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MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS Thin Airfoil Theory Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

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OVERVIEW: THIN AIRFOIL THEORY In words: Camber line is a streamline Written at a given point x on the chord line dz/dx is evaluated at that point x Variable is a dummy variable of integration which varies from 0 to c along the chord line Vortex strength = ( ) is a variable along the chord line and is in units of In transformed coordinates, equation is written at a point, 0. is the dummy variable of integration –At leading edge, x = 0, = 0 –At trailed edge, x = c, = The central problem of thin airfoil theory is to solve the fundamental equation for ( ) subject to the Kutta condition, (c)=0 The central problem of thin airfoil theory is to solve the fundamental equation for ( ) subject to the Kutta condition, ( )=0

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SUMMARY: SYMMETRIC AIRFOILS

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Fundamental equation of thin airfoil theory for a symmetric airfoil (dz/dx=0) written in transformed coordinates Solution –“A rigorous solution for ( ) can be obtained from the mathematical theory of integral equations, which is beyond the scope of this book.” (page 324, Anderson) Solution must satisfy Kutta condition ( )=0 at trailing edge to be consistent with experimental results Direct evaluation gives an indeterminant form, but can use L’Hospital’s rule to show that Kutta condition does hold.

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SUMMARY: SYMMETRIC AIRFOILS Total circulation, , around the airfoil (around the vortex sheet described by ( )) Transform coordinates and integrate Simple expression for total circulation Apply Kutta-Joukowski theorem (see §3.16), “although the result [L’= ∞ V ∞ 2 ] was derived for a circular cylinder, it applies in general to cylindrical bodies of arbitrary cross section.” Lift coefficient is linearly proportional to angle of attack Lift slope is 2 /rad or 0.11/deg

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EXAMPLE: NACA SYMMETRIC AIRFOIL Bell X-1 used NACA (6% thickness) as horizontal tail Thin airfoil theory lift slope: dc l /d = 2 rad -1 = 0.11 deg -1 Compare with data –At = -4º: c l ~ –At = 6º: c l ~ 0.65 –dc l /d = 0.11 deg -1 dc l /d = 2

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SUMMARY: SYMMETRIC AIRFOILS Total moment about the leading edge (per unit span) due to entire vortex sheet Total moment equation is then transformed to new coordinate system based on After performing integration (see hand out, or Problem 4.4), resulting moment coefficient about leading edge is – /2 Can be re-written in terms of the lift coefficient Moment coefficient about the leading edge can be related to the moment coefficient about the quarter-chord point Center of pressure is at the quarter-chord point for a symmetric airfoil

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EXAMPLE: NACA SYMMETRIC AIRFOIL Bell X-1 used NACA (6% thickness) as horizontal tail Thin airfoil theory lift slope: dc l /d = 2 rad -1 = 0.11 deg -1 Compare with data –At = -4º: c l ~ –At = 6º: c l ~ 0.65 –dc l /d = 0.11 deg -1 Thin airfoil theory: c m,c/4 = 0 Compare with data c m,c/4 = 0

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CENTER OF PRESSURE AND AERODYNAMIC CENTER Center of Pressure: Point on an airfoil (or body) about which aerodynamic moment is zero –Thin Airfoil Theory: Symmetric Airfoil: Aerodynamic Center: Point on an airfoil (or body) about which aerodynamic moment is independent of angle of attack –Thin Airfoil Theory: Symmetric Airfoil:

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CAMBERED AIRFOILS: THEORY In words: Camber line is a streamline Written at a given point x on the chord line dz/dx is evaluated at that point x Variable is a dummy variable of integration which varies from 0 to c along the chord line Vortex strength = ( ) is a variable along the chord line and is in units of In transformed coordinates, equation is written at a point, 0. is the dummy variable of integration –At leading edge, x = 0, = 0 –At trailed edge, x = c, = The central problem of thin airfoil theory is to solve the fundamental equation for ( ) subject to the Kutta condition, (c)=0 The central problem of thin airfoil theory is to solve the fundamental equation for ( ) subject to the Kutta condition, ( )=0

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CAMBERED AIRFOILS Fundamental Equation of Thin Airfoil Theory Camber line is a streamline Solution –“a rigorous solution for ( ) is beyond the scope of this book.” Leading term is very similar to the solution result for the symmetric airfoil Second term is a Fourier sine series with coefficients A n. The values of A n depend on the shape of the camber line (dz/dx) and

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EVALUATION PROCEDURE

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PRINCIPLES OF IDEAL FLUID AERODYNAMICS BY K. KARAMCHETI, JOHN WILEY & SONS, INC., NEW YORK, APPENDIX E

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CAMBERED AIRFOILS After making substitutions of standard forms available in advanced math textbooks We can solve this expression for dz/dx which is a Fourier cosine series expansion for the function dz/dx, which describes the camber of the airfoil Examine a general Fourier cosine series representation of a function f( ) over an interval 0 ≤ ≤ The Fourier coefficients are given by B 0 and B n

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ADVANCED CALCULUS FOR APPLICATIONS, 2 nd EDITION BY F. B. HILDEBRAND, PRENTICE-HALL, INC., ENGLEWOOD CLIFFS, N.J., 1976

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CAMBERED AIRFOILS Compare Fourier expansion of dz/dx with general Fourier cosine series expansion Analogous to the B 0 term in the general expansion Analogous to the B n term in the general expansion

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CAMBERED AIRFOILS We can now calculate the overall circulation around the cambered airfoil Integration can be done quickly with symbolic math package, or by making use of standard table of integrals (certain terms are identically zero) End result after careful integration only involves coefficients A 0 and A 1

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CAMBERED AIRFOILS Calculation of lift per unit span Lift per unit span only involves coefficients A 0 and A 1 Lift coefficient only involves coefficients A 0 and A 1 The theoretical lift slope for a cambered airfoil is 2 , which is a general result from thin airfoil theory However, note that the expression for c l differs from a symmetric airfoil

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CAMBERED AIRFOILS From any c l vs. data plot for a cambered airfoil Substitution of lift slope = 2 Compare with expression for lift coefficient for a cambered airfoil Let L=0 denote the zero lift angle of attack –Value will be negative for an airfoil with positive (dz/dx > 0) camber Thin airfoil theory provides a means to predict the angle of zero lift –If airfoil is symmetric dz/dx = 0 and L=0 =0

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SAMPLE DATA: SYMMETRIC AIRFOIL Lift Coefficient Angle of Attack, A symmetric airfoil generates zero lift at zero

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SAMPLE DATA: CAMBERED AIRFOIL Lift Coefficient Angle of Attack, A cambered airfoil generates positive lift at zero

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SAMPLE DATA Lift coefficient (or lift) linear variation with angle of attack, a –Cambered airfoils have positive lift when = 0 –Symmetric airfoils have zero lift when = 0 At high enough angle of attack, the performance of the airfoil rapidly degrades → stall Lift (for now) Cambered airfoil has lift at =0 At negative airfoil will have zero lift

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AERODYNAMIC MOMENT ANALYSIS Total moment about the leading edge (per unit span) due to entire vortex sheet Total moment equation is then transformed to new coordinate system based on Expression for moment coefficient about the leading edge Perform integration, “The details are left for Problem 4.9”, see hand out Result of integration gives moment coefficient about the leading edge, c m,le, in terms of A 0, A 1, and A 2

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AERODYNAMIC MOMENT SUMMARY Aerodynamic moment coefficient about leading edge of cambered airfoil Can re-writte in terms of the lift coefficient, c l –For symmetric airfoil dz/dx=0 A 1 =A 2 =0 c m,le =-c l /4 Moment coefficient about quarter-chord point –Finite for a cambered airfoil For symmetric c m,c/4 =0 –Quarter chord point is not center of pressure for a cambered airfoil –A 1 and A 2 do not depend on c m,c/4 is independent of –Quarter-chord point is theoretical location of aerodynamic center for cambered airfoils

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CENTER OF PRESSURE AND AERODYNAMIC CENTER Center of Pressure: Point on an airfoil (or body) about which aerodynamic moment is zero –Thin Airfoil Theory: Symmetric Airfoil: Cambered Airfoil: Aerodynamic Center: Point on an airfoil (or body) about which aerodynamic moment is independent of angle of attack –Thin Airfoil Theory: Symmetric Airfoil: Cambered Airfoil:

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ACTUAL LOCATION OF AERODYNAMIC CENTER NACA x A.C. < 0.25c NACA x A.C. > 0.25 c x/c=0.25

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IMPLICATIONS FOR STALL Flat Plate Stall Leading Edge Stall Trailing Edge Stall Increasing airfoil thickness

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LEADING EDGE STALL NACA 4412 (12% thickness) Linear increase in c l until stall At just below 15º streamlines are highly curved (large lift) and still attached to upper surface of airfoil At just above 15º massive flow-field separation occurs over top surface of airfoil → significant loss of lift Called Leading Edge Stall Characteristic of relatively thin airfoils with thickness between about 10 and 16 percent chord

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TRAILING EDGE STALL NACA 4421 (21% thickness) Progressive and gradual movement of separation from trailing edge toward leading edge as is increased Called Trailing Edge Stall

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THIN AIRFOIL STALL Example: Flat Plate with 2% thickness (like a NACA 0002) Flow separates off leading edge even at low ( ~ 3º) Initially small regions of separated flow called separation bubble As a increased reattachment point moves further downstream until total separation

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NACA 4412 vs. NACA 4421 NACA 4412 and NACA 4421 have same shape of mean camber line Theory predicts that linear lift slope and L=0 same for both Leading edge stall shows rapid drop of lift curve near maximum lift Trailing edge stall shows gradual bending-over of lift curve at maximum lift, “soft stall” High c l,max for airfoils with leading edge stall Flat plate stall exhibits poorest behavior, early stalling Thickness has major effect on c l,max

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AIRFOIL THICKNESS

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AIRFOIL THICKNESS: WWI AIRPLANES English Sopwith Camel German Fokker Dr-1 Higher maximum C L Internal wing structure Higher rates of climb Improved maneuverability Thin wing, lower maximum C L Bracing wires required – high drag

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OPTIMUM AIRFOIL THICKNESS Some thickness vital to achieving high maximum lift coefficient Amount of thickness influences type of stall Expect an optimum Example: NACA 63-2XX, NACA looks about optimum c l,max NACA

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MODERN LOW-SPEED AIRFOILS NACA 2412 (1933) Leading edge radius = 0.02c NASA LS(1)-0417 (1970) Whitcomb [GA(w)-1] (Supercritical Airfoil) Leading edge radius = 0.08c Larger leading edge radius to flatten c p Bottom surface is cusped near trailing edge Discourages flow separation over top Higher maximum lift coefficient At c l ~1 L/D > 50% than NACA 2412

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MODERN AIRFOIL SHAPES RootMid-SpanTip Boeing 737

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