MAE 3241 AERODYNAMICS AND FLIGHT MECHANICS Thin Airfoil

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, q 0. 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 ∞ 2 G] 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 p/rad or 0. 11/deg

EXAMPLE: NACA 65 -006 SYMMETRIC AIRFOIL dcl/da = 2 p • Bell X-1 used NACA 65 -006 (6% thickness) as horizontal tail • Thin airfoil theory lift slope: dcl/da = 2 p 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 • Bell X-1 used NACA 65 -006 (6% thickness) as horizontal tail • Thin airfoil theory lift slope: dcl/da = 2 p 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 • Compare with data 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 – Thin Airfoil Theory: • Symmetric Airfoil:

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, q 0. 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. KARAMCHETI, JOHN WILEY & SONS, INC. , NEW YORK, 1966. APPENDIX E

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 B 0 and Bn

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

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

ADVANCED CALCULUS FOR APPLICATIONS, 2 nd 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 B 0 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 A 0 and A 1

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 p, 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 = 2 p • Compare with expression for lift coefficient for a cambered airfoil • Let a. 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 a. L=0=0

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

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

SAMPLE DATA Lift (for now) • 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 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 A 0, A 1, and A 2

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 • A 1=A 2=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 – A 1 and A 2 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 – Thin Airfoil Theory: • Symmetric Airfoil: • Cambered Airfoil:

ACTUAL LOCATION OF AERODYNAMIC CENTER x/c=0. 25 NACA 23012 x. A. C. < 0. 25 c x/c=0. 25 NACA 64212 x. A. 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 a. 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 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 -2 XX, NACA 63 -212 looks about optimum NACA 63 -212 cl, max

MODERN LOW-SPEED AIRFOILS NACA 2412 (1933) Leading edge radius = 0. 02 c NASA LS(1)-0417 (1970) Whitcomb [GA(w)-1] (Supercritical Airfoil) Leading edge radius = 0. 08 c 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