1、NASA TECHNICAL NOTEO!ZI-ZNASA TN D-8090A VORTEX-LATTICE METHODFOR THE MEAN CAMBER SHAPESOF TRIMMED NONCOPLANAR PLANFORMSWITH MINIMUM VORTEX DRAGJohn E. LamarLanglo, Research CenterHamDton, Va. 23665_.QO_UT/O4zZ_7 S ._gl _NA1iONAL AERONAUTICS AND SPACE ADMINISTRATION - WASHINGTON, D. C. JUNE 1976Prov
2、ided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1. Report No. 2. Government Accession No. 3. Recipients Catalog No.NASA TN D-80904. Title and SubtitleA VORTEX-LATTICE METHOD FOR THE MEAN CAMBERSHAPES OF TRIMMED NONCOPLANAR PLANFORMS WITHMINIMUM VORTEX DR
3、AG7. Author(s)John E. Lamar9. Performing Organization Name and AddressNASA Langley Research CenterHampton, Va. 2366512. Sponsoring Agency Name and AddressNational Aeronautics and Space AdministrationWashington, D.C. 205465. Report DateJune 19766. Performing Organization Code8. Performing Orgamzation
4、 Report No.L-1052210. Work Unit No.505-06-II-0511. Contract or Grant No.13. Type of Report and Period CoveredTechnical Note14. Sponsoring Agency Code15 Supplementary Notes16. AbstractA new subsonic method has been developed by which the mean camber surface can bedetermined for trimmed noncoplanar pl
5、anforms with minimum vortex drag. This methoduses a vortex lattice and overcomes previous difficulties with chord loading specification.This method uses a Trefftz plane analysis to determine the optimum span loading for mini-mum drag, then solves for the mean camber surface of the wing, which will p
6、rovide therequired loading. Sensitivity studies, comparisons with other theories, and applications toconfigurations which include a tandem wing and a wing-winglet combination have been madeand are presented.17. Key Words (Suggested by Author(s)Mean camber surfaceSubsonic flowVortex-lattice methodInt
7、eracting surfacesOptimization19. Security Classif. (of this reportl 20. Security Classif. (of this page)Unclassified Unclassified18. Distribution StatementUnclassified- UnlimitedSubject Category 0221. No. of Pages18522. Price“$7.00For sale by the National Technical Information Service, Springfield,
8、Virginia 22161Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-A VORTEX-LATTICE METHOD FOR THE MEAN CAMBER SHAPESOF TRIMMED NONCOPLANAR PLANFORMSWITH MINIMUM VORTEX DRAGJohn E. LamarLangley Research CenterSUMMARYA new subsonic method has been develope
9、d by which the mean camber surface canbe determined for trimmed noncoplanar planforms with minimum vortex drag. Thismethod uses a vortex lattice and overcomes previous difficulties with chord loadingspecification. This method uses a Trefftz plane analysis to determine the optimum spanloading for min
10、imum drag, then solves for the mean camber surface of the wing, whichwill provide the required loading. Pitching-moment or root-bending-moment constraintscan be employed as well at the design lift coefficient.Sensitivity studies of vortex-lattice arrangement have been made with this methodand are pr
11、esented. Comparisons with other theories show generally good agreement.The versatility of the method is demonstrated by applying it to (1) isolated wings, (2) wing-canard configurations, (3) a tandem wing, and (4) a wing-winglet configuration.INTRODUCTIONConfiguration design for subsonic transports
12、usually begins with the wing, afterwhich the body and its effects are taken into account, and then the tails are sized andlocated by taking into account stability and control requirements. With the advent ofhighly maneuverable aircraft having closely coupled lifting surfaces, there has been anincrea
13、sed interest in changing the design order so that multiple surfaces could bedesigned together to yield a trimmed configuration with minimum induced drag at somespecified lift coefficient. Such a combined design approach requires that the mutual inter-ference of the lifting surfaces be considered ini
14、tially.Single planform design methods are available to optimize the mean camber surface,better called the local elevation surface, for wings flying at subsonic speeds (for example,ref. 1) and at supersonic speeds (for example, refs. 2 and 3). The design method pre-sented in reference 1 was developed
15、 from an established analysis method (Multhopp type),also presented in reference 1, by using the same mathematical model, but the designProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-methodsolves for the local meanslope rather than the lifting press
16、ures. In the usualimplementationof reference 1, the design lifting pressures are taken to be linear chord-wise, but must be represented in this solution by a sine series which oscillates aboutthem. An examplepresentedherein demonstratesthat corresponding oscillations mayappearin pressure distributio
17、ns measured onwings which havebeendesignedby themethodof reference 1. The methoddevelopedherein overcomes this oscillatory liftingpressure behavior by specifying linear chord loadings at the outset.The developmentapproachused in the two-planform designproblem will be simi-lar to that used for a sing
18、le planform. The analytic methodemployed,selected becauseof its geometric versatility, is the noncoplanartwo-planform vortex-lattice methodofreference 4.Thedesignprocedure is essentially an optimization or extremization problem.Subsonicmethods(for example, seerefs. 5 and6) are available for determin
19、ing the spanload distributions onbent lifting lines in the Trefftz plane, but they donot describe thenecessarylocal elevation surface. This is one of the objectives of the present methodwhich will utilize the Lagrange multiplier technique (also employedin refs. 2 and 3). Themethodof reference 4 is u
20、sedto provide the neededgeometrical relationships betweenthecirculation andinducednormal flow for complexplanforms, as well as to computethe lift,drag, andpitching moment.This paper also presents the results of precision studies and comparisons withother methodsand data. Severalexamplesof solutions
21、for configurations of recent inter-est are also presented. The FORTRANcomputer program written to perform the compu-tation is described (appendixA), alongwith details of the program input data (appendixB)andoutputdata (appendixC). Listings andtypical running times of exampleconfigura-tions are given
22、 (appendixD), anda FORTRANprogram listing is provided (appendixE).AppendixF provides details concerningthe changesneededto substitute a root-bending-momentconstraint for the basic constraint on configuration pitching-moment balance.SYMBOLSThe geometric description of planforms is basedon the body-ax
23、is system. (Seefig. 1 for positive directions.) For computationalpurposesthe planform is replaced by avortex lattice which is in a wind-axis system. Both the bodyaxes andthe wind axeshavetheir origins in the planform planeof symmetry. (Seesketch (a) for details.) The axissystem of a particular horse
24、shoevortex is wind oriented and referred to the origin of thathorseshoevortex (fig. 1). For the purpose of the computer program, the length dimensionis arbitrary for a given case; anglesassociatedwith the planform are always in degrees.(The variable namesusedfor input data in the computer program ar
25、e described inappendixB.)2Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-A/,nmFw,/, n - Fv,l, n tan _blelement of influence function matrix A, , which con-4ntains induced normal flow at lth point due to nth horseshoe vortex of unitstrength; total nu
26、mber of elements is N x N2 2AR aspect ratiofractional chord location where chord load changes from constant value tolinearly varying value toward zero at trailing edgeai,bi,c i coefficients in spanwise scaling polynomialb wing spanC Boroot-bending-moment coefficient about X-axis,C D drag coefficient
27、,Dragq_SrefRoot bending momentq_Sref(b/2)CD, o drag coefficient at C L = 0C L lift coefficient, Liftq_SrefC m pitching-moment coefficient about Y-axis,Pitching momentq_SrefCrefC N normal-force coefficient, Normal forceq_SrefACp lifting pressure coefficientc chordclCrefsection lift coefficientreferen
28、ce chordProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Fw,l,n Fv,/,nFw,l,n, Fv,l,nGinfluence function which geometrically relates inducedeffect of nthhorseshoevortex to quantity which is proportional to induceddown-wash or sidewashat slope point l (
29、see sketch (a) and also eqs. (5)and (6)sum of influence function Fw,/, n or Fv,/, n at slope point l onplanform caused by two symmetrically located horseshoe vortices,left wing panel vortex denoted by n and right wing panel vortexdenoted by N+ 1 - n (see fig. 1)function to be extremized (see eq. (19
30、)_- Nca + 0.75; i (brackets indicate “take the greatest integer“)KLnNmNcNsqo_SrefS4maximum number of spanwise scaling terms (see eqs. (25) to (27)liftpitching moment about coordinate originfree-stream Mach numbernumber of span stations where pressure modes are defined as used inreference 1maximum nu
31、mber of elemental panels on both sides of configuration; maxi-mum number of chordal control points at each of m span stations as usedin reference 1number of elemental panels from leading to trailing edge in chordwise rowtotal number of (chordwise) rows in spanwise direction of elemental panelson con
32、figuration semispanfree-stream dynamic pressurereference areahorseshoe vortex semiwidth in plane of horseshoe (see fig. 2)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-UX,Y,Zc/4X3c/4y*,z*_nfree-stream velocityaxis system of given horseshoe vortex (
33、see fig. 1)body-axis system for planform (see fig. 1)wind-axis system for planform (see sketch (a)distance along X-, Y-, and Z-axis, respectively=distance along X-, Y-, and Z-axis, respectivelyincremental movement of X-Y coordinate origin in streamwise directionmidspan x-location of quarter-chord of
34、 elemental panelmidspan _-location of three-quarter-chord of elemental panely and z distances from image vortices located on right half of plane ofsymmetry, as viewed from behind, to points on left panelcanard height with respect to wing plane, positive downlocal elevation normalized by local chord,
35、 referenced to local trailing-edgeheight, positive down/th elemental local slope in vector _z/_x) of N/2 elements (see eq. (1)angle of attack, degPrandtl-Glauert correction factor to account for effect of compressibility insubsonic flow, _1 - Moo2vortex strength of nth element in vector (F) of N/2 e
36、lementsindependent variable in extremization processProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-77A40“, 0 “_Subscripts:Ci,j,kle6incidence angle, positive leading edge up, degnondimensional spanwise coordinates, b/-“-2nondimensional spanwise coord
37、inate based on local planform semispanplanform leading-edge sweep angle in X-Y plane, degLagrange multiplier (see eq. (19)distance along local chord normalized by local chordfractional chordwise location of point where mean camber height is to becomputed (see eq. (28)dihedral angle from trailing vor
38、tex to point on left panel being influenced;measured from left panel, _ measured from right panelconstraint function (see eqs. (20) and (21); also horseshoe vortex dihedralangle in Y-Z plane on left wing panel, deghorseshoe vortex dihedral angle on right wing panel, qS = -_b, degquarter-chord sweep
39、angle of elemental panel; because of small angleassumption, also used as sweep angle of spanwise horseshoe vortex fila-ment in X-Y plane, deg= tan- 1/_-_ )canarddesignindices to vary over the range indicatedleading edgeProvided by IHSNot for ResaleNo reproduction or networking permitted without lice
40、nse from IHS-,-,-l_n associated with slope point and horseshoe vortex, reslSectively, ranging from1 to N/2L left trailing legR right trailing legroot-chord locationv vortexw wingMatrix notation:()column vectorsquare matrixXgangle of attackUr- Flow angle of attack determined at each slope point/“w-.,
41、 / v- Typical spanwise_t /_ vortex filamentVortex-lattice trailing filameSketch (a)THEORETICAL DEVE LOPMENTThis section presents the application of vortex-lattice methodology to the mean-camber-surface design of two lifting pIanforms which may be separated vertically andhave dihedral. For a given pl
42、anform, local vertical displacements of the surfaces withrespect to their chord lines in the wing axis (see sketch (a) are assumed to be negligi-ble; however, vertical displacements of the solution surfaces due to planform separation7Provided by IHSNot for ResaleNo reproduction or networking permitt
43、ed without license from IHS-,-,-or dihedral are included. The wakes of these bent lifting planforms are assumedto lie intheir respective extendedbent chord planeswith no roll up. For a two-planform configu-ration the resulting local elevation surface solutions are thosefor which boththe vortexdrag i
44、s minimized at the designlift coefficient and the pitching momentis constrained tobe zero aboutthe origin. For an isolated planform nopitching-moment constraint isimposed. Thus, the solution is the local elevation surface yielding the minimum vortexdrag at the designlift coefficient. Lagrange multip
45、liers together with suitable interpo-lating andintegrating procedures are used to obtain the solutions. The details of thesolution are given in the following five subsections.Relationship BetweenLocal Slopeand CirculationFrom reference 4, the distributed circulation over a lifting system is related
46、to thelocal slopebyr “1where the matrix LAJ is the aerodynamic influence coefficient matrix based on the panelingtechnique described in reference 4. This matrix has elements ofm FAl,n =lIFw,l,n(X,Y,Z,S,_P,dP) - v,l,n(X ,Y,Z,S,g2,dp) tan c_llwhich, because of the assumed spanwise symmetry of loading,
47、 leads to F Fw,l,n(X ,y,z,s,_ ,_b) - w,/,n(X ,y,z,s,_ ,4_)lef t paneland-I- F _ “w,l, N+l-n(x YZS_“_b)right panelFv,t,n- (x,y,z,s,_P,_b) - Fv,/,n(X,y,z,s,gT,_)lef t panel(2)(3)+ Fv,l,N+ l_n(x, y, z, s, gJ, qS)right panel (4)8Provided by IHSNot for ResaleNo reproduction or networking permitted withou
48、t license from IHS-,-,-whereFw(x,y,z,s,O ) =(y tan , - x) cos(x) 2 + (y sin 4)2 + cos2 do(y2 tan2 $ + z2 sec2 _ - 2yx tan _) - 2z cos _ sin 4_(Y + x tan _)(-J(x + s cos 20 from a modified version ofthe program. The results from N c 20 are provided so that the solution convergenceand its rate can be examined. The table clearly shows that the results of the presentmethod are more positive than, but tend toward, the exact ones with increasing values ofN c but at a slower rate as Nc increases. Though not shown herein, it was observedthat the result
