Engineeringnotes

I 412 J. AiRCRAFT UNI II VOL. 24, NO. ~ ;I wher !' ~ and !j EngineeringNotes . H quae

~ ENGINEERING NOTES are short manuscripts describing new developments or important results of 1 preliminary nature. These Notes cannot exceed 6 manuscript pages and 3figures; a page of text may be subs/ftuted for a figure and vice versa. After informal review by the edilers, they may be published within afew months of the date of receipt. Style requirements are the same as for regular contributions (see inside back cever). Hen

I T I: Optimum Flap Schedules and Minimum prove the aerodynamic efficiency (LID). Some examples of aircraft usingsuch devices are F-4E, F-5E, F-16, F-is, etc. Dur- the Drag Envelopes for Combat Aircraft ing a maneuver, the flaps automatica!ly follow a predetermined idea deflection schedule which is a function of Mach number and win angle of attack. The flap deflection schedule is mainly deter- .1CJ B. Rajeswari* mined through extensive wind-tunnel tests. lea, National Aeronautical Laboratory, Bangalore, India In this paper, a simple analytical method based on linear Ref I and theory is developed to determine the optimum flap schedule K.R. Prabhut for both leading- and trailing-edge flaps. Vor tI Kentron, Inc., Hampton, Virginia Problem Formulation tior and Method of Solution cid, wit Nomenclature If the drag polars are plotted for various flap deflections {3 and "I, then the envelope of these polars defines the dis! A = aspect ratio lift b = wing span minimum drag envelope. The corresponding deflections {3oPl I = local chord and 'Yepl'which minimize the lift-dependent drag CDL' define C(1/) the optimum flap schedule. CD = total drag coefficient CDL = lift-dependent drag coefficient Lift-DependentProfile Drag = minimum drag envelope CDLm For a cambered airfoil, the drag polar can be fairly well CDPL = lift-dependent profile drag coefficient represented by the relation CDV = vortex drag coefficient CL = lift coefficient Cd =Cdmin +k(CI - C1i)2 (1) CLu,CL6CL~ = wing lift curve slope with respect to angle of attack, leading-edge flap deflection angle, To a first approximation, C1i and Cd can be taken to be and trailing-edge flap deflection angle, functions of camber alone. Extending these arguments to respectively each spanwise section of a three-dimensional wing (i.e., Cd = sectional drag coefficient assuming that the wing is composed of a series of two- minimum profile drag Cdmin = dimensional airfoils of varying camber and thickness), the C1 = section lift coefficient lift-dependent profile drag CVPL can be determined by in- C1i = lift coefficient at minimum drag-also ideal tegrating Eq. (1) across the span and is written as lift coefficient F =quantity proportional to CDPL 1 bl2 Kp =lift-dependent profile drag factor r (2) CDPL -b12 Kp(CI-CI)2C dy=KpF KT = lift-dependent drag factor of minimum drag =8 J envelope k = lift-dependent drag factor for a section where Kp is a constant (being independent of camber), and LID = lifHo-drag ratio bl2 S wing area 1 = F=- (CI-C1i)2cdy (3) Y =spanwisestation S. . l-b12 IX =angle of attack (with respect to wing chordal plane) For a wing at an incidence ex, CI(-I1) and C1. can be ex- IXiO(1/) = distribution of total induced angle of pressed in accordance with linear theory as 1 incidence (3 = leading-edge flap deflection angle Cr (1/) =Ola+02{3+ °3'Y l' = trailing-edge flap deflection angle r(1/) (4) = spanwise loading CI.=04[3+asl'1 1/ =yl(bl2)-nondimensional spanwise station !1Cp = chordwise loading {3oPt =optimum leading-edge flap deflection angle I'opt =optimum trailing-edge flap deflection angle 30 Introduction / \ LEADING EDGE FLAP / SCHEDULE / / j ING leading- and trailing-edge flaps are usually de- / / ployed to improve the lifting ability of wings, especially t20 / W ~ / during takeoff and landing. In recent times, however, these TRAILINGEDGE FLAP c:i. / ~ flaps have been used also during maneuver conditions to im- / ) SCHEDULE to // //"'- - REFERENCE /./ /"'- - -- COMPUTED /;"'// (INCOMPRESSIBLE) Received Nov. 6, 1986; revision received Dec. 18, 1986. Copyright " I , , @ American Institute of Aeronautics and Astronautics, Inc., 1987.All 0 4 B 12 16 20 rights reserved. 'Scientist, Aerodynamics Division. «: -... t Aerospace Engineer. Fig.l Flap sehedule of F-18 aircraft. ENGINEERING NOTES 413 JVNE1987

,!tere al"'" a5 are c~nstants: Substituting these in Eq, (3) The total lift and vortex drag coefficients are \~ d carrying out the mtegratIOn, F can be expressed as a 1IJ1 " C d adratlCIII l.., (3, an ')': + 1 £ttl r . (7) C L =A j -, r ( I) dll F==A,CL2 +A2{J2+A3'Y2 +A4{3CL +A5{3'Y +A6'YCL (5) +1 /:"CPfl CDv=A r(i/)O:j()(ll) dl) (8) Bere, CL ==CL"CX+CL{3{3+CL,' for small ex, (3, and 'Y. 1 -I ';01(' - t/te.Y .' 1.'/OP'.e'). ...td ,.,{IV rhe lift-dependent profile drag factor Kp is determined by :bpC'. Within the limits of linear theory, CDvcan be expressed as a use of data correlation curves given in Refs. 1 and 2. The quadratic 1e5of !~:allift coefficient Clj and zero lift incidence on stations of .~p (10'- t :ng with deflected flaps are calculated using the local loading e~ cc.v CDV =B, CL 2 + B2{32+ B3"(2 + B4{3CL+ B5{3"(+ B6"(CL (9) jS.e J1!i!1ed ~we and C, (1/) formulae given in Ref. 3 modified to include a Ping-edgeflap. The details of these calculations are givenin :eJeCef( jll1d lead where B,. B2"'" B6 are constants. rttJfJ1~deCe(- Rei. 4. .rtf) }jlJ Total Lift-DependentDrag lil1ell( ortex Drag . Since both CDv and F are quadratic in CL, {3, and 'Y,the .JorteiWlc V rJcKie's method3 for calculation of spanwise load distribu- 5C~ total lift-dependent drag is also a quadratic and can be ex- ~p - J1on wings with spanwise discontinuities in angle of in- pressed as tl'~ence and/or wing chord has been adopted here for wings C\b plain leading- and trailing-edge flaps. Once the load CDL =CDv+KpF=C,CL2 ;tstribution r (1/) is known, the span wise distribution of local 'OjJ5(3 li~t C, (1/) is related to r (1J)by + C2{32 + C3"(2 + C4{JCL + C5{3"(+ C6'YCL (10) ~cCCJ, ,/Ie CI (I) =(2b/c)r(l) (6) ~{jjJ~p~ where CI, C2,..., C6 are constants. .ljOde~jJ 1"' Flap Schedule and Minimum Drag Envelope To get the flap schedule, CDi' Eq. (10) is minimized with respect to {J and "( for a given CL. The first derivatives of -ejj ~¥ 1.0 - - - (tNCOMPRES COMPUTED SIBLE) MINIMUM DRAG CDL with respect to {J and 'Yare equated to zero. The -0 REFERENCE j(1i ENVELOPE resulting values of {3oPtand 'Yoptare substituted in Eq. (10), ( and the minimum drag envelope is given by 08 M =0.6) , ""-

~i jj}' Results Figure 1 gives a comparison of leading- and trailing-edge (~} flap schedules and Fig. 2 the resulting minimum drag 0.02 0'04 0-06 0.()8 0.10 012 0 envelope for F-18 aircraft. 5 The comparison of flap Co --- schedules between theory and experiment for the F-18 air- L craft (Fig. 1) does not seem to be very good. A possible reason for this could be the relative insensitivity of the drag Fig.2 F-18 minimum drag envelope. coefficient to flap deflection angle, at least around the flap angles for minimum drag and at the lift coefficients under consideration. Figure 2 shows that for the undeflected flap case, the tI!-' - THEORY estimated drag departs from the experimental one for L1. BAND IN WHICH OPTIMUM CL >0.4, indicating the limits of the linear theory. However, DEFLECTION LIES - EXPTL. with the flaps deflected (both leading and trailing edge), the t agreement between experiment and estimation is remarkably ~ good even for CL of about 0.9. This can possibly be at- )7 tributed to the ability of the leading-edge flaps in maintain- ing attached flow at these high CL values. ~ 30[ A comparison of the drag envelope for the F-16 aircraft ~ 20 with and without programmable leading-edge flaps was w /7j:-):' made. The decrease in CDL when flaps are employed is t') [ quoted as 18070in Ref. 6, which compares well with about ~ 1°, :J--)--' 15% obtained from the present method. Figure 3 displays t') :z another comparison between theory and tests conducted at :J -~/ " --L. ~' the National Aeronautical Laboratory on an 'aircraft model ~ 0 02 04 06 08 1.0 t- (aspect ratio A =3.2) at a Mach number of 0.5. These tests LIFT COEFFICIENT CL were done with and without trailing-edge flaps deflected. Figure 3 shows that the trailing-edge flap deflection schedule Fig. 3 Trailing-edge flap schedule. is predicted reasonably well by the theory.

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Conclusions Vg = ground velocity along track Fo A simple method based on linear theory has been Vgc = ground velocity across track linea Vw = wind velocity developed for the determination of trailing- and leading-edge flap deflection schedules to obtain minimum lift-dependent Vz =component of airspeed in vertical direction = coordinates drag (or equivalently maximum lift-to-drag ratio). The resul- x,y,z = velocities tant lift-dependent drag polar can also be determined. Exten- x,y,i. whel = accelerations sive comparisons with available experimental results have x,y,z the ( a proved the general validity of the method. It is expected that = factor used to linearize equations of motion this method would be useful in the preliminary design phase ao ~ average value of a during time of flight (j of an aircraft and also in reducing later on the quantum of =drift angle P wind-tunnel testing needed to determine flap schedules. = air density Vne Pave = average density during time of flight 0 Acknowledgments = dive angle The first author wishes to acknowledge Dr. K. Yegna Introduction Sine Narayan and Dr. H.N. V. Dutt for valuable suggestions and encouragement during the preparation of this paper. RRORS in determining the initial conditions at bomb E release, i.e. airspeed, altitude, vertical velocity, drag coef- .6 References ficient and density, can greatly affect the accuracy at which a bomb reaches the desired impact point. Sensitivity coefficients I Subsonic Lift Dependent DragDue to Boundary Layer of Plane Symmetric Section Wings, ESDU, Royal Academy of Engineering, determined from the solution of the equations of motion 60031-32, 1966. define how errors in initial conditions propagate in errors in 2RAE Data Sheets, Aerodynamics, Vol. 2, Wings, Royal the downrange and crossrange impact point. Because of the Academy of Engineering, 01.01.01, 1953 and 01.01.05, 1955. drag term, the equations of motion are nonlinear and thus are Af 3McKie, J., "The Estimation of the Loading on Swept Wings not amenable to analytic solutions. However, by using a ter with Extending Chord Flaps at Subsonic Speeds," Royal Academy simplified aerodynamic trajectory (SAT) where the drag term of Engineering, TR-69034, Feb. 1969. is linearized the sensitivity coefficients can be determined in Ai 4Rajeswari, B. and Prabhu, K.R., "Estimation of Flap Schedule closed form. and Minimum Drag Envelope," National Aeronautical Laboratory, Bangalore, India, AE-TM-3-83, April 1983, 1977. an 5Siewert , R. F. and Whitehead, R.E., "Analysis of Advanced Approach Variable Camber Concepts," AGARD-CP-241, 1977. Referringto Figure I, the downrangeand crossrangeimpact 6Webb, T.S., Kent, D.R., and Webb, J.B., "Correlation of F-16 points respectivelycan be determinedby w Aerodynamics and Performance Predictions with Early Flight Test Results," AGARD-CP-242. D[=DR + VgTJ- (VaTJcosO-RB) coso (I) ar and

C[= CR + VgcTJ+ (VaTJcosO - RB) sino (2)

By taking the differentials of Eqs. (1) and (2), the total Bombing Error Sensitivities Using downrange and cross range error in the impact point can be determined based on the contributions of individual errors in a Simplified Aerodynamic Trajectory initial conditions. For the downrange impact point

/1D[ = ilDR + ARB + (/1 Vg - /1 Va cosO) TJ (3) WilliamD. Siuru Jr. * University of Colorado at Colorado Springs and for the crossrange Colorado Springs, Colorado /1C[=!1CR+!1 VgcTJ+ (VaTJ cosO-RB)!1o (4)

Nomenclature = drag coefficient of bomb CD z C[ = crossrange impact point CR = crossrange release point d = diameter of bomb D[ = downrange impact point x DR = downrange release point E =e~aTJ ho = altitude at bomb release ~ Sin e g = gravitational constant v. KD = drag factor m = mass of bomb RB = ballistic range y Tf =time of fall Va ~ airspeed of bomb at release Vave =average velocity during time of flight

Received Dec. 17, 1986; revision received Jan. 21,1987. Copyright @ American Institute of Aeronautics and Astronautics, Inc., 1987. x All rights reserved. <~. c,.> 'Senior Research Associate, Space and Flight Systems Laboratory. Associate Fellow AIAA. Fig. 1 Coordinate system and geometry for impact prediction.