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Low-Speed Aerodynamics Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 Learning Objectives

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Low-Speed Aerodynamics Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 Learning Objectives Low-Speed Aerodynamics Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 Learning Objectives • 2D lift and drag • Reynolds number effects • Relationships between airplane shape and aerodynamic characteristics • 2D and 3D lift and drag • Static and dynamic effects of aerodynamic control surfaces Reading: Flight Dynamics Aerodynamic Coefficients, 65-84 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 2-Dimensional Aerodynamic Lift and Drag 2 1 Wing Lift and Drag • Lift: Perpendicular to free-stream airflow • Drag: Parallel to the free-stream airflow 3 Longitudinal Aerodynamic Forces Non-dimensional force coefficients, CL and CD, are dimensionalized by dynamic pressure, q, N/m2 or lb/sq ft reference area, S, m2 of ft2 ⎛ 1 2 ⎞ Lift = C q S = C ρV S L L ⎝⎜ 2 ⎠⎟ ⎛ 1 2 ⎞ Drag = C q S = C ρV S D D ⎝⎜ 2 ⎠⎟ 4 2 Circulation of Incompressible Air Flow About a 2-D Airfoil Bernoullis equation (inviscid, incompressible flow) (Motivational, but not the whole story of lift) 1 p + ρV 2 = constant along streamline = p static 2 stagnation Vorticity at point x Vupper (x) = V∞ + ΔV(x) 2 ΔV(x) γ (x) = V (x) V V(x) 2 2−D lower = ∞ − Δ Δz(x) Circulation about airfoil Lower pressure on upper surface c c ΔV(x) Γ = γ (x)dx = dx 2−D ∫ 2−D ∫ 0 0 Δz(x) 5 Relationship Between Circulation and Lift Differential pressure along chord section ⎡ 1 2 ⎤ ⎡ 1 2 ⎤ Δp(x) = pstatic + ρ∞ (V∞ + ΔV (x) 2) − pstatic + ρ∞ (V∞ − ΔV (x) 2) ⎣⎢ 2 ⎦⎥ ⎣⎢ 2 ⎦⎥ 1 ⎡ 2 2 ⎤ = ρ∞ (V∞ + ΔV (x) 2) − (V∞ − ΔV (x) 2) 2 ⎣ ⎦ = ρ∞V∞ΔV (x) = ρ∞V∞Δz(x)γ 2−D (x) 2-D Lift (inviscid, incompressible flow) c c Lift = Δp x dx = ρ V γ (x)dx = ρ V Γ ( )2−D ∫ ( ) ∞ ∞ ∫ 2−D ∞ ∞ ( )2−D 0 0 1 2 ! ρ∞V∞ c(2πα ) thin, symmetric airfoil + ρ∞V∞ (Γcamber ) 2 [ ] 2−D 1 2 ! ρ V c C α + ρ V Γ 6 ∞ ∞ ( Lα ) ∞ ∞ ( camber )2−D 2 2−D 3 Lift vs. Angle of Attack 2-D Lift (inviscid, incompressible flow) ⎡1 2 ⎤ (Lift) ! ρ∞V∞ c CL α + ⎡ρ∞V∞ (Γcamber ) ⎤ 2-D ⎣⎢2 ( α )2−D ⎦⎥ ⎣ 2−D ⎦ = [Lift due to angle of attack] + [Lift due to camber] 7 Typical Flow Variation with Angle of Attack • At higher angles, – flow separates – wing loses lift • Flow separation produces stall 8 4 What Do We Mean by 2-Dimensional Aerodynamics? Finite-span wing –> finite aspect ratio b AR = rectangular wing c b × b b2 = = any wing c × b S Infinite-span wing –> infinite aspect ratio 9 What Do We Mean by 2-Dimensional Aerodynamics? Assuming constant chord section, the 2-D Lift is the same at any y station of the infinite-span wing 1 2 1 2 Lift3−D = CL ρV S = CL ρV (bc) [Rectangular wing] 3−D 2 3−D 2 1 2 Δ(Lift3−D ) = CL ρV cΔy 3−D 2 ⎛ 1 2 ⎞ 1 2 lim Δ Lift = lim C ρV cΔy ⇒ "2-D Lift" C ρV c ( 3−D ) ⎜ L3 D ⎟ ! L2 D Δy→ε>0 Δy→ε>0 ⎝ − 2 ⎠ − 2 10 5 Effect of Sweep Angle on Lift Unswept wing, symmetric airfoil, 2-D lift slope coefficient Inviscid, incompressible flow Referenced to chord length, c, rather than wing area ⎛ ∂CL ⎞ CL = α = α CL = (2π )α [Thin Airfoil Theory] 2−D ⎜ ⎟ ( α )2−D ⎝ ∂α ⎠ 2−D Swept wing, 2-D lift slope coefficient Inviscid, incompressible flow CL = α CL = (2π cosΛ)α 2−D ( α )2−D 11 Classic Airfoil Profiles • NACA 4-digit Profiles (e.g., NACA 2412) – Maximum camber as percentage of chord (2) = 2% – Distance of maximum camber from leading edge, (4) = 40% – Maximum thickness as percentage of chord (12) = 12% • Clark Y (1922): Flat lower surface, 11.7% thickness – GA, WWII aircraft – Reasonable L/D – Benign theoretical stall characteristics – Experimental result is more abrupt Fluent, Inc,12 2007 6 Typical Airfoil Profiles Positive camber Neutral camber Negative camber Talay, NASA SP-367 13 Airfoil Effects • Camber increases zero-α lift coefficient • Thickness – increases α for stall and softens the stall break – reduces subsonic drag – increases transonic drag – causes abrupt pitching moment variation • Profile design – can reduce center-of- pressure (static margin, TBD) variation with α – affects leading-edge and trailing-edge flow separation Talay, NASA SP-367 14 7 NACA 641-012 Chord Section Lift, Drag, and Moment (NACA TR-824) Rough ~ Turbulent CD CL, 60 flap Smooth ~ Laminar CL “Drag Bucket” CL, w/o flap Cm Cm, w/o flap Cm, 60 flap α CL 15 Historical Factoid Measuring Lift and Drag with Whirling Arms and Early Wind Tunnels Whirling Arm Experimentalists Otto Lillienthal Hiram Maxim Samuel Langley Wind Tunnel Experimentalists Wright Frank Wenham Gustave Eiffel Hiram Maxim Brothers 16 8 Historical Factoid Wright Brothers Wind Tunnel 17 Flap Effects on Aerodynamic Lift • Camber modification • Trailing-edge flap deflection shifts CL up and down • Leading-edge flap (slat) deflection increases stall α • Same effect applies for other control surfaces – Elevator (horizontal tail) – Ailerons (wing) – Rudder (vertical tail) 18 9 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 19 Parasitic Drag, CDo • Pressure differential, viscous shear stress, and separation 1 Parasitic Drag = C ρV 2S D0 2 Talay, NASA SP-367 20 10 Reynolds Number and Boundary Layer ρVl Vl Reynolds Number = Re = = µ ν where ρ = air density, kg/m2 V = true airspeed, m/s l = characteristic length, m µ = absolute (dynamic) viscosity = 1.725 ×10−5 kg / m i s ν = kinematic viscosity (SL) = 1.343×10−5 m / s2 21 Reynolds Number, Skin Friction, and Boundary Layer Skin friction coefficient for a flat plate Friction Drag C = Wetted Area: Total surface f qS area of the wing or aircraft, wet subject to skin friction where Swet = wetted area Boundary layer thickens in transition, then thins in turbulent flow −1/2 C f ≈ 1.33Re [laminar flow] −2.58 ≈ 0.46(log10 Re) [turbulent flow] 22 11 Effect of Streamlining on Parasitic Drag CD = 2.0 CD = 1.2 CD = 0.12 CD = 1.2 CD = 0.6 Talay, NASA SP-367 DRAG 23 https://www.youtube.com/watch?v=ylh1CPqBwEw Subsonic CDo Estimate (Raymer) 24 12 Historical Factoid Wilbur (1867-1912) and Orville (1871-1948) Wright • Bicycle mechanics from Dayton, OH • Self-taught, empirical approach to flight • Wind-tunnel, kite, and glider experiments • Dec 17, 1903: Powered, manned aircraft flight ends in success 25 Historical Factoids • 1906: 2nd successful aviator: Alberto Santos-Dumont, standing! – High dihedral, forward control surface • Wrights secretive about results until 1908; few further technical contributions • 1908: Glenn Curtiss et al incorporate ailerons – Separate aileron surfaces at right – Wright brothers sue for infringement of 1906 US patent (and win) • 1909: Louis Bleriot's flight across the English Channel 26 13 Description of Aircraft Configurations 27 A Few Definitions Republic F-84F Thunderstreak 28 14 Wing Planform Variables Aspect Ratio Taper Ratio b AR = rectangular wing c tip chord c λ = tip = 2 b × b b croot root chord = = any wing c × b S Delta Wing Swept Trapezoidal Wing Rectangular Wing 29 Wing Design Parameters • Planform – Aspect ratio – Sweep – Taper – Complex geometries – Shapes at root and tip • Chord section – Airfoils – Twist • Movable surfaces – Leading- and trailing-edge devices – Ailerons – Spoilers • Interfaces – Fuselage – Powerplants – Dihedral angle 30 Talay, NASA SP-367 15 Mean Aerodynamic Chord, c Mean aerodynamic chord (m.a.c.) ~ mean geometric chord b 2 1 2 c tip chord c = c (y)dy λ = tip = S ∫ c root chord −b 2 root ⎛ 2⎞ 1+ λ + λ 2 = ⎜ ⎟ croot [for trapezoidal wing] ⎝ 3⎠ 1+ λ 31 Location of Mean Aerodynamic Chord and Aerodynamic Center • Axial location of the wings subsonic aerodynamic center (a.c.) – Determine spanwise location of m.a.c. – Assume that aerodynamic center is at 25% m.a.c. Trapezoidal Wing Elliptical Wing Mid - chord line 32 16 3-Dimensional Aerodynamic Lift and Drag Insect Wing (flat plate) Delta Wing 33 • Washout twist Wing Twist Effects – reduces tip angle of attack – typical value: 2 - 4 – changes lift distribution (interplay with taper ratio) – reduces likelihood of tip stall – allows stall to begin at the wing root • separationburble produces buffet at tail surface, warning of stall – improves aileron effectiveness at high α Talay, NASA SP34-367 17 Aerodynamic Strip Theory • Airfoil section may vary from tip-to-tip – Chord length – Airfoil thickness – Airfoil profile – Airfoil twist • 3-D Wing Lift: Integrate 2-D lift coefficients of airfoil sections across finite span Incremental lift along span dL = CL (y)c(y)qdy 2−D Aero L-39 Albatros dCL (y) = 3−D c(y)qdy dy 3-D wing lift b/2 L3 D = CL (y)c(y)q dy − ∫ 2−D −b/2 35 Bombardier Dash 8 Handley Page HP.115 Effect of Aspect Ratio on 3-Dimensional Wing Lift Slope Coefficient (Incompressible Flow) High Aspect Ratio (> 5) Wing #∂C & 2π AR # AR & C % L ( = = 2π % ( Lα $ ∂α ' AR+ 2 $ AR+ 2 ' 3−D Low Aspect Ratio (< 2) Wing π AR # AR& C = = 2π % ( Lα 2 $ 4 ' 36 18 Effect of Aspect Ratio on 3-D Wing Lift Slope Coefficient (Incompressible Flow) All Aspect Ratios (Helmbold equation) π AR CL = α ) 2 , # AR& +1+ 1+% ( . *+ $ 2 ' -. 37 Effect of Aspect Ratio on 3-D Wing Lift Slope Coefficient All Aspect Ratios (Helmbold equation) Wolfram Alpha (https://www.wolframalpha.com/) plot(pi A / (1+sqrt(1 + (A / 2)^2)), A=1 to 20) π AR CL = α ) 2 , # AR& +1+ 1+% ( .
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