11、ll.derivatives for cruciform arrangementswith high-aspect-ratiosurfaces have also been reported.The purpose of the present paper is to provide theoretical esthatesof the lateral force, the rolling moment, and the yawing nmment producedby the sideslippingmotion of a tail arrangement consistingof a tr
12、iangu-lar vertical surface attached to a symmetrical triaqar horizontal sur-face. The leading edges of the tail surfaces are subsonic; the trailingedges, supersonic. Consideration has also been given to the applicationof the results presented herein to other plan-form shapes of the tailsurfaces and
13、possible wing-vertical-tail coinations.The analysis is performed within the framework of linearizedsupersonic-flowtheory. Inasmuch as the linearizedperturbated flowwithin the Mach cone from the apex of the tail is conical (the arrange-ment is a conical body), the analysis reduces to the solution of
14、a sin-gular integral equation associatedwith a two-dimensional “mixedtype”boundary-value problem. The solution is obtainedby an application ofthe general method-sfor evalwtti these integral eqmtions that havebeen propounded by Muskheldshvi13 in reference 6.sYmQItsThe orientation of the tail arrangem
15、entwith respect to the X, Y,and Z body axes and the positive directions of the velocities, forces,and moments are indicated in figure 1.x, Y, z body-axes coordinatesYl, q rectangulm coordinates in plane parallel to YZ-planev =T)+iz =X+iyf)m,y,a linearized velocity-potentialfunctionu, v, w x-, Y-, an
16、d Z-components of perturbation velocity, respec-tively (v and w are also defined in the v-plane asbeing parallel to the q- and -sxes, respectively)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3071z = x + iyf3,complex velocities, Uc = u + i
17、u*, vc = v + ifi,and Wc =W+iw+harmonic conjugates of the u-, v-, and w-velocities,respectively,free-stream velocity a71 free-streamMach nuniberfree-stream densityfree-stream dynamic pressure, ; pvzpressurepressureangle ofangle ofdifference across surfaceIcoefficientattack, radianssideslip, radiansco
18、mnon root chord of vertical and horizontal tailsemispan of horizontal tailtransformed semispan of horizontal tail in v-planespan of vertical tailtransformedtransfomnedtransformedspan of vertical tail in v-planesemispan of horizontal tail in z-plsmespan of vertical tail in z-planearbitrary real const
19、antsarea of horizontal tailarea of vertical tail,. -. _._. u- .- . . - . -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 MACA TN 30717 amgle in plane of horizontal tail between a ray throughorigin and X-axisangle between leading edge of horizontal
20、 tail and X-axist =tan7AHto=tanyo=TE angle in plane of vertical tail between a ray throughorigin and X-axiso angle between leading edge of vertical tail and X-axisr =tan EAvro .tan co=A= %2aspect ratio of horizontal tail, =4tm70L%v aspect ratio of verticalJ.Ltail, %2=2taneo%(-%I=k=()B*A=* + B*A=* EA
21、=B2AV2 - - 4 1 16 16 4cn(u/k)dn(u/k)Jacobian elliptic functions of argument u andmodulus ksn(u/k) JE, E complete elliptic integrals of second kind with moduli kand jl - k2, respectivelyProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,NACA m 3071K, K
22、complete elliptic integrals of first kind with moduli kand 11 - k2, respectivelyG=L-KkJ -4=., -_ E(k)1 *F-?YLNCzlateral force, see figure 1rolling mment, see figure 1yawing moment, see figure 1Ylateral-force coefficient, ()af3nc%=T $-+0rolling-momentyawing-momentSubscripts:Hvhorizontal tailvertical
23、tail(L$2Lcoefficient,*ANCoefficient,*%v-/ _ _. ,- . . - - Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 NAC!Am 3071.ANALYSISGeneral ConsideationsThe object of the ensuing analysis is to determine the aerodynamicpressures and correspondingforces a
24、nd moments acting on the surfacesof the tail arrangement sketched b fig-me 1 that sre produced by thesideslippingnmtion of the tail. The leading edges of the horizontaland vertical surfaces are subsonic (withinthe ch cone from the apexof the system) and the trailing edges are supersonicand at zero a
25、ngleof sweep. It is stipulatedthat the tail surfaces sre of zero caniberand vanishingly small thickness. It is apparent that this tail con-figuration in sideslip attitude is equivalent (by rotation) to a righttrianguhr wing at an angle of attack with a triangular end plate orfin at zero geometric an
26、gle of attack attached to its streanmd.seedge.Such an arrangement is sketched in figure 2, and for conveniencethisorientation of the tail arrangement is considered in the followinganalysis. With the orientation shown in figure 2, the surface approxi-mately in the horizontal plane at a constant geome
27、tric angle of attackis tetatively called th-wing” and theand at zero geometric angle of attack isThe analysis is based on linearizedflow theory. Specifically, solutions ofpotential equationsurface h the verti therefore, these termsvanish because our primary interest is the evaluation of the rate ofc
28、hange of the aerotamic forces and moments as a approaches zero.Because of the conical geometry of the wing-fin arrangement, thefollowing analysis to determine the required solution for the pressureemploys the concepts of conicsl-flowtheory. This concept lies thatall disturbance-velocityquantities su
29、ch as u, v, and w remain con-stant along rays emanating from the origin (apex of arrangement)andhence become functions of only two independentvariables that specifythe direction of the ray.Busemsmn (ref. 7) initially showed that the assumption of conicalflow implies mathematicallythat the disturbanc
30、e-velocityfield withinthe Mach cone of the system is governedby an elliptic differentialequation, and by a transformationof coordinatesthis equation reducesto the two-dimensionalLaplace equation with respect to either the u,v, or w perturbation velocity. The problem of obtaining a solutionto equatio
31、n (1) or (2) therefore reduces to one of obtainhg a solutionto Laplaces differential equation in two dimensions subject to certain1 -_ ._ - - . - _ -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACA TN 3071boundary conditions. These consideratio
32、nslead naturally to a mode ofsolution using complex-functionmethods (refs. 8 to 10) and associatedintegral-ecyxationconcepts. The following sectionspresent an analysisand solution of the wing-fin problem based on these procedures.Prescribed Flow ConditionsA sketch of the wing-fin arrangement showing
33、 the body axes used inthe analysis is presented in figure 3. Denoted also in this figure arethe prescribed values of the disturbance velocities u, v, and w andtheir spatial derivatives in the plane of the wing and plane of the fin.These prescribed values of the velocities sad their derivatives arede
34、termined from a knowledge of the boundary conditions,the symmetryconditions,and the equations of irrotationality.The boundary conditionsare as follows:On the Mach cone surface,U.v. w. oon the wing surface,and on the fti surface,( )v X, O*, Z = oFrom symmetry considerations (see ref. 10) it can be sh
35、ownthe plane of the wing the antisynmetricu- and v-velocities arethat inzero offthe wing. In the pe of the fin, however, the tangential velocitiesare not zero off the fin because the arrangement lacks symmetrywithrespect to the XZ-plane; in fact, these velocities must be continuousacross this region
36、.The use in the equations of irrotationalityof the given boundaryvalues of the velocities, together with values of the velocities deter-mined from symetry conditions,produces the additionalprescribedvalues of the velocity derivatives denoted in figure 3 and needed inthe analysis. It is also stipulat
37、ed that, as the leading edges are.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2T NACA TN 3071 9approached, the disturbance velocities become locally binite as the-1/2 power; that is, the flow around the subsonic leading edges behavesin the same m
38、anner as the flow around the leading edges of thin flatplates in an incompressibleflow (see refs. 9 and 10). This stipula-tion on the type of edge singularity can be used in order to obtaina unique solution to the integral equation of the boundary-value problemthat is solved subsequently (see append
39、ix A).Transformation of Supersonic Conical Flow toTwo-Dimensional IncompressibleFlowThe transformation of the supersonic conical-flow equation in u,v, or w to the two-dimensionalL that is, the u-, v-, and w-velocities have the(0z=u- E) zv=v w= ()Yzx x ?Now if the X-, Y-, and Z-coordinates me transfo
40、rmed in the followingmanner:Y1 =B;1 =B:(4).,( _. .- _ . . - - -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10 NACA TN 3071then equations (2) defining the u-, v-, and w-velocities are transformedinto elliptic differential eqpations in the two vari
41、ables yl and Z1.Apication of the noncotiormsl transformation (see ref. I-1)transformsv-plane inY1 i1 1 - %2- Y12v =q+ig= -t-i(5)1 f-z 1 la-zthe u-, v-, and w-velocities from the ylzl-plme to thewhich each of the disturbance velocities satisfiesLaplacesequation; that is, in the v-plane,+1.l=oA=o#w=ow
42、here # tithe placian operator ($+$). The effect of thetransformation expressed by equation (5) is to deform the dotily con-nected smnukr region between the Mach cone and the body in the ylzl-plane(see fig. 4) into a doubly connected open-slit region in the v-plane (seefig. 5). The continuous cut (l,
43、rn,-1) in the v-plane correspondsto theMach circle in the ylzl-plane. If the v-plane is considered as composedof two sheets, then the region external to the Mach circle in theylzl-plane transforms into the lower sheet of the v-surface and is con-nected to the upper sheettransformed Mach circle.zonta
44、l and vertical sxes,ment is preserved (exceptIt shouldbe pointedthrough the branch cut (l,co,-1),that is, theThe transformation dues not distort the hori-and hence the shape of the wing-fin arrange-for scale) in the v-plane.out that, since the wing-fin contour is trans-.formed to one sheet (the Wer
45、sheet) of the double-sheeted v-surface,the correspondencebetween the velocities in the crossflowplane of theoriginal XM%ystem within the Mach circle and the upper sheet of thev-surface is 1:1. Furthermore, since the transformation is conformal inthe neighborhood of the q- and -axes, the prescribed v
46、alues of theProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3071 “ uvelocities and their spatial derivatives which are constant along thewing plane and fin plane in the original space remain constant alongthe q- and (-axes in the v-plane.If t
47、he complex velocities uc = u + iu*, Vc . v + iv+, andWC = w + i# are considered in the v-plane, then from the analysis ofHayes and Multhopp (refs. 9 and l-l)the ensuing comparabilityrelations(which take the place of the equations of continuity and irrotationality)provide the necessary relations in t
48、he v-plaae for attempted solutions:dvc = - :ducdwc = -iB d%v(6a)(6b)In terms of the real parts of Uc, Vc, and Wc, the relations of equa-tions (6) are as foows (see ref. 11):For =0 and -11,au_ baq “J:+) b=%) (3)equation (14) relates the #- andu-velocities along the transformed fin contour in the z-plane.Integration of eqyations (13) and (14),with equations(9) and (m)for v and # taken into account, produces the
