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

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 x is a dummy variable of integration which varies from 0 to c along the chord line Vortex strength g=g (x) is a variable along the chord line and is in units of In transformed coordinates, equation is written at a point, q0. q is the dummy variable of integration At leading edge, x = 0, q = 0 At trailed edge, x = c, q =p The central problem of thin airfoil theory is to solve the fundamental equation for g (x) subject to the Kutta condition, g(c)=0 The central problem of thin airfoil theory is to solve the fundamental equation for g (q) subject to the Kutta condition, g(p)=0

SUMMARY: SYMMETRIC AIRFOILS

SUMMARY: SYMMETRIC AIRFOILS Fundamental equation of thin airfoil theory for a symmetric airfoil (dz/dx=0) written in transformed coordinates Solution “A rigorous solution for g(q) 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 g(p)=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.

SUMMARY: SYMMETRIC AIRFOILS Total circulation, G, around the airfoil (around the vortex sheet described by g(x)) Transform coordinates and integrate Simple expression for total circulation Apply Kutta-Joukowski theorem (see §3.16), “although the result [L’=r∞V ∞2G] 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 2p/rad or 0.11/deg

EXAMPLE: NACA 65-006 SYMMETRIC AIRFOIL dcl/da = 2p Bell X-1 used NACA 65-006 (6% thickness) as horizontal tail Thin airfoil theory lift slope: dcl/da = 2p rad-1 = 0.11 deg-1 Compare with data At a = -4º: cl ~ -0.45 At a = 6º: cl ~ 0.65 dcl/da = 0.11 deg-1

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 q After performing integration (see hand out, or Problem 4.4), resulting moment coefficient about leading edge is –pa/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

EXAMPLE: NACA 65-006 SYMMETRIC AIRFOIL cm,c/4 = 0 Bell X-1 used NACA 65-006 (6% thickness) as horizontal tail Thin airfoil theory lift slope: dcl/da = 2p rad-1 = 0.11 deg-1 Compare with data At a = -4º: cl ~ -0.45 At a = 6º: cl ~ 0.65 dcl/da = 0.11 deg-1 Thin airfoil theory: cm,c/4 = 0

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

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 x is a dummy variable of integration which varies from 0 to c along the chord line Vortex strength g=g (x) is a variable along the chord line and is in units of In transformed coordinates, equation is written at a point, q0. q is the dummy variable of integration At leading edge, x = 0, q = 0 At trailed edge, x = c, q =p The central problem of thin airfoil theory is to solve the fundamental equation for g (x) subject to the Kutta condition, g(c)=0 The central problem of thin airfoil theory is to solve the fundamental equation for g (q) subject to the Kutta condition, g(p)=0

CAMBERED AIRFOILS Fundamental Equation of Thin Airfoil Theory Camber line is a streamline Solution “a rigorous solution for g(q) 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 An. The values of An depend on the shape of the camber line (dz/dx) and a

EVALUATION PROCEDURE

PRINCIPLES OF IDEAL FLUID AERODYNAMICS BY K PRINCIPLES OF IDEAL FLUID AERODYNAMICS BY K. KARAMCHETI, JOHN WILEY & SONS, INC., NEW YORK, 1966. APPENDIX E

PRINCIPLES OF IDEAL FLUID AERODYNAMICS BY K PRINCIPLES OF IDEAL FLUID AERODYNAMICS BY K. KARAMCHETI, JOHN WILEY & SONS, INC., NEW YORK, 1966. APPENDIX E

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(q) over an interval 0 ≤ q ≤ p The Fourier coefficients are given by B0 and Bn

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

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

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

CAMBERED AIRFOILS Compare Fourier expansion of dz/dx with general Fourier cosine series expansion Analogous to the B0 term in the general expansion Analogous to the Bn term in the general expansion

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 A0 and A1

CAMBERED AIRFOILS Calculation of lift per unit span Lift per unit span only involves coefficients A0 and A1 Lift coefficient only involves coefficients A0 and A1 The theoretical lift slope for a cambered airfoil is 2p, which is a general result from thin airfoil theory However, note that the expression for cl differs from a symmetric airfoil

CAMBERED AIRFOILS From any cl vs. a data plot for a cambered airfoil Substitution of lift slope = 2p Compare with expression for lift coefficient for a cambered airfoil Let aL=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 aL=0=0

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

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

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

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 q 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, cm,le, in terms of A0, A1, and A2

AERODYNAMIC MOMENT SUMMARY Aerodynamic moment coefficient about leading edge of cambered airfoil Can re-writte in terms of the lift coefficient, cl For symmetric airfoil dz/dx=0 A1=A2=0 cm,le=-cl/4 Moment coefficient about quarter-chord point Finite for a cambered airfoil For symmetric cm,c/4=0 Quarter chord point is not center of pressure for a cambered airfoil A1 and A2 do not depend on a cm,c/4 is independent of a Quarter-chord point is theoretical location of aerodynamic center for cambered airfoils

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

ACTUAL LOCATION OF AERODYNAMIC CENTER x/c=0.25 NACA 23012 xA.C. < 0.25c x/c=0.25 NACA 64212 xA.C. > 0.25 c

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

LEADING EDGE STALL NACA 4412 (12% thickness) Linear increase in cl until stall At a just below 15º streamlines are highly curved (large lift) and still attached to upper surface of airfoil At a 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

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

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

NACA 4412 vs. NACA 4421 NACA 4412 and NACA 4421 have same shape of mean camber line Theory predicts that linear lift slope and aL=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 cl,max for airfoils with leading edge stall Flat plate stall exhibits poorest behavior, early stalling Thickness has major effect on cl,max

AIRFOIL THICKNESS

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

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 63-212 looks about optimum NACA 63-212 cl,max

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 cp Bottom surface is cusped near trailing edge Discourages flow separation over top Higher maximum lift coefficient At cl~1 L/D > 50% than NACA 2412

MODERN AIRFOIL SHAPES Boeing 737 Root Mid-Span Tip http://www.nasg.com/afdb/list-airfoil-e.phtml