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N 6 4 ^775 Aiaa Transport Aircraft Design https://ntrs.nasa.gov/search.jsp?R=19640017838 2020-03-24T06:29:34+00:00Z N 6 4 ^775 - (ACCESSION NUMB ) B 0 S )CODE) IPAGES) -J ( I. )CATEORY) (NASA CR OR EMX OR AD NUMBER) A NUMERICAL METHOD OF ESTIMATING AND OPTIMIZING THE SUPERSONIC AERODYNAMIC CHARACTERISTICS OF WINGS OF ARBITRARY PLANFOBM Wilbur D. Middleton and Harry W. Carlson NASA Langley Research Center Langley Station, Hampton, Va. AIAA Paper No. 64-590 AIAA TRANSPORT AIRCRAFT DESIGN & OPERATIONS MEETING SEATTLE, WASHINGTON AUGUST 10-12, 1964 rst publication rights reserved by American Institute of Aeronautics and Astronautics, 1290 Avenue of the Americas, New York, N. V. 10019. Abstracts may be published without permission if credit is given to author and to AiAA. (Price - AIAA Member 50, Non-Member $1.00). A NUMERICAL METHOD OF ESTIMATING AND OPTIMIZING THE SUPERSONIC AERODYNAMIC CHARACTERISTICS OF WINGS OF ARBITRARY PLANFORM Wilbur D. Middleton and Harry W. Carlson Aerospace Engineers, Supersonic Mechanics Section Full-Scale Research Division NASA Langley Research Center Introduction Symbols The continuing and recently accentuated search A(L,N) leading-edge grid element weighting for aerodynamically efficient supersonic transport factor designs necessitates not only the fullest use of existing technology, but also requires the develop- b wing span ment of new analytical methods of evaluating poten- tially efficient configurations. Because of the CD drag coefficient large portion of total airplane drag associated wita the generation of lift, there is need for a CD,min zero-lift drag coefficient of flat-wing thorough study of design concepts offering the pos- configuration sibility of reduction of this drag component. drag coefficient due to lift, The wing planforn, which is of primary impor- CD - CD,mifl tance in its effect on drag at lifting conditions, in past studies has been largely restricted to pitching-moment coefficient straight line leading and trailing edges. Such Cm limitations have resulted mainly from a lack of analytical methods for estimating the aerodynamic m,o,F zero-lift pitching-moment coefficient characteristics of wings of arbitrary planform. of flat-wing configuration This deficiency may now be overcome by application of modern digital computers to the solution of tAi = Cm - linearized theory integral equations for wing plan- forms which may employ curved or cranked leading CL lift coefficient and trailing edges. Computer programs such as those discussed in this paper greatly expand the lift-curve slope per degree angle of possibilities for the development of truly effi- CL,a, attack cient supersonic airplane configurations. .- C, pressure coefficient U' This paper illustrates the application of numerical methods to wing camber surfaces of arbi- lifting pressure coefficient trary planform, to obtain the surface shape tC required to support a specified pressure distribu- tion or, inversely, to obtain the pressure distri- overall length of wing measured in bution on wings of specified shape. Two special streamwise direction cases are noted and demonstrated: in the direct solution, the surface shape corresponding to an L,N designation of influencing grid optimum combination of loadings (for least drag at elements a specified lift) may be obtained; in the inverse solution, the pressure distribution on a flat wing designation of field-point grid of arbitrary plariform may be obtained. elements Used in combination, the numerical methods M Mach number allow the determination of linear theory drag-due- to-lift polars and lift-moment relationships for R average value of influence function wings of arbitrary planform which may employ any within a grid element specified surface shape. With attention given to the real flow considerations that limit the appli- x,y,z Cartesian coordinate system, x-axis cation of linear theory,- studies of the aerody- streamwise namic characteristics of arbitrary wing planforms with various surface shapes may be analytically Xcp x-coordinate of wing center of pressure conducted. z c camber surface ordinate To illustrate the use of the method, a set of examples is presented for a typical planform - zc,le series. The agreement obtained between the numer- ical method and experiment is shown in a second set wing angle of attack, deg of examples, employing both flat and warped wing surfaces. a;Lt. CAS'Lc FILE I. H fikl^^ ' = - 1 the summation term, which adds directly to the pre- scribed lifting pressure coefficient at the field dummy variables of integration for x element to define the necessary surface slope. In and y, respectively this equation, the factor A(L,N) accounts for partial grid elements, being equal to the element designates a region of integration length in the x-direction. bounded by the wing planform and the fore Mach cone from the point x,y Equation (2), a rearrangement of equation (1), provides a solution to the inverse problem, that of A wing leading-edge sweepback angle solving for the lifting pressure coefficient corre- sponding to a specified surface shape. 3 In this Subscripts: equation the summation term utilizes previously determined lifting pressure coefficients, which are computed following a prescribed order of calcula- F tion, i.e., from apex aft. With this procedure, and utilizing the fact that R(r,L*) is zero, no W unknown pressure coefficients arise in the numeri- designates various drag-componènts cal summations. WF (see fig. 7) The wing surface slope variation required in FF equation (2) is provided by supplemental calcula- tions, which consist of determining the streamwise inclination of all grid elements from a set of cam- Discussion ber surface ordinates. The flat wing at a small angle of attack represents a special case of equa- A typical wing planform described by a rec- tangular Cartesian coordinate system is illustrated tion (2), in which L*,N*) is a constant. in figure 1. Overlaid upon the wing planform is bX the grid system used in the numerical solution of the linear theory integral equation. A mosaic of The precision of the numerical method in whole and partial grid elements approximates the defining the theoretical surface shape required to actual wing planform and surface shape. In support a specified pressure distribution on wings accordance with the assumptions of linear theory, of arbitrary planform has been illustrated in an the wing has negligible thickness, and lies essen- NASA report- 2 Similar illustrations involving the tially in the z = 0 plane. inverse solution are contained in an NASA prospec- tive report, 3 from which a typical example is In the numerical approach, the grid elements, included here to show the type of detailed pressure identified by L and N, are arranged such that distributions obtained from the numerical method, L is numerically equal to x and N is numeri- and the departures from more rigorous solutions. cally equal to J3y, where x and Ay take on only In figure 3 numerical solution pressure distribu- integer values. Partial grid elements along the tions for a flat double-delta planform at two Mach wing leading and trailing edges are used to permit numbers are compared with results from a super- a closer approximation to the actual wing planform. position analysis. Agreement between the two The grid system of figure 1 is rather coarse for pressure distributions is generally quite good, illustrative purposes; in actual usage, many more although the numerical solution does not produce grid elements are employed. the sharp pressure peaks along Mach lines that characterize the superposition analysis. Compari- With respect to a specified field point x,y, sons of total wing lift coefficient and center of the upstream region of influence T (bounded by pressure between the two solutions show reasonably the fore Mach cone from x,y and the wing leading good agreement as follows: edge), is approximated by the shaded grid elements of figure 1. Each of these elements has associ- M = l.kl M = 1.667 ated with it an influence factor R which relates the effect of the element and its average pressure CL,m xcp/ l CL,a xcp/l to the surface slope required to obtain _a specified lifting pressure at x,y. The factor R is deter- Superposition 0.0511 0.682 O.O461 Not mined from an approximate solution to the linear analysis given theory integrals over the region bounded by the individual elements. Numerical .0507 .687 .04149 .686 solution The variation of R within the fore Mach lines from the field point and the corresponding The use of the numerical methods to obtain grid element (L*,N) is illustrated in figure 2. camber surfaces for wings having a prescribed For a set of elements at a constant L* - L value, loading, and pressure distributions for flat wings the sum of the R values is zero, the single neg- of the same planform is illustrated in figures ative value at N* = N balancing all the others. and 5 for a series of wings having the same span Where L* = L, only a single element is contained and length. The three wings have delta, ogee, and within the Mach lines, and R = 0. blunt ogive planfornis, oriented with respect to the apex Mach lines such that the sweepback parameter The basic equation relating the required sur- (0 cot A) of the delta wing is 0.70. face slope at x,y to a prescribed pressure dis- tribution, written in the form of the numerical In figure 4, the surface shape and loading solution, is equation (1) of figure 2.2 The effect distributions are shown for an optimum combination of the upstream elements within T is included in (having least drag at a specified lift coefficient) of a uniform, linear spanwise, and linear chordwise optimum or other specified loading is desired. The loading, calculate'd by the numerical method.
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