1、I.,- 11.AERODYNAMICSFOR AERONAUTICSTECHNICAL NOTE 3105SLENDERHAVING SWEPT TRAILING EDGESBy Harold MirelsLewis FlightPropulsion LaboratoryCleveland, Ohio“,WashingtonMarch 1954.COMBINATIONS.-y -km am.= . . . .-. . Provided by IHSNot for ResaleNo reproduction or networking permitted without license fro
2、m IHS-,-,-.I tuHL! nhY nh-, tiNACA TN3105TABLE OF coIiTm$TsPageSUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lINTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1SECTION 1 -BASICCONCEFTS . . . . . . . . . . . . . . . . . . . . . 3l.l EquationofMotion. . . . . . . .
3、. . . . . . . . . . . . . 31.2 Asymptotic FormofCrossflow . . . . . . . . . . . . . . . . 41.3Lift, Drag, and Moments . . . . . . . . . . . . . . . . . . 61.4 Symmetry inPlanarProblems . . . . . . . . . . . . . . 10SECTION 2 - GENERATINGFUNCTIONS . . . . . . . . . . . . . . . . ll2.1Evaluation of Br
4、anch Points . . . . . . . . . . . . . . ll2.2 Behavior of Flow Near Boundary Edges . . . . . . . . . . . . 122.3 Detemnination of GeneratingFunctions . . . . . . . . . . 14SECTION 3 -INTEGRAL EXl?RXSSIONSFOROW FIELD . . . . . . . . . 2O3.1 Inteal Expressions Direct Problem) . . . . . . . . . . . 203
5、.2 IntegralExpressions Inverse Problem) . . . . . . . . .213.3 IntegralEquations (DirectProblem) . . . . . . . . . . . . 22SECTION 4 - TOFSWEPl? WINGS (DIRECTPROBIJIM) . . . . . . . .234.1 Load Distribution . . . . . . . . . . . . . . . . . . .234.21ntegral Equations for S . . . . . . . . . . . . .
6、. .264.3 Limiting Solutions of Integral Equations . . . . . . . . . . 294.4 Numerical Solutions . . . . . . . . . . . .30SECTION 5 - ROLL13JGSWEXTG (DCT PROBLEM) . . . . . . . . .315.1 Load Distribution and Rolling Moment . . . . . . . . . . . . 315.21ntegral Equation for R . . . . . . . . . . . . .
7、 . 335.3 Limiting Solutions of IntegralEquation . . . . . . . . . . 335.4 Numerical Solutions . . . . . . . . . . . . . . . .33SECTION 6 - PITCEUNG SWEPT WING (DCT PROBLEM) . . . . . . . . . . 356.1 Load Distribution . . . . . . . . . . . . . . . . . . . 356.21ntegrsJ_Equation for Q. . . . . . . . .
8、 . . . . . . . . .366.3 lXmiting Solutions of IntegralEquation . . . . . . . . . . 376.4 Numerical Solutions . . . . . . . . . . . . . . . . 38SECTION 7 - LIl?IOF SWEFT WINGS (INVERSEPROWM) . . . . . . 4O7.1 Determination of Crossflow . . . . . . . . . . . . . . . 407.2 Equation of Trailing Edge . .
9、 . . . . . . . . . . . . . . .427.3 Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43-. .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ii NACA TN 3105PageSECTION 8 - ROLIZNG SWEPT WING (DIVERSEPROBIWM ) . . . . . . . .448.1 Determinat
10、ion of Crossflow . . . . . . . . . . . . . . . . 448.2 Equation of Trailing Edge . . . . . . . . . . . . . . . . . 458.3 Rolling Moment . . . . . . . . . . . . . . . . . . . . 45SECTION 9 - PITCHING SWEFT WING ( PROBLEM) . . . . . . .469.1 Determination of Cros6fluw . . . . . . . . . . . . . . . . 4
11、69.2 Equation of Trailing Edge . . . . . . . . . . . . . . . . . 469.3 Pitching Moment . . . . . . . . . . . . . . . . . . . . 47SECTION 10 -WING-BODY COMKUWTIONS. . . . . . . . . . . .4710.l Joukowski Transformation . . . . . . . . . . . . . . . . . 4810.2 Lift of Swept Wing on CylindricalBody . .
12、. . . . . . . . 50SECTION Xl_- UNSTEADY TWO-DIMENSIONALINCOMPFUZSSIB13FLOWS . . . . . 52ll.l General Considerations . . . . . . . . . . . . . . . . . . 5211.2 Two-DimensionalAirfoils . . . . . . . . . . . . . . . . . 55.3 Wagner Problem . . . . . . . . . . . . . . . . . . . . 56SECTION 12 - CONCLUDI
13、NG REMARKS . . . . . . . . . . . . . . . .58APPENDIXESA- SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . 59B -FINITE PART OF IMPROEER INTEGRALS . . . . . . . . . . . .63c - ELLIPTIC INTEGRALS . . . . . . . . . . . . . . . . . . . . . . 65D- LIMITING SOLUTIONS OF INTEGRAL EQUATIONS . .
14、. . . . . . . . . 68E- COMPARISOI?WIm m 13 . . . . . . . . . . . . . . .72REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 73FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . 76. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS
15、-,-,-NATIONALAXRODYNAMCCS OF,ADVISORY COMMITTEEFOR AERONAUTICSTECHNICAL NOTE 3105SLENDER WINGS AND WING-BODY COMBINATIONSHAVING SWEPT TRAILING EDGESBy Harold ltlrelssLIMMARYA general method, based on two-dimensionalcrossflow concepts, ispresented for obtainingthe lift and moments on highly swept win
16、gs.Emphasis is placed on obtaining solutionsfor wings having swepttrailing edges. The methd is applicable for all problems where thevelocity boundary conditions can be made homogeneousby differentia-tion in the streamwiseor spanwisedirections.Lift, roll, and pitch solutions,for highly swept wings, a
17、re pre-sented. Both direct problems (where the plan form is given) and in-verse problems (where the shed vortex sheet is given) are considered.The solutions of the direct problems are expressed in terms of func-tions which are evaluated from integral equations. Some limitingsolutions of the integral
18、 equations are indicated. Numerical resultsare given for wings having paraILel leading and trailing edges.The transformationof a wing-body problem to an equivalent iso-lated wing problem is discussedand the application for finding thelift of a wing-body combinationis indicated.Application of the met
19、hod forsolvingunsteady two-dtiensionalincompressibleflow problems is also indicated. In particular, theWagner problem is formulated in terms of the techniques developedherein.INTRODUCTIONIn 1924, Muand inv - ,2)3/2(4.1.4)PJwhere indicates the infinitepart of the improper integral (appendixB). Integr
20、atim of equation (4.1.4)yields (see appendix C)(4.1.5)3Strictly speaking, equation (1.1.2)is applicablewhen the bound-ary conditionsare satisfied on the wing surface. When the boundaryconditions are satisfied in the z = O plane, as is done herein, theappropriate expressionfor pressure isP- PO=-PO (
21、)uuo+mwuo+:for a configurationat angle of attack a. At any rate, the loading isproportional to u, for a wing of zero thickness, since sll the otherterms are symmetricwith respect to z.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3105 25whe
22、re F(*,k) and E(t,kt) are incompleteeiptic integrals of thefirst and second kind, respectively,with smplitude j3and moduluskt.These have the values(4.1.6)To satisfy the condition of zero loading in the wake, equation (4.1.5)is set equal to zero for y = yl. This gives a relation between aud A2. If S,
23、 a function of x, is intrduced according to .()W2 A. +A2y22z s= Y2(Y22 - Y12)then equation (4.1.5)becomeswhere K and E* me complete elliptic integrals of first and Eecondkind with modulus k!. Equation (4.1.7)describes the spanwisemri-ation of loading onas a scd.e factor.the wing. The unlmown functio
24、n S appears onlyThe generatingfunction can now be written as16A/(P - Y22W - Yf)Fran equations (1.3.4)and (1.2.5)and the asymptotic(4.1.8),the lift per unit x is, in terms of S,From equations (4.1.7)and (1.3.11the suction forceedge isdFx ()Y22 - Y12= _Q2 22 Y2The function S remains to be determined.(
25、4.1.8)form of equati(4.1.9)at the leading(4.1.10).- . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-26 NACA TN 31054.2 Integral equationsfor S. - The function S is determinedfran equations (3.3.1)or (3.3.2). Each of these equations is con-sidered
26、separately. -(a)Equation (3.3.1): Substitutioninto equation (3.3.1)yields .oy=l.p.i.uo/*(4.2,PJ 0(4.2.lb)where b is the value of y2 at x . c and y b is assumed. Thepath of inteation for equations (4.2.1)is indicated in figure 9(a).Transformingfrom to y2 as the miable of integration givesFran equatio
27、n (4.2.2)it is found that S +1 for y2b. Thus the result isProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3105 27L 5YY2o= (Et y2 . Y& W )Y1 dyl+ .(Y2-Y2)(yz - y22) “ SY2 6-Y2W2b(4.2.3)It is noted that the integrand of equation (4.2.3)contains
28、both theunknown function S and its derivative.(b)Equation (3.3.2: Substitutinginto equation (3.3.2)andnoting that y b yield-11-+11-= I.P.iL+ 2(4 $s+K2-Y12HC2 -Y22).(4.2.4)Z=oside of equation 4.2.4)asIt is desirable to express the right-handa function of x plus a function of y so that the functions o
29、f xor y can be equated. Taking the imaginarypart of the right-handside and utilizing the finite-part technique result in an area ofintegrationas indicated in figure 10(a). This integration canbedecomposed into two separate integrationsas indicated in figures10(b) and lO(c). The right-hand side of equation (4.2.4)thenbecomes- . . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
