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

2-Dimensional Aerodynamic Lift and Drag

1 Wing Lift and Drag

• Lift: Perpendicular to free-stream airflow

• Drag: Parallel to the free-stream airflow

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)

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

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

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

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

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

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 kinematic viscosity (SL) 1.343 10−5 m / s2 ν = = ×

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)

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

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

13 Description of Aircraft Configurations

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

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

16 3-Dimensional Aerodynamic Lift and Drag

Insect Wing (flat plate)

Delta Wing

• 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 AR 2 AR 2 $ ∂α '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 ' -.

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+% ( . *+ $ 2 ' -.

19 Wing-Fuselage Interference Effects

• Wing lift induces – Upwash in front of the wing affects canard – Downwash behind the wing affects aft tail – Local angles of attack modified, affecting net lift and pitching moment • Flow around fuselage induces upwash on the wing, canard, and tail

from Etkin 39

Longitudinal Control Surfaces

Wing-Tail Configuration

Flap Elevator

Delta-Wing Configuration

Elevator 40

20 Angle of Attack and Control Surface Deflection

• Horizontal tail with elevator control surface

• Horizontal tail at positive angle of attack

• Horizontal tail with positive elevator deflection

Control Flap Carryover Effect on Lift Produced By Total Surface

from Schlichting & Truckenbrodt

C c LδE vs. f C x + c Lα f f

42 c f (x f + c f )

€

21 Bell X-1 Aileron Carryover Effect M = 0.13, Re = 1.2 x 106

q p p − ( − o ) SP ! , pressure coefficient q

Area proportional to lift

NACA-RM-L53L18, 1954 43

Lift due to Elevator Deflection Lift coefficient variation due to elevator deflection

∂CL Sht CL ! = τ htηht CL δ E ∂δ E ( α )ht S ΔC = C δ E L Lδ E

τ ht = Carryover effect

ηht = Tail efficiency factor

CL = Horizontal tail lift-coefficient slope ( α )ht

Sht = Horizontal tail reference area Lift variation due to elevator deflection ΔL = C qSδ E Lδ E 44

22 Example of Configuration and Flap Effects

Next Time: Induced Drag and High-Speed Aerodynamics

Reading: Flight Dynamics Aerodynamic Coefficients, 85-96 Airplane Stability and Control Chapter 1 Learning Objectives Understand drag-due-to-lift and effects of wing planform Recognize effect of angle of attack on lift and drag coefficients How to estimate Mach number (i.e., air compressibility) effects on aerodynamics Be able to use Newtonian approximation to estimate lift and drag

23 Supplementary Material

Thin Airfoil Theory

Downward velocity, w, at xo due to vortex at x Differential Integral

γ (x)dx 1 1 γ (x) dw xo = w xo = dx ( ) ( ) ∫0 2π (xo − x) 2π (xo − x) Boundary condition: flow tangent to mean camber line

w(xo ) ⎛ dz ⎞ = α − V ⎝⎜ dx ⎠⎟ xo McCormick, 1995 48

24 Thin Airfoil Theory Integral equation for vorticity

1 1 γ (x) ⎛ dz ⎞ dx = α − ⎜ ⎟ 2 V ∫0 x x ⎝ dx ⎠ π ( o − ) xo

Coordinate transformation 1 x = (1− cosθ ) 2

Solution for vorticity Coefficients

1 π dz ∞ A0 = α − dθ ⎡ 1+ cosθ ⎤ π ∫0 dx γ = 2V A0 + An sinnθ ⎢ ∑ ⎥ π ⎣ sinθ n=1 ⎦ 2 dz An = cosnθ dθ π ∫0 dx

McCormick, 1995 49

Thin Airfoil Theory

Lift, from Kutta-Joukowski theorem

1 L = ρVγ x dx = 2π A + π A ∫ ( ) 0 1 0 For thin airfoil with circular arc

A0 = α, A1 = 4zmax C = 2πα + 4πz = C α + C [Circular arc] L2−D max Lα Lo =C α [Flat plate] Lα

∂C C = L = 2π Lα ∂α McCormick, 1995 50

25 Effect of Aspect Ratio on 3-Dimensional Wing Lift Slope Coefficient

• High Aspect Ratio (> 5) Wing • Wolfram Alpha

plot(2 pi (a/(a+2)), a=5 to 20)

• Low Aspect Ratio (< 2) Wing • Wolfram Alpha

plot(2 pi (a / 4), a=1 to 2)

Aerodynamic Stall, Theory and Experiment • Flow separation produces stall • Straight rectangular wing, AR = 5.536, NACA 0015 • Hysteresis for increasing/decreasing α

Anderson et al, 1980 52

26 Maximum Maximum Lift of Lift Coefficient, Rectangular Wings CL max

Schlicting & Truckenbrodt, 1979

Angle of Attack for CL max

Aspect Ratio ϕ : Sweep angle δ: Thickness ratio 53

Maximum Lift of Delta Wings with Straight Trailing Edges

Maximum Lift Angle of Attack Coefficient, CL max for CL max

Aspect Ratio Aspect Ratio λ: Taper ratio Schlicting & Truckenbrodt, 1979 54

27 Typical Effect of Reynolds Number on Parasitic Drag • Flow may stay attached farther at high Re, reducing the drag

from Werle*

* See Van Dyke, M., An Album of Fluid Motion55 , Parabolic Press, Stanford, 1982

Aft Flap vs. All-Moving Control Surface • Carryover effect of aft flap – Aft-flap deflection can be almost as effective as full surface deflection at subsonic speeds – Negligible at supersonic speed • Aft flap – Mass and inertia lower, reducing likelihood of mechanical instability – Aerodynamic hinge moment is lower – Can be mounted on structurally rigid main surface

28 Historical Factoid Samuel Pierpoint Langley (1834-1906)

• Astronomer supported by Smithsonian Institution • Whirling-arm experiments • 1896: Langley's steam-powered Aerodrome model flies 3/4 mile • Oct 7 & Dec 8, 1903: Manned aircraft flights end in failure

Multi-Engine Aircraft of World War II

Consolidated B-24 Boeing B-29 Boeing B-17

Douglas A-26 • Large W.W.II aircraft had unpowered controls: – High foot-pedal force North American B-25 – Rudder stability problems arising from balancing to reduce pedal force

• Severe engine-out problem Martin B-26 for twin-engine aircraft

29 Medium to High Aspect Ratio Configurations

Cessna 337 DeLaurier Ornithopter Schweizer 2-32

Vtakeoff = 82 km/h hcruise = 15 ft

Vcruise = 144 mph hcruise = 10 kft

Mtypical = 75 mph hmax = 35 kft

Mcruise = 0.84 hcruise = 35 kft

• Typical for subsonic aircraft

Boeing 777-300 59

Uninhabited Air Vehicles

Northrop-Grumman/Ryan Global Hawk General Atomics Predator

Vcruise = 70-90 kt hcruise = 25 kft

Vcruise = 310 kt hcruise = 50 kft

30 Stealth and Small UAVs

Lockheed-Martin RQ-170 General Atomics Predator-C (Avenger)

Northrop-Grumman X-47B InSitu/Boeing ScanEagle

http://en.wikipedia.org/wiki/Stealth_aircraft 61

Subsonic Biplane

• Compared to monoplane – Structurally stiff (guy wires) – Twice the wing area for the same span – Lower aspect ratio than a single wing with same area and chord – Mutual interference – Lower maximum lift – Higher drag (interference, wires) • Interference effects of two wings – Gap – Aspect ratio – Relative areas and spans – Stagger

31 Some Videos Flow over a narrow airfoil, with downstream vortices http://www.youtube.com/watch?v=zsO5BQA_CZk Flow over transverse flat plate, with downstream vortices

http://www.youtube.com/watch?v=0z_hFZx7qvE Laminar vs. turbulent flow http://www.youtube.com/watch?v=WG-YCpAGgQQ&feature=related

Smoke flow visualization, wing with flap http://www.youtube.com/watch?feature=fvwp&NR=1&v=eBBZF_3DLCU/ 1930s test in NACA wind tunnel

http://www.youtube.com/watch?v=3_WgkVQWtno&feature=related 63