TRANSONIC WING DESIGN PROCEDURE

Abstract: The conventional steps of defining performance requirements of an aircraft wing and conducting general sizing of the wing are followed by the new steps of determining the aerodynamic sweep angle of the wing with respect to an actual location of a shock wave on the wing. A two dimensional Mach number for the wing airfoil is calculated by multiplying the three dimensional Mach number by the cosine of the aerodynamic sweep angle. A two dimensional lift coefficient for the wing airfoil is calculated by dividing the three dimensional lift coefficient by the square of the cosine of the aerodynamic sweep angle. Airfoil shape in two dimensions is determined on the basis of the two dimensions is determined on the basis of the two dimensional Mach number and the two dimensional lift coefficient. The shape of the wing in three dimensions is then defined by placing the airfoil in the wing along an arc constructed by skewed chord lines perpendicular to local sweep lines of the wing at a series of locations along a chord of the wing.

Description: TRANSONIC WING DESIGN PROCEDURE The present invention relates to aircraft wing design procedures. More particularly, the invention relates to procedures for designing high performance wings for transonic operation. Transonic refers to that aircraft speed regime below the speed of sound where flow expansion (and acceleration) about an aircraft results in supersonic flow regions and thus a mixed (subsonic-supersonic) flow design problem with embedded shock waves. Typically, once performance requirements have been defined, a general sizing study is conducted to define the basic wing shape (sweep, taper, thickness and aspect ratio) . A two dimensional airfoil design is then developed in accordance with the designer's experience, iterative wind tunnel testing, or with the aid of a computer using numerical and/or analytical techniques. The two dimensional design is then used as the basis for three dimensional wing shaping. It is almost always the case, that when the three dimensional wing, developed from the two dimensional wing section, undergoes wind tunnel or flight testing. The performance characteristics do not coincide with the original performance predictions based on two dimensional section characteristics. Although various reasons are often postulated for why a particular design has failed to meet the performance requirements, such explanations are usually not helpful in providing information as to how the wing can be modified for better performance. Instead, it is usually necessary to experimentally change the design repeatedly, and to perform repeated wind tunnel testing to arrive at a satisfactory design. Such trial and error approaches are extremely costly, do not always produce a wing having acceptable performance, and raise significant issue concerning the usefulness of conventional aircraft wing design methods. Three-dimensional analytical or computational techniques have naturally evolved, but an inability to predict drag levels, a requirement for estimating wing performance, compromises implementation for wing design. In accordance with the invention, a wing is designed by first using the conventional steps of defining performance requirements and using the requirements to conduct sizing of the wing. According to the invention, the next step is determining the aerodynamic sweep angle of the wing with respect to an actual location of a shock wave on the wing. This step is followed by calculating a two dimensional Mach number for the wing airfoil section by multiplying the three dimensional Mach number by the cosine of the aerodynamic sweep angle and calculating a two dimensional lift coefficient by the square of the cosine of the aerodynamic sweep angle. Two dimensional airfoil shape is then determined on the basis of the two dimensional Mach number and the two dimensional lift coefficient requirement. The shape of the wing in three dimensions is then defined by placing the airfoil in the wing along an arc constructed of skewed chord lines perpendicular to local sweep lines of the wing at a series of locations along a chord of the wing. According to the invention, -the method set forth above may be performed for each of a plurality of locations along the span of a wing from the root to the tip. The invention also encompasses evaluating the performance of the resulting wing in three dimensions by two-dimensional flow simulation techniques. The steps of evaluating and modifying the design may be performed repeatedly until desired predicted performance characteristics are achieved. In order that the invention may be readily carried into effect, it will now be described with reference to the accompanying drawings, wherein: FIG. 1 is a two-dimensional section of an airfoil shape in the. X.Y plane; Fig. 2 is a conceptual view of a portion of an infinite, sheared, non-tapered wing panel used in conventional design analysis; Fig. 3 is a conceptual view of a conventional, finite tapered wing; Fig 4 is a conceptual view of a finite, tapered wing with a "bowed" airfoil cut, according to the invention. FIGS. 5A to 5F illustrate pressure distribution correlation using two dimensional airfoil analysis coupled with sweep-taper theory, for the wing of FIG. 5G, for a series of Mach numbers, according to the invention; FIG. 6 is a table setting forth the conditions which obtain in FIG. 5A to FIG. 5F; and for comparison purposes the comparable values using simple sweep theory; and FIG. 7 is a graph showing the drag prediction improvement achieved by using sweep-taper theory according to the invention. It is well known that although a jet aircraft may be traveling at a speed lower than the speed of sound, the flow of air over its wings may include embedded supersonic regions. Aircraft speeds in the transonic region cause a shock wave to exist over the wing. The shock wave causes increasing levels of drag as speed is increased, thus providing a limit to aircraft performance. It is therefore desirable, when designing a wing, to have a chordwise pressure profile with small and gradual changes in pressure, rather than the sharp, large changes indicative of a severe shock wave. Conventional wing design takes into account only ' two dimensional airflow effects with a simple adjustment for wing sweep. According to such conventional procedure the wing is designed on the basis of an airfoil shape 10 following contours defined by a set of a co-ordinates in the X.Y plane (FIG. 1) . However, a wing must be defined by contours in three dimensional space. Wings used in the transonic region have taper and sweep. Chord length taper is used for structural reasons to reduce wing weight. Sweep reduces compressibility drag by decreasing the air flow velocity magnitude that goes through the shock wave. In two dimensional airfoil design, an attempt is made to contour the section pressure profile so that the shock wave is as weak as possible. Section thickness, lift and Mach number define the limits of how closely this ideal is reached. However, the range equation generally indicates that maximum range may be obtained if the aircraft is flown at a speed where a shock wave of some magnitude is created. Designed two dimensional characteristics are generally lost when sweep and taper are introduced. This degrades the performance of the wing, as the pressure profile is different than as designed and drag is generally higher than anticipated. FIG. 2 illustrates, conceptually, a portion of an infinite swept-back wing panel 11 disposed at a sweep angle A with respect to a direction perpendicular to the velocity of air flow oo over the wing. According to simple sweep theory, the velocity can be resolved into two components. A first component is one parallel to the wing panel. This component is not illustrated because it is assumed to have no appreciable effect on wing performance. Of greater interest is the normal component M^ of the air flow velocity perpendicular to the leading edge 12 (and trailing edge 14 which is parallel to leading edge 12) of wing panel 11. Only I , determines the nature of the pressure field on wing panel 11. Thus, simple sweep theory would permit the designer to analyze the airfoil taken from the wing in a straight line in a direction parallel to M-_, analyze the wing at a speed of Moocos / determine the pressure 'field, and scale the 2 pressure field back by dividing by cos A to define the pressure field on wing panel- 11. Simple sweep theory permits the designer to work in two dimensions, but to convert the results back to a three dimensional environment via sweep angle. For example, by extracting an airfoil normal to the sweep line, analyzing at two dimensional conditions given by simple sweep theory, and converting the resultant pressure back to the streamwise direction, it follows that where: α) C = pressure coefficient; and P Λ = angle of sweep. However, in simple sweep theory, no matter what the location of the shock wave on wing panel 11, the sweep angle is always assumed to be the same; it is linked to a geometric feature. Thus, the concept of simple sweep theory must be extended to finite, tapered wings wherein the sweep of the wing varies between the leading edge and the trailing edge. FIG. 3 illustrates, conceptually, the change in sweep angle for a finite, tapered wing 16 from an angle of A at the leading edge 18, to an angle of Λ_ at the trailing edge 20. The sweep of the shock wave on the wing depends on its position thereon. As the aircraft is maneuvered, causing the wing angle-of-attack to increase to Mach number to increase, the shock wave will develop and move back on the wing, reducing the sweep angle and increasing drag. Conventional design procedures generally call for an assumption that the location of the one quarter chord spanwise generator 22 of wing 16 determines the sweep angle for the two dimensional to three dimensional conversion. This geometric or planform sweep angle approach does not take into account the aerodynamics of the wing and contributes to deviations in performance from the initial design criteria when a wing is tested in a wind tunnel or in flight. Conventional wing design, and wing design according to the present invention, starts with the initial step" of defining wing performance requirements. Once performance requirements have been adequately defined, the next step in both conventional wing design, and wing design according to the present invention, is sizing of the wing. The basic parameters of wing area, loading, sweep, taper, thickness of the wing and aspect ratio necessary to meet the defined performance are determined. After such determination, the next step in conventional wing design is to determine the shape of the two dimensional airfoil on the basis of the defined parameters. There are many approaches to such detailed contouring of the two dimensional airfoil to achieve a particular pressure characteristic by analytical or numerical procedures. These approaches are well known in the art. A general discussion is provided by A. B. Haines, in "Computer-Aided Design: Aerodynamics, 1 ' The Aeronautical Journal, Vol. 83, No. 819, March, 1979. According to a first aspect of the present invention, the conditions under which two dimensional design is carried out are defined. For a certain wing lift requirement, it is necessary to define the lift requirement for the airfoil and the design Mach number for the airfoil. According to the present invention, the wing is designed on the basis of the discovery that the effective sweep of the wing is tied to the location of the shock wave on the wing. This location can be "designed in" as may be most advantageous for the use for which the wing is intended. This avoids the prior art difficulty of designing an airfoil shape ideal for conditions which the wing never actually experiences during use. Thus, according to the invention, the actual aerodynamic sweep angle of the wing due to the actual desired location of the shock wave on the wing is chosen or defined. After this aerodynamic sweep has been determined, the speed or Mach number for the wing, in two dimensions, is calculated by multiplying the three dimensional speed by the cosine of the aerodynamic sweep angle: M 2-D = M 3-D cos A Shockwave (2) The coefficient of lift of the airfoil is defined by dividing the three-dimensional lift coefficient by the square of the cosine of the aerodynamic sweep; (3) ° L 2-D " CL 3- D ' COs2 Λ Shockwave The two dimensional airfoil is then designed using cut-and-try, computer-aided design, or iterative design procedures, starting with the desired pressure field (shock wave location) , as is well known in the art. For example, experience with the hodograph design method suggests that supercritical airfoils may be designed by satisfying the following empirical relationship: M T 1iT0nT + t/c - 0.95 (4) Standard solutions generally require that an aerodynamic influence coefficient matrix be developed, inverted, and multiplied through by the column vector representing boundary conditions for the wing mean surface according to: P w - (5 ) where : P = wing panel pressure difference WW = aerodynamic influence coefficient matrix ^ z c = airfoil/wing camber slope; and ΈΓ a = angle of attack The program may be modified to run in an inverse mode. The influence coefficient matrix is thus multiplied through by desired spanwise and chordwise load distributions before inversion wherein: Requested loading is chordwise loading of the airfoil, scaled for sweep, across the entire wing span. A subsonic condition is selected to insure that panel method predictions are not adversely compromised by compressibility effects. A second aspect of the invention relates to how the three dimensional wing is shaped based on the designed two dimensional airfoil, so that the three dimensional wing maintains the desired two dimensional pressure field, thus producing the required lift and minimum drag. FIG. 4 illustrates, conceptually, the manner in which the two dimensional airfoil is placed in the three-dimensional wing 16. For the swept-back wing of FIG. 3 or FIG. 4, the effective velocity increases as the flow progresses downstream. In addition, the effective flow component acts on surface geometry that cannot be obtained by any straight line cut through the wing. The equivalent two dimensional airfoil in an arbitrary wing is generated by constructing chord lines perpendicular to the local sweep lines at a number of locations along the wing chord. Expressions for constructing the effective airfoil shape can be obtained using the ratio of bowed chord C 2 _ D and streamwise chord length: Cs.treamwise _X_ and • 2-D = 'streamwise '2-D streamwise (8) where: X = Abscissa of point in X, Y plane Y = ordinate of point in X, Y plane C = chord length of wing section = local sweep of the spanwise generator loc /ILE = sweep at leading edge = sweep at trailing edge Thus, the three dimensional shape of wing 16 of FIG. 4 is defined by placing the airfoil in the wing along an arc 24 constructed of skewed chord lines 26A, 26B, 26C and 26D perpendicular to local sweep lines, which are leading edge 18 and spanwise generators 28B, 28C and 28D, respectively, of the wing 16 at a series of locations along a chord of wing 16. Generally, it will also be required that the necessary twist distribution be included in the wing geometry. The root of a swept-back wing will generally have a positive angle, while the tip will have a negative angle relative to the root. After the wing shape has been defined by placing the airfoil in the wing, streamwise cuts can be taken from the designed shape, in a manner known in the art to define the wing profile for lofting the wing. However, since the designed airfoil is found along are 24, the shock wave will appear at the proper place, and airfoil section performance of the three-dimensional wing will be very close to the performance of the two dimensional airfoil. This is in sharp contrast to prior art design procedures wherein the designed airfoil will generally be placed in the wing perpendicular to one particular spanwise generator, such as, for example, the one quarter chord spanwise generator. It will be understood that, in general, the nature of tne designed airfoil will change from wing root to wing tip. For example, the root airfoil may have a thickness in the order of twelve percent, while the tip airfoil may have a thickness in the order of eight percent. In this case, the design procedure outlined above will be repeated for several stations along the wing span, as, in effect, different airfoils are present at the different stations. Thus, it is understood that for more complex wing geometries, where the airfoil shape must change along the wing span, the design procedure of the present invention will be repeated for a number of points along the wing span. FIGS. 5A to 5F illustrate a correlation study which validates the design procedure according to the invention as set forth above. The effective two dimensional airfoil is extracted from an experimental wing illustrated in FIG. 5G having planform 30 with a leading edge sweep angle of 40° and a wing taper ratio ofλ= c tjr / c rθ ot = °- 4 designed for an equivalent 2-D condition of = 1.7 at M = 0.689 using equations 7 and 8 above. Table 1 (FIG. 6) provides the chord location and sweep angle of the wing shock wave for the six cases of FIG. 5A to FIG. 5F taken along streamwise chord 32. These sweep angles are the actual aerodynamic sweep angles used to convert with equation 1, the free stream Mach number (M = 0.90) to the effective normal, or 2-D Mach number. In each case, the airfoil lift coefficient scaled for sweep, was required to match the wing lift coefficient. When no shock wave exists (M = 0.70), the quarter-chord sweep angle is used. As noted in FIG. 5A to FIG. 5F, agreement between the pressure field of the experimental three dimensional wing and, for example, the two dimensional Korn-Garabedian analysis coupled with sweep-taper theory, according to the present invention, is good. According to a third aspect of the present invention, it is possible to improve the performance evaluation of a designed three-dimensional wing. Using conventional design evaluation procedures, including flow simulation techniques, the two dimensional drag is a fixed number with lift which is simply added to other wing drag components to predict total wing drag. According to the present invention, it is recognized that as the shock wave develops under more severe conditions, it moves back on the wing, lowering the effective sweep angle and increasing drag. Thus, the wing does not really have a geometric sweep, but an aerodynamic sweep which changes in magnitude with the speed manuever condition of the wing. Airfoil profile drag C__ can be computer as follows: «3 C Dp = (C d 2 _ D " C d > cos Λ Shockwave < 9 > U " " 2-D The total drag C-. is given by: where: Cj = two dimensional coefficient of drag α 2-D Cj = two dimensional coefficient of f -D frictional drag C D = configuration zero - lift drag; and C.. lif -induced drag FIG. 7 illustrates the drag prediction improvement achieved for the experimental wing by implementing sweep-taper theory according to the present invention. It will be noted that for a given lift coefficient, actual drag found by wind tunnel experiments is higher than that predicted when simple sweep theory provides the airfoil profile drag levels. However, when sweeptaper theory according to the present invention is used to provide airfoil profile drag levels, agreement between theory and experiment is excellent. It is possible that the computed three dimensional wing performance may not be as close as is desired to the predicted performance, even after the design procedure according to the present invention has been used, as it may require more than one attempt to find the optimum combination of shock location (effective sweep) , pressure field, and drag. In that event, the design can be changed. If an analysis technique such as the Bauer-Gar bedian-Korn-Jameson (BGKJ) code (or other codes as set forth in the paper by Haines mentioned above) indicates that the desired performance has not been obtained, the position of the shock wave or the design speed or airfoil shape, for example, can be changed. Thus, there can be numerous design iterations leading to a selected design, before any actual prototype is constructed and tested in a wind tunnel. The design procedure of the present invention may be used for a wide variety of aircraft wings to assure that two dimensional design performance is obtained on a three dimensional wing. Almost all aircraft now in service and planned in the future, with the exception of supersonic transports and supersonic regime fighters, operate in the transonic region, during almost all flight time except for take-off and landing. Thus, the present invention may be applied to the design of wings for transports such as airliners and executive jets, and to highly maneuverable fighters which operate in the transonic region by merely " changing the design requirements in a manner well known in the art. Although shown and described in what is believed to be the most practical and preferred embodiment, it is apparent that departures from the specific method and design described and shown will suggest themselves to those skilled in the art and may be made without departing from the spirit and scope of the invention. I, therefore, do not wish to restrict myself to the particular construction described and illustrated, but desire to avail myself of all modifications that may fall within the scope of the appended claims.

Claims: WHAT IS CLAIMED IS: 1. In a method for designing a transonic aircraft wing including the steps of defining performance requirements of the wing and using said requirements to conduct general sizing of the wing, the improvement comprising: determining aerodynamic sweep angle of the wing with respect to an actual location of a shock wave on the wing; calculating a two dimensional Mach number for the wing airfoil by multiplying the three dimensional Mach number by the cosine of the aerodynamic sweep angle; calculating a two dimensional lift coefficient for the wing airfoil by dividing the three dimensional lift coefficient by the square of the cosine of the aerodynamic sweep angle; determining airfoil shape in two dimensions on the basis of the two dimensional Mach number and the two dimensional lift coefficient; and defining section shape of the wing in three dimensions by placing the airfoil in the wing aiong an arc constructed of skewed chord lines perpendicular to local sweep lines of the wing at a series of locations along a chord of the wing. 2. The method of Claim 1, wherein the aerodynamic sweep angle is determined by the actual location of the Shockwave. 3. The method of Claims 1 or 2, further comprising the steps of calculating a twist for the wing as a function of distance from a root thereof, and modifying the shape of the wing in three dimensions in accordance with the calculated twist. 4. The method of any one of the preceding claims, further comprising performing said method for each of a plurality of locations between a root and a tip of said wing. 5. The method of any one of the preceding claims, further comprising evaluating the performance of the wing in three dimensions by two-dimensional flow simulation techniques to produce predicted performance results; and modifying the wing in accordance with said predicted performance results. 6. The method of Claim 5 wherein said steps of evaluating and modifying are performed repeatedly. 7. The method of Claims 5 or 6 wherein said step of evaluating the performance includes estimating the profile drag in accordance with the relationship: .__3 Dp " (C d« . " C d. > cos Λ Shockwave and "U r 2-D C D = C Do + C Di + C Dp where: C γ . = total drag C~ = airfoil profile drag Cj = two dimensional coefficient of drag α 2-D C, - two dimensional coefficient of frictional f 2-D drag C Do - configuration zero - lift drag; and C_. = lift-induced drag. 8. The method of any one of the preceding claims, wherein said step of placing the airfoil in the wing is performed in accordance with the relationship: + (1 1 - ( - C cos Λ-l,oc (T + tan Aloc tan Λ χE ) and Y 2-D - streamwise '2-D 'streamwise J where: X = Abscissa of point in X, Y plane Y = ordinate of point in X, Y plane C = chord length of wing section Aloc = local sweep of the spanwise generator A LE = sweep at leading edge A TE = sweep at trailing edge

Inventors: BOPPE CHARLES W, BOPPE, CHARLES, W.
Applicants: GRUMMAN AEROSPACE CORP, GRUMMAN AEROSPACE CORPORATION
Application year: 1987
Main classification: B64C3
IPC codes: B64C3