Induced Drag and High-Speed Aerodynamics Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 • Drag-due-to-lift and effects of wing planform • Effect of angle of attack on lift and drag coefficients • Mach number (i.e., air compressibility) effects on aerodynamics • Newtonian approximation for lift and drag Reading: Flight Dynamics Aerodynamic Coefficients, 85-96 Airplane Stability and Control Chapter 1 Copyright 2018 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE331.html http://www.princeton.edu/~stengel/FlightDynamics.html 1 Induced Drag 2 1 Aerodynamic Drag 1 1 Drag = C ρV 2S ≈ C + εC 2 ρV 2S D 2 ( D0 L ) 2 2 1 ≈ ⎡C + ε C + C α ⎤ ρV 2S ⎣⎢ D0 ( Lo Lα ) ⎦⎥ 2 3 2 Induced Drag of a Wing, εCL § Lift produces downwash (angle proportional to lift) § Downwash rotates local velocity vector clockwise in figure § Lift is perpendicular to velocity vector § Axial component of rotated lift induces drag § But what is the proportionality factor, ε? 4 2 Induced Angle of Attack C C sin , where Di = L α i α i = CL πeAR, Induced angle of attack C = C sin C πeAR C 2 πeAR Di L ( L ) ! L 5 Three Expressions for Induced Drag of a Wing C 2 C 2 1+δ C L L ( ) C 2 Di = ! ! ε L πeAR π AR e = Oswald efficiency factor = 1 for elliptical distribution δ = departure from ideal elliptical lift distribution 1 (1+δ ) ε = = πeAR π AR 6 3 Spanwise Lift Distribution of Elliptical and Trapezoidal Wings Straight Wings (@ 1/4 chord), McCormick Spitfire TR = taper ratio, λ P-51D Mustang For some taper ratio between 0.35 and 1, trapezoidal lift distribution is nearly elliptical 7 Induced Drag Factor, δ 2 CL (1+ δ ) C = Di π AR • Graph for δ (McCormick, p. 172) Lower AR 8 4 Oswald Efficiency Factor, e C 2 C = L Di πeAR Empirical approximations for e Pamadi AR λ κ = cosΛLE R = 0.0004κ 3 − 0.008κ 2 + 0.05κ + 0.86 1.1C e ≈ Lα RCL + (1− R)π AR Raymer α e ≈ 1.78(1− 0.045AR0.68 ) − 0.64 [Straight wing] 0.68 0.15 e ≈ 4.61(1− 0.045AR )(cosΛLE ) − 3.1 [Swept wing] 9 Maximum Lift-to-Drag Ratio Maximize L/D by proper choice of CL L C C ∂(L / D) = L = L = 0 D C C C 2 C D Do + ε L ∂ L C + εC 2 − C 2εC C − εC 2 ∂(L / D) ( Do L ) L ( L ) ( Do L ) = 0 = 2 = 2 2 2 ∂CL C + εC C + εC ( Do L ) ( Do L ) C 1 Do (L / D) = (CL ) = max (L/D)max 2 εC ε Do 10 5 Lift and Drag Coefficients Over Large Angles of Attack (0< α < 90) All coefficients converge to Newtonian-like values at very high angle of attack Low-AR wing has less drag than high-AR wing at given α 11 Lift vs. Drag for Large Variation in Angle of Attack (0< α < 90) Subsonic Lift-Drag Polar Low-AR wing has less drag than high-AR wing, but less lift as well High-AR wing has the best overall subsonic L/D 12 6 Lift-to-Drag Ratio vs. Angle of Attack • Performance metric for aircraft • High-AR wing: Best overall subsonic L/D • Low-AR wing: Best L/D at high angle of attack L C q S C = L = L D CDq S CD 13 € Newtonian Flow and Aerodynamic Forces 14 7 Newtonian Flow • No circulation • Cookie-cutter flow • Equal pressure across bottom of a flat plate • Flow brought to a halt at the surface 15 Newtonian Flow Normal Force ⎛ Mass flow rate⎞ Normal Force = ⎝⎜ Unit area ⎠⎟ × (Change in velocity) Projected Area × ( ) × (Angle between plate and velocity) N = (ρV )(V − 0)(Ssinα )(sinα ) = ρV 2 Ssin2 α ( )( ) 2 2 ⎛ 1 2 ⎞ CN = 2sin α = 2sin α ρV S ( )⎝⎜ 2 ⎠⎟ α = Incident flow angle ⎛ 1 2 ⎞ ≡ C ρV S 16 N ⎝⎜ 2 ⎠⎟ 8 Newtonian Forces on a Flat Plate Normal Force Coefficient 2 CN = 2sin α Lift and Drag Lift = N cosα 2 CL = (2sin α )cosα Drag = N sinα C = 2sin3 α D 17 Aerodynamic Force Estimation for a Hypersonic Aircraft Integrate differential normal force over the aircraft surface, accounting for varying surface incidence (i.e., angle) to the flow ! f $ ! X $ # x & # B & f fB = ∫ # y &dx dydz = # YB & Surface# & # & f Z "# z %& " B % 18 9 Application of Newtonian Flow Space Shuttle in • But where does the airflow go? Supersonic Flow • Hypersonic flow (M ~> 5) – Shock wave close to surface (thin shock layer), merging with the boundary layer – Flow is ~ parallel to the surface High-Angle-of- – Separated upper surface flow Attack Research Vehicle (F-18) • All Mach numbers at high angle of attack – Separated flow on upper (leeward) surfaces Flat Plate, Re = 50,000 19 Historical Factoid Conversions from Propellers to Jets Douglas XB-43 Douglas XB-42 Mixmaster Northrop XB-49 Northrop YB-35 Convair B-36 Convair YB-60 20 10 Historical Factoid Jets at an Awkward Age • Performance of first jet aircraft outstripped stability and control North American B-45 technology – Lacked satisfactory actuators, sensors, and control electronics Lockheed P-80 – Transistor: 1947, integrated circuit: 1958 • Dramatic dynamic variations over larger flight envelope – Control mechanisms designed to Douglas F3D lighten pilot loads were subject to instability • Reluctance of designers to embrace change, fearing decreased reliability, increased cost, and higher weight 21 Convair XP-81 Mach Number Effects True Airspeed Mach Number = Speed of Sound Supersonic Bullet, Ernst Mach 1888 Schlieren 1838-1916 photograph 22 11 Drag Due to Pressure Differential S 0.029 S C = C base ≈ base M < 1 Hoerner Dbase pressurebase S ( ) [ ] Swet S C friction Blunt base Sbase pressure drag 2 ⎛ Sbase ⎞ < 2 ⎜ ⎟ (M > 2, γ = specific heat ratio) γ M ⎝ S ⎠ C Dincompressible CD ≈ (M <1) “The Sonic Barrier” wave 1− M 2 Prandtl C Dcompressible factor ≈ (M >1) M 2 −1 CD ≈ M ≈ 2 (M >1) M 2 −1 23 Shock Waves in Supersonic Flow Air Compressibility Effect • Drag rises due to pressure increase across a shock wave • Subsonic flow – Local airspeed less than sonic (i.e., speed of sound) everywhere • Transonic flow – Airspeed less than sonic at some points, greater than sonic elsewhere • Supersonic flow • Critical Mach number – Local airspeed greater than – Mach number at which local sonic virtually everywhere flow first becomes sonic – Onset of drag-divergence – Mcrit ~ 0.7 to 0.85 24 12 Effect of Chord Thickness on Wing Lockheed P-38 Pressure Drag Lockheed F-104 • Thinner chord sections lead to higher Mcrit, or drag-divergence Mach number 25 Air Compressibility Effect on Wing Drag Sonic Booms http://www.youtube.com/watch?v=gWGLAAYdbbc Transonic Supersonic Subsonic Incompressible 26 13 Pressure Drag on Wing Depends on Sweep Angle M crit M = unswept critswept cos Λ Talay, NASA SP-367 27 Historical Factoid From Straight to Swept Wings • Straight-wing models were redesigned with swept wings to reduce compressibility effects on drag and increase speed • Dramatic change in stability, control, and flying qualities North American FJ-1 and Republic F-84B Thunderbird and Grumman F9F-2 Panther and FJ-4 Fury F-84F Thunderstreak F9F-6 Cougar 28 14 Airbus A320 Supercritical Wing NASA Supercritical Wing F-8 • Richard Whitcombs supercritical airfoil – Wing upper surface flattened to increase Mcrit – Wing thickness can be restored • Important for structural efficiency, fuel storage, etc. (–) (+) Pressure Distribution on Supercritical Airfoil ~ Section Lift 29 Subsonic Air Compressibility and Sweep Effects on 3-D Wing Lift Slope • Subsonic 3-D wing, with sweep effect π AR CL = α + 2 . $ AR ' -1+ 1+& ) 1− M 2 cosΛ 0 - &2cosΛ ) ( 1 4 )0 ,- % 1 4 ( /0 sweep angle of quarter chord Λ1 4 = 30 € 15 Subsonic Air Compressibility Effects on 3-D Wing Lift Slope Subsonic 3-D wing, sweep = 0 plot(pi A / (1+sqrt(1 + ((A / 2)^2) (1 - M^2))), A=1 to 20, M = 0 to 0.9) 31 Subsonic Air Compressibility Effects on 3-D Wing Lift Slope Subsonic 3-D wing, sweep = 60 plot(pi A / (1+sqrt(1 + (A ^2) (1 – 0.5 M^2))), A=1 to 20, M = 0 to 0.9) 32 16 Lift-Drag Polar for a Typical Bizjet • L/D = slope of line drawn from origin – Single maximum for a given polar – Two solutions for lower L/D (high and low airspeed) – Available L/D decreases with Mach number • Intercept for L/Dmax depends only on ε and zero-lift drag Note different scales for lift and drag 33 Wing Lift Slope at M = 1 Approximation for all wing planforms π AR # AR& C = = 2π % ( Lα 2 $ 4 ' 34 17 Supersonic Effects on Arbitrary Wing and Wing-Body Lift Slope • Impinging shock waves • Discrete areas with differing M and local pressure coefficients, cp • Areas change with α • No simple equations for lift slope m = tanγ tan µ Schlicting & Truckenbrodt, 1979 , sweep angles of shock and leading edge, from x axis 35 γ µ = Supersonic Compressibility Effects on Triangular Wing Lift Slope Supersonic delta (triangular) wing Supersonic leading edge Subsonic leading edge 4 2π 2 cot Λ C = CL = Lα α M 2 − 1 (π + λ) where λ = m(0.38 + 2.26m − 0.86m2 ) m = cot ΛLE cotσ sweep angle of leading edge, from y axis Λ LE = 36 18 Historical Factoid Fighter Jets of the 1950s: “Century Series” • Emphasis on supersonic speed McDonnell F-101 North American F-100 Convair F-102 Republic F-105 Lockheed F-104 37 Transonic Drag Rise and the Area Rule • Richard Whitcomb (NASA Langley) and Wallace Hayes (Princeton) • YF-102A (left) could not break speed of sound in level flight; F-102A (right) could 38 19 Transonic Drag Rise and the Area Rule Cross-sectional area of total configuration should gradually increase and decrease to minimize transonic drag Talay, NASA SP-367 Sears-Haack Body Area Rule http://en.wikipedia.org/wiki/Sears-Haack_body https://en.wikipedia.org/wiki/Area_rule 39 Secondary Wing Structures • Vortex generators, fences, vortilons, notched or dog-toothed wing leading edges – Boundary layer control – Maintain attached flow with increasing α – Avoid tip stall McDonnell-Douglas F-4 LTV F-8 Sukhoi Su-22 40 20 Leading-Edge Extensions • Strakes or leading edge extensions – Maintain lift at high α – Reduce c.p.