The Physics of Flight100 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.

Sir George Cayley, 1773-1857• Showed lift is proportional to velocity squared and sin α, 1804• Wrote a three-part paper on “Aerial Navigation,” 1809-1810• Designed the first successful glider

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

Otto Lilienthal (1848-1896)• 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,” 1886.• First person killed in an aircraft accident, 1896.

Orville and Wilbur Wright (1871-1948; 1867-1912)• 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

Wright Brothers’ 1902 Gliderfirst full 3-axis controlled flight

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

• 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.

Status of Aerodynamic Theory in the Early 1900s

2m

1ρ U p constant2 + =

p F / A is the pressure=

F = force, A = Area

Since it follows that U1 > U2 so p1 < p2. Therefore, F2 > F1.(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.

Bernoulli’s Theorem and the Lift on a Wing

1 2>l l

ρm mass density, U velocity= =

The usual argument:

Vorticity Circulation

ω U=∇×rrr Γ c U d= ⋅∫

r rl

Theorem: ω ( U) 0∇⋅ =∇⋅ ∇× =rr r rr

Vortex lines are continuous

Theorem: Γ c sU d U dA= ⋅ = ∇× ⋅∫ ∫r r rr r

l

Γ ωs dA= ⋅∫rr

Some Key Concepts

Basic Equations and Assumptions• Continuity equation

• Navier Stokes equation

• The assumption of incompressibility If ρm = constant, then

• The vorticity equation

Convection Diffusion• Reynolds Number

ρ ρmm( U) 0

t∂

+∇⋅ =∂

rr

ρ ρ ν 2m m

U 1(U )U p U ( U)t 3

∂ + ⋅ ∇ =−∇ + ∇ + ∇ ∇⋅ ∂

rr r r rr r r r

U 0∇⋅ =rr

νω ω ω2(U )t

∂= ∇ × × + ∇

∂

rrr

r r

νN| convection | ULR| diffusion |

=

Reynolds Number

Typical values (at sea level)

1.2 x 1020.3 in5 mphHousefly

3.9 x 1031 in5 mphButterfly

6.2 x 1044 in20 mphSeagull

2.5 x 1058 in40 mphModel airplane

1.6 x 1063 ft60 mphGlider

4.7 x 1065 ft100 mphLight plane

1.1 x 10820 ft 600 mphCommercial Jet

RNLU

• The inviscid assumption, If , then

(Euler’s equation)

• Kelvin’s theoremIf ρm = constant and = 0, then

• Velocity potentialsIf Γ = 0 in the upstream flow, then at all points in the flow. It follows then that

• Laplace’s equationFrom the continuity equation, , one then has

Basic Equations and Assumptions (con’t)

Γ ωs

dA constant= ⋅ =∫rr

ω U 0=∇× =rrr

ρmU (U )U pt

∂+ ⋅∇ =−∇ ∂

rr rr r

ΦU=∇r r

Φ ΦΦ2 2

22 20, or 0

x y∂ ∂

∇ = + =∂ ∂

U 0∇ ⋅ =rr

ν 0=NR 1

ν

Complex Potentials

)y,x(i)y,x()z(F ΨΦ +=

02 =∇ Φ 02 =∇ Ψ

If z = x + iy, any analytic complex functionprovides a solution to Laplace’s equation

Examples:

Uniform flow At an angle α Dipole Vortex0U z αi

0U ze−

z0µ Γ

π0i nz2 l

and

Flow Around a CylinderHorizontal flow At an angle α With circulation

2

0aU zz

+

α

2i

0aU z ez

− +

Γπ

20

0ia zU z n

z 2 a + +

l

The Blasius Force Equation

ρ

Γπ

π Γ

ρ Γ

2mx y c

x yn

00 n

n

2 2c 0 0

x

y m 0 0

F iF i W dz ,2

where W (z) dF/dz U iU is the complex velocity.

1 1W (z) U i b2 z z

W dz 2 i Res (W ) i2UF 0F U

− = ∫

= = −

= + + ∑

∫ = ∑ = −

=

=

The Joukowski Transformation, 1910The Joukowski transformation

2cz ' z z= +

transforms a cylinder into a flat plate

,where z’ = x’ + iy’,

Flow Around a Flat Plate Airfoil

Since Γ0 = 0, by the Blasius theorem Fy = 0.

The dilemma: Since the upstream vorticity is zerothe circulation must be zero, so there can be no lift.

But a flat plate airfoil produces lift, so there must be circulation.

The Kutta-Joukowski Condition, 1910

( )

( )

ρ Γ

π α

π αρ

ρ

y m 0 0

L 2m 0

2m

12

12

F U

L A 2 sin

LC 2 sinAU

U

=

=

= =

The flow must be smooth and continuous at the trailing edge. Requires a circulation, Γ π α0 04 U asin .=

The Joukowski Family of Airfoils

( )π α β βL 0 0tC 2 1 0.77 sin( ), 2h /= + + =l

l

Comparison with Experimental Data

Wind Tunnel Observations

α = 5˚

α = 10˚

α = 15˚

Kutta conditionsatisfied

Slight flowseparation

Complete flowseparation(stall)

The Maximum Lift Coefficient

Origin of the Circulation

νω ω ω2x (Ux )t∂ =∇ + ∇∂

rrr r r

1L 2

1L

Where is the Vorticity?(In the boundary layer)

Finite Wing Span Effects(continuity of )

Note: Γ1 = Γ2 = Γ0, vortex trails cannot be avoided.

ωr

Vortex Trails

Vortex Trails

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)

Induced Drag, Di

Upstream At the wing

Di = component of L in the U0 direction

angle of the downflow at the wingδ y 0U /U =

The Elliptical Wing Theorem Prandtl, 1918-1919

Problem: Minimize Di, while holding L constant

ρ Γs

m 0s

L U (z)dz−

= ∫

ρ Γ Γs sm

is s

d (z ')D dzdz '4 dz z z '− −

= − ∫ ∫−

Answer: , i.e., an ellipseΓ Γ1/ 22

c(z) 1 (z / s) = −

The Total Drag ForceInduced Drag

π ρ

2

i 2 2m

1 A LD 4 1s U A2

=

Note, A/s2 = 1/Aspect ratio

ρ 2f m Df

1D U A C2=

Viscous Drag

Some Elliptical Wings

Winglets

Viscous Drag ReductionLaminar Flow Airfoils, NACA 1930s

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

Compressibility EffectsShock Wave

Detached Shock Attached ShockMach Cone

Mach Number(Ernst Mach, 1889)

M = U0/VS

δ δsv ~p , 1/5=

Thin Wing Theory(Theodor von Karman, 1940s)

2xx '

1 M=

−

( ) Φ Φ2 22

2 2 01 Mx y

∂ ∂+ =−∂ ∂

Laplace’s equation modified for compressibility effects

M < 1Elliptical differential equation

M > 1Hyperbolic differential equation

Solution: transform to M = 0 Solution: wave equation

Φ2

1y xM 1

f −−

=

The Flat Plate Airfoil

αL 2

4CM 1

=−

Subsonic (M < 1) Supersonic (M > 1)

π αL 22C sin

1 M=

−

Note the change in the center of pressure, from ./4 to /2l l

The Critical Mach Number

A Shock at the Critical Mach Number

Sweepback (Busemann, 1935)

Sweepback increases the critical Mach number

First use of sweepbackMe-262, 1941

Supercritical Airfoil(Whitcomb, 1971)

Boeing 777 (Mc = 0.85)

Conventional Airfoil Supercritical Airfoil

A Wind Tunnel in Your Computer

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