1、NASA Contractor Report 35 15 NASA CR 3515 *; wing axial force coefficient wing normal force coefficient wing drag coefficient wing lift coefficient wing pitching-moment coefficient pressure coefficient pressure coefficient on the cambered wing at 0“ angle of attack pressure coefficient on the flat w
2、ing of 1“ angle of attack component of C due to pure camber loading (the contribution with no leading-edgE:ingularity) component of C due to flat wing loading (the contribution with a leading%ge singularity) location correction factor for program perturbation velocity (see equation 8) normal force i
3、ntegration factor for basic pressure loading of flat wing at 1“ angle of attack acting on the flat surface (see equation 22). Also used as normal force integration factor for flat wing contribution to the basic cambered wing loading at 0“ angle of attack acting on the camber surface; and as axial fo
4、rce integra- tion factor for basic pressure loading of flat wing at 1“ angle of attack acting on the camber surface normal force integration factor for the pure camber contribu- tion to the basic cambered wing loading at 0“ angle of attack acting on the camber surface (see equation 25) axial force i
5、ntegration factor for basic pressure loading of flat wing at 1“ angle of attack acting on the camber surface (see equation 30) Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-i j S S Au A”C AUf us VI W V xs Y, z X Xi8 Xi Ax AxC ,AxR,Ax L index of win
6、g element longitudinal position within the wing program grid system (see figure I) index of wing element lateral position within the wing program grid system (see figure 1) constants used in definition of camber surface slope constants used in curve fitting of program perturbation velocities and pre
7、ssure coefficients for integration purposes Mach number Reynolds number linearized theory downwash velocity influence function (see equation 4) wing reference area distance along section camber line longitudinal perturbation velocity difference across the wing lifting surface as a fraction of the fr
8、ee stream velocity value of Au for the cambered wing at 0“ angle of attack value of Au for the flat wing at 1“ angle of attack perturbation velocities in the x, y, and z directions, respectively free stream velocity Cartesian coordinates distance in the x direction measured from the wing leading edg
9、e X values at leading and trailing edge of wing element at element semispan values of x at which camber surface z ordinates are specified longitudinal spacing of grid lines used in establishment of program wing grid system longitudinal distances employed in the influence function R see sketch (d) Pr
10、ovided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-(ALJ) (Au=), (Aum)o,f E 0 “zt A “le limiting value of leading-edge thrust parameter Au/?-at the wing leading edge limiting value of leading-edge thrust parameter Auflat the wing leading edge for the cambe
11、red wing at 0“ angle of attack limiting value of leading-edge thrust parameter Au/?-at the wing leading edge for the flat wing at 1“ angle of attack angle of attack of wing (in degrees unless otherwise specified) JET7 angle between a line tangent to the wing section camber surface and the camber sur
12、face reference plane value of E at wing leading edge angle of attack of wing giving a local theoretical leading-edge thrust of zero for a specified wing spanwise station sweep angle of element quarter chord line sweep angle of wing leading edge DEVELOPMENT OF COMPUTATIONAL SYSTEM Development of this
13、 method begins with what is believed to be a unique approach to the theoretical analysis of wings at subsonic speeds. Among the features are linearized theory solutions by pure iteration, and the use of leading-edge singularity parameters to identify separate velocity distribution components with an
14、d without singularities. The later feature permits more accurate determination of leading-edge thrust distribution for wings with twist and camber and provides for improved pressure distribution integration techniques. The linearized theory solution will be described first, and then attention will b
15、e given to the empirical determination of attainable leading-edge thrust and detached vortex flow forces used in the estimation of overall wing performance. Program Grid System and Hing Definition . The linearized theory solutions are obtained by an iterative solution of influence equations for an a
16、rray of trapezoidal wing elements representing the actual wing planform as depicted in figure 1. Here only a small number of 6 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-elements are shown for the purpose of illustration; in practice several hun
17、dred elements would be employed. The elements are superimposed on a rectangular grid so that the inboard and outboard element chords lie along unit values of tl?z spanwise parameter my and the midspan leading and trailing edges lie on unit values of the chordwise parameter x/Ax. The scaling of the w
18、ing from model or airplane dimensions to program dimensions is chosen to provide the desired number of elements in the spanwise direction. The distance Ax controls the chordwise spacing of the elements; it is selected by specification of an element aspect ratio which is constant for all but the lead
19、ing-edge and trailing-edge elements. Element corner points at the wing leading and trailing edges are found by inter- polation of the scaled program input planform definition. These. points determine the leading-edge sweep of the first element and the trailing-edge sweep of the last element in each
20、chordwise row identified by the index j(Ay). Sweep angles for elements between the leading- and trailing-edge elements are found from simple Sketch (a) geometry for a superimposed arrow wing planform as indicated in sketch (a). Each element is assigned a number as indicated in figure 1 and a record
21、is kept of the number assigned to the leading- and trailing-edge elements in each chordwise row. The index i(x/ax) is used in determining the order of solution; elements are selected first according to advancing values of the i index then according to advancing values of the j index. The order of so
22、lution thus marches front to rear and inboard to outboard. The wing surface slopes are obtained by a curve fit of interpolated program input camber surface coordinates. The curve fit equation has the form: 2 = z. + k,(x - xi, + k2(x - x;)2 (1) Provided by IHSNot for ResaleNo reproduction or networki
23、ng permitted without license from IHS-,-,-As shown in sketch (b), the inter- polated input camber surface ordinates are chosen so as to place one ordinate xi at or ahead of the element leading edge, one ordinate xi within the element and one ordinate x,j at or behind the element trailing edge. With
24、the con- stants kl and k2 chosen to pass the curve through these three points, they can then be used in definition of the element surface slope expressed as: A-. uL = kl + k2 x; dx z -c -x e ZE I “; xi I - x element L Sketch (b) (2) where X e is distance from element leading edge and kl and k2 are r
25、edefined to correspond to the new origin Stored values of kl,e and k2,e allow subsequent recalculation of surface slopes anywhere within the element. The slope at the element three-quarter point is used in satisfying boundary conditions. As will be discussed subsequently, the program repeats the bas
26、ic linearized theory solution for two wing surfaces. One of these wing surfaces has the slopes described above; the other has a constant slope equal to the tangent of 1 degree angle of attack (dz/dx = -0.01745). Linearized Theory Solution Each trapezoidal element used to represent the wing is assume
27、d to have an associated horseshoe vortex with a bound leg along the quarter chord line and trailing legs extending to infinity along the extensions of the inboard and outboard chords as shown in sketch (c). At any point in the plane of the wing, the downwash velocity created by the vortex is given b
28、y: 8 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Vortex ; = $ Au ce Sketch .(c) (3) where Au is the longitudinal perturbation velocity difference across the wing surface C e is the element average chord, and Ris the influence factor In terms of t
29、he geometry system used here, R-is given as: (Afjy to. 5) + AXR tanA/6 + (ABY +05) (ABY -0.5) + AXL tan/l/f3 (A)c) + (ABY -05) 1 1 (A)* = E Tii;au*C; - ,Ax) +0.18 sin (5 V) for Xl/Ax 0.5 and (8) - f = 1 + 0.36 (1s25 “) xl/AX +(I . 18 sin (j-25 - XIAX 1.5 7T) for x,Ax ,o . 5 The singularity parameter
30、 obtained when the location of the velocity is defined by the factor f is shown at the bottom of figure 3. It was found that the simple correction derived from the two-dimensional results appeared to be equally valid in three dimensions. Typical program results for constant chord wings (right hand p
31、anel only) of various sweep angles at M = 0 are shown in figure 4. The singularity parameter is shown as a function of chordwise position for a midspan section. It is seen that there is no erratic behavior of the first two elements. Results for other sweep angles between 0“ and 80“ and other Mach nu
32、mbers up to 0.8 were similar. Convergence of the Iterative Solution It was found that the iterative solution converged quite rapidly to a reasonable approximation of fully converged results as estimated by extrapolation and as given by vortex lattice matrix inversion methods. However; when stringent
33、 convergence criteria are applied, as is required to obtain accurate leading-edge singularity information, a large number of iterations may be necessary. An 13 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-example for a 40 degree leading-edge sweep
34、 constant chord wing (right hand panel only) is given in figure 5. The first and second iterations give the general character of the solution. More than four iterations are required before suffi- ciently accurate leading-edge perturbation velocities are provided. The program convergence criteria pre
35、viously discussed is met after the tenth iteration in this example. For more complex planforms and for severely cambered wings more iterations will be required. For some of the examples shown later up to 50 iterations were required. Superposition of Cambered and Flat Wing Solutions In this program ,
36、 results covering a range of angles of attack are obtained by combining the solution for the input cambered wing (considered to be at 0“ angle of attack) with a solution for a flat wing of the same planform at 1“ angle of attack. An example of these basic solutions for a 40“ swept leading- edge cons
37、tant chord wing (one panel only) is shown in figure 6. The mean camber surface is defined as an arc of a circle with a radius selected to give a leading edge slope of dz/dx =0.0875 (a 5“ angle). Results for the cambered wing are given at the top of the figure, and results for the flat wing are given
38、 at the bottom. Note that the cambered wing as well as flat wing displays a leading-edge singularity. Figure 7 shows results for other angles of attack obtained by combining the cambered and the flat wing solutions by use of the expression: Au = Au, + Auf :; ;o The angle of attack of 1.8O was chosen
39、 for this illustration because at or near that angle the leading-edge singularity vanishes. The velocity distribution for this case may be considered to be a pure camber loading. For this constant cur- vature surface, the velocity distribution closely follows a curve defined by: Au = kc J 14 Provide
40、d by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-or (0) A distribution of this form will be used in the subsequent analysis of leading edge thrust characteristics. Theoretical Section Thrust Characteristics Figure 8 illustrates how the.angle of attack for a
41、vanishing singularity at a given spanwise station may be found directly. Singularity parameters in the form AU- are shun for the first three elements of both the cambered wing at CY. = 0“ and flat wing at CL = lo. From previous observations of the nature of cambered and flat wing velocity distributi
42、ons, it is reasonable to assume a leading-edge singularity parameter of the form: AUm = kf $ Jtan2nl,+82(AUfl)o12 (13) 15 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-For the flat wing: ( AJm, = (Aum sin a 0,f sin 1“ and (14) With the definition o
43、f azt and ct f the section thrust coefficient may be found for any angle of attack by ise of the expression: sina Ct = Ct,f - Slnazt 2 sin 1“ 1 (15) The preceding derivations for the evaluation of section thrust characteris- tics are based on the assumption of constant curvature of the camber surfac
44、e in the region of the leading edge (a linear variation in the surface slope) for at least the first two elements. For application of the method to severely cambered surfaces, the wing must be composed of a large enough number of elements to pro- vide nearly constant curvature over these first two e
45、lements behind the leading edge. Section Aerodynamic Coefficients Section aerodynamic coefficients are found by integration of the section pressure distributions, for which the pressure coefficient is assumed to be given by Cp = 2Au. Since perturbation velocities are obtained by superposition of cam
46、bered and flat w ing solutions,the pressure coefficient may be expressed as C P = cp,c +c sin a p,f sin 1“ 06) . . or C P = 2 Auc + 2 Auf “5; 70 (17) 16 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-As shown in sketch (f) the pressure acting on the
47、 airfoil camber surface produces an incremental section normal force given by: dcN = Cpds COSE= Cp dx and an incremental section axial force given by: dcA = Cp ds sinE = -C (dz, dx P dx Sketch (f) The section coefficients may thus be expressed as: 1 N = 2; rC I Cp dx 0 C +) dx P dx (18) (9) In order
48、 to account for leading-edge singularities where appropriate and to avoid them where not appropriate, the integrations are performed by parts. Normal force coefficient. - The total section normal force coefficient (exclusive of thrust or vortex forces) is given by: N = N,C + N,F % ; (20) cN F, the section normal force coefficient generated by the flat wing pressure diitribution for 1“ angle of attack is obtained by the integration 17 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-depicted in sketch (g). Wit
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