The Physics of Flight 100 Years Since the Wright Brothers

Transcription

1 The Physics of Flight 100 Years Since the Wright Brothers by Donald A. Gurnett Colloquium presented in the Dept. of Physics and Astronomy, University of Iowa, Iowa City, Iowa, December 12, 2003.

2 Sir George Cayley, Showed lift is proportional to velocity squared and sin α, 1804 Wrote a three-part paper on Aerial Navigation, Designed the first successful glider

3 A replica of Caley s glider, flown by Derek Piggott, 1973

4 Otto Lilienthal ( ) Measured the lift and drag of a wing Made over 2000 flights in a glider, some as far as 350m Wrote a book The flight of birds as the basis for the art of flying, First person killed in an aircraft accident, 1896.

5 Orville and Wilbur Wright ( ; ) Knew of Cayley and Lilienthal s work Made one of the first wind tunnels, 1901 Invented wing warping as a method of roll control,1902

6 Wright Brothers 1902 Glider first full 3-axis controlled flight

7 Orville Wright, Dec. 17, 1903 first controlled powered flight

8 Status of Aerodynamic Theory in the Early 1900s All of the basic equations of fluid mechanics were known. However, no acceptable theory existed for the lift force on a wing. Worst than that, the existing theories predicted that the lift was exactly zero.

9 Bernoulli s Theorem and the Lift on a Wing The usual argument: 1 ρ U 2 p constant m + = 2 ρ m = mass density, U= velocity p = F/A is the pressure F = force, A = Area l l Since it follows that U 1 > U 2 so p 1 < p 1> 2 Therefore, F 2. 2 > F 1. (WRONG). One cannot assume that the stagnation point (where the streamlines separate) is exactly at the leading edge. One must solve for the entire flow pattern over the wing, which varies considerably with the angle of attack.

10 Some Key Concepts Vorticity r r = U r ω Circulation Γ = c Ud r r l r r r r r Theorem: ω = ( U) = 0 Vortex lines are continuous r r r r r = = UdA r r Γ = da Theorem: Γ cudl s s ω

11 Basic Equations and Assumptions Continuity equation Navier Stokes equation ρ m The assumption of incompressibility If ρ m = constant, then The vorticity equation Reynolds Number ρ t m r r + ( ρ U) = 0 m r U r r r r r 2 1 r r r + (U )U = p + ρmν U + ( U) t 3 r ω t R N r r U= 0 r ω r r = (U ) + ν r ω Convection Diffusion convection UL = diffusion ν 2

12 Reynolds Number Typical values (at sea level) U L R N Commercial Jet 600 mph 20 ft 1.1 x 10 8 Light plane 100 mph 5 ft 4.7 x 10 6 Glider 60 mph 3 ft 1.6 x 10 6 Model airplane 40 mph 8 in 2.5 x 10 5 Seagull 20 mph 4 in 6.2 x 10 4 Butterfly 5 mph 1 in 3.9 x 10 3 Housefly 5 mph 0.3 in 1.2 x 10 2

13 Basic Equations and Assumptions (con t) The inviscid assumption, RN 1 If ν = 0, then r U r r r r ρ m + (U )U = p t Kelvin s theorem If ρ m = constant and ν = 0, then r r Γ = s ω da = constant (Euler s equation) Velocity potentials r r r If Γ = 0 in the upstream flow, then ω = U= 0 at all points in the flow. It follows then that r r U= Φ Laplace s equation r r From the continuity equation, U = 0, one then has Φ Φ Φ = 0, or + = x y

14 Complex Potentials If z = x + iy, any analytic complex function provides a solution to Laplace s equation F(z) = Φ (x,y) + iψ(x,y) Examples: 2 and Φ = 0 Ψ = 0 2 Uniform flow At an angle α Dipole Vortex Uz 0 Uze 0 iα µ i Γπ 0 nz 2 l 0 z

15 Flow Around a Cylinder Horizontal flow At an angle α With circulation 2 0 a U z z + α 2 i 0 a U z e z + Γ π i a z U z n z 2 a + + l

16 The Blasius Force Equation ρm 2 Fx ify = i c W dz, 2 where W(z) = df/dz = U iu is the complex velocity. Γ0 1 1 W(z) = U0 + i + bn 2π z n z 2 2 c 0Γ0 W dz= 2πi Res(W ) = i2u F = 0 F x = ρ U Γ y m 0 0 x n y

17 The Joukowski Transformation, 1910 The Joukowski transformation z' = z+ 2 c z,where z = x + iy, transforms a cylinder into a flat plate

18 Flow Around a Flat Plate Airfoil Since Γ 0 = 0, by the Blasius theorem F y = 0. The dilemma: Since the upstream vorticity is zero the circulation must be zero, so there can be no lift. But a flat plate airfoil produces lift, so there must be circulation.

19 The Kutta-Joukowski Condition, 1910 The flow must be smooth and continuous at the trailing edge. Requires a circulation, Γ0= 4πU0asin α. F = ρ Γ y m 0 0 L= C L ( 1 2 ρ ) m U 2 U L = = ( 1 ) 2 ρ U A m 0 2 A2πsinα 2πsinα

20 The Joukowski Family of Airfoils ( t ) CL = 2π sin( α+ β0), β0= 2h/ l l

21 Comparison with Experimental Data

22 Wind Tunnel Observations α = 5 Kutta condition satisfied α = 10 Slight flow separation α = 15 Complete flow separation (stall)

23 The Maximum Lift Coefficient

24 Origin of the Circulation r r r ω r r = x(ux ω ) + ν 2 t ω L L

25 Where is the Vorticity? (In the boundary layer)

26 Finite Wing Span Effects (continuity of ω r ) Note: Γ 1 = Γ 2 = Γ 0, vortex trails cannot be avoided.

27 Vortex Trails

28 Vortex Trails

29 The Induced Downflow at the Wing Note: The downflow velocity at the wing is exactly (1/2) of the downstream value (for a straight wing)

30 Induced Drag, D i Upstream At the wing D i = component of L in the U 0 direction δ U y/u 0= angle of the downflow at the wing

31 The Elliptical Wing Theorem Prandtl, D i L= ρ U Γ(z)dz s m 0 s s s ρm d ΓΓ(z') = dzdz' 4 s sdz z z' Problem: Minimize D i, while holding L constant 1/ 2 2 Answer: Γ(z) = Γ c 1 (z / s), i.e., an ellipse

32 The Total Drag Force Induced Drag Viscous Drag 1 A L Di = 4 π 2 s 1 2 ρmua 2 2 D Note, A/s 2 = 1/Aspect ratio 1 = ρ 2 2 f mu ACDf

33 Some Elliptical Wings

34 Winglets

35 Viscous Drag Reduction Laminar Flow Airfoils, NACA 1930s

36 First Use of a Laminar Flow Airfoil, P-51, 1940

37 Shock Wave Compressibility Effects δ δ v ~p, = 1/5 s Mach Number (Ernst Mach, 1889) M = U 0 /V S Mach Cone Detached Shock Attached Shock

38 Thin Wing Theory (Theodor von Karman, 1940s) Laplace s equation modified for compressibility effects = Φ Φ ( ) 1 M x y M < 1 Elliptical differential equation M > 1 Hyperbolic differential equation Solution: transform to M = 0 Solution: wave equation x' = x 1 M 2 1 Φ = f y x 2 M 1

39 The Flat Plate Airfoil Subsonic (M < 1) Supersonic (M > 1) C L = 2π 1 M 2 sinα C L = 4α 2 M 1 Note the change in the center of pressure, from l/4 to l/2.

40 The Critical Mach Number

41 A Shock at the Critical Mach Number

42 Sweepback (Busemann, 1935) Sweepback increases the critical Mach number First use of sweepback Me-262, 1941

43 Conventional Airfoil Supercritical Airfoil (Whitcomb, 1971) Supercritical Airfoil Boeing 777 (M c = 0.85)

44 A Wind Tunnel in Your Computer

45 Course Advertisement Consider taking Mechanics of Continua, 29:211 12:15 to 1:30 p.m., T-Th, 618 Van Allen Hall Topics covered include the fundamental equations of fluid mechanics, incompressible and compressible flows in 2 and 3 dimensions, wave propagation, shock waves, instabilities, turbulence, and boundary layer physics Prerequisites: working knowledge of vector calculus, i.e., curl, divergence, gradient and associated identities

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