MAE 3241: AERODYNAMICS AND
FLIGHT MECHANICS
Introduction to Finite Wings and Induced Drag
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
AIRFOILS VERSUS WINGS
• Upper surface (upper side of wing): low pressure
– Recall discussion on exactly why this is physically
– Recall discussion on how to show this mathematically
AIRFOILS VERSUS WINGS
• Upper surface (upper side of wing): low pressure
• Lower surface (underside of wing): high pressure
AIRFOILS VERSUS WINGS
• Upper surface (upper side of wing): low pressure
• Lower surface (underside of wing): high pressure
• Flow always desires to go from high pressure to low pressure
• Flow ‘wraps’ around wing tips
FINITE WINGS
Front View
Wing Tip
Vortices
FINITE WINGS
FINITE WINGS
FINITE WINGS
AMERICAN AIRLINES FLIGHT 587
•
American Airlines Flight 587 crashed into the Belle Harbor neighborhood of Queens in
New York City shortly after takeoff from John F. Kennedy International Airport on
November 12, 2001. This was the second deadliest U.S. aviation accident to date.
•
On November 12, 2001, about 0916:15 eastern standard time, American Airlines flight 587,
an Airbus Industries A300-605R, N14053, crashed into a residential area of Belle Harbor,
New York, shortly after takeoff from John F. Kennedy International Airport, Jamaica, New
York. Flight 587 was a regularly scheduled passenger flight to Las Americas International
Airport, Santo Domingo, Dominican Republic, with 2 flight crewmembers, 7 flight
attendants, and 251 passengers aboard the plane. The plane’s vertical stabilizer and rudder
separated in flight and were found in Jamaica Bay, about 1 mile north of the main wreckage
site. The plane’s engines subsequently separated in flight and were found several blocks
north and east of the main wreckage site. All 260 people aboard the plane and 5 people on
the ground were killed, and the plane was destroyed by impact forces and a postcrash fire.
•
The official National Transportation Safety Board (NTSB) report of October 26, 2004,
stated that the cause of the crash was the overuse of the rudder to counter wake turbulence
AMERICAN AIRLINES FLIGHT 587
•
The A300-600, which took off just minutes after a Japan Airlines Boeing 747 on the same runway, flew
into the larger jet's wake, an area of very turbulent air. The first officer attempted to keep the plane
upright with the rudder. The strength of the air flowing against the moving rudder stressed the aircraft's
vertical stabilizer and eventually snapped it off entirely, causing the aircraft to lose control and crash.
•
Investigators were concerned regarding the manner in which the tail fin separated. The tail fin is
connected to the fuselage with six attaching points, each set has two sets of nuts, one made out of
composite material, another from aluminum which is connected by a titanium bolt, however damage
analysis showed the bolts and aluminum lugs were intact but not the composite lugs. There were fears
that the composites were faulty because they are used in other areas of the plane including the engine
mounting and the wings, however examinations of construction and the materials gave the plane a
clean bill of health.
Airbus and American are currently disputing the extent to which the two parties are responsible for the
disaster. American charges that the crash was mostly Airbus's fault, because the A300 was designed
with unusually sensitive rudder controls. Most aircraft require increased pressure on the rudder pedals
to achieve the same amount of rudder control at a higher speed. The Airbus A300 and later A310 do not
operate on a fly-by-wire flight control system, instead using conventional mechanical flight controls.
The NTSB determined that "because of its high sensitivity, the A300-600 rudder control system is
susceptible to potentially hazardous rudder pedal inputs at higher speeds."[3]
•
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Airbus charges that the crash was mostly American's fault, because the airline did not train its pilots
properly about the characteristics of the rudder. Aircraft tail fins are designed to withstand full rudder
in one direction at maneuvering speed. However, they are not usually designed to withstand an abrupt
shift in rudder from one direction to the other. Most American pilots believed that the tail fin could
withstand any rudder movement at maneuvering speed.
http://www.ntsb.gov/events/2001/AA587/default.htm
MINIMUM SEPARATION RULES
• An aircraft of a lower wake vortex category must not be allowed to take off less
than two minutes behind an aircraft of a higher wake vortex category
• If the following aircraft does not start its take off roll from the same point as the
preceding aircraft, this is increased to three minutes
EXAMPLE: 737 WINGLETS
FINITE WING DOWNWASH
• Wing tip vortices induce a small downward component of air velocity near
wing by dragging surrounding air with them
• Downward component of velocity is called downwash, w
Chord line
Local relative wind
•
Two Consequences:
1. Increase in drag, called induced drag (drag due to lift)
2. Angle of attack is effectively reduced, aeff as compared with V∞
ANGLE OF ATTACK DEFINITIONS
Relative Wind, V∞
ageometric
ageometric: what you see, what you would see in a wind tunnel
Simply look at angle between incoming relative wind and chord line
This is a case of no wing-tips (infinite wing)
ANGLE OF ATTACK DEFINITIONS
aeffective
aeffective: what the airfoil ‘sees’ locally
Angle between local flow direction and chord line
Small than ageometric because of downwash
The wing-tips have caused this local relative wind to be angled downward
ANGLE OF ATTACK DEFINITIONS
a geometric  a effective  a induced
ageometric: what you see, what you would see in a wind tunnel
Simply look at angle between incoming relative wind and chord line
aeffective: what the airfoil ‘sees’ locally
Angle between local flow direction and chord line
Small than ageometric because of downwash
ainduced: difference between these two angles
Downwash has ‘induced’ this change in angle of attack
INFINITE WING DESCRIPTION
LIFT
Relative Wind, V∞
•
LIFT is always perpendicular to the RELATIVE WIND
•
All lift is balancing weight
FINITE WING DESCRIPTION
Finite Wing Case
•
•
a geometric  a effective  a induced
Relative wind gets tilted downward under the airfoil
LIFT is still always perpendicular to the RELATIVE WIND
FINITE WING DESCRIPTION
Finite Wing Case
a geometric  a effective  a induced
Induced Drag, Di
•
•
•
Drag is measured in direction of incoming relative wind (that is the
direction that the airplane is flying)
Lift vector is tilted back
Component of L acts in direction parallel to incoming relative wind →
results in a new type of drag
3 PHYSICAL INTERPRETATIONS
a geometric  a effective  a induced
1. Local relative wind is canted downward, lift vector is tilted back so a
component of L acts in direction normal to incoming relative wind
2. Wing tip vortices alter surface pressure distributions in direction of
increased drag
3. Vortices contain rotational energy put into flow by propulsion system to
overcome induced drag
INDUCED DRAG: IMPLICATIONS FOR WINGS
V∞
a eff  a
Finite Wing
Infinite Wing
(Appendix D)
CL  cl
C D  cd
HOW TO ESTIMATE INDUCED DRAG
Di  L sin a i
Di  La i
•
•
•
•
Local flow velocity in vicinity of wing is inclined downward
Lift vector remains perpendicular to local relative wind and is tiled back
through an angle ai
Drag is still parallel to freestream
Tilted lift vector contributes a drag component
TOTAL DRAG ON SUBSONIC WING
Profile Drag
Profile Drag coefficient
relatively constant with
M∞ at subsonic speeds
Also called drag due to lift
D  D friction  D pressure  Dinduced
D  D profile  Dinduced
Di
C D  cd , profile 
q S
Look up
May be calculated from
Inviscid theory:
Lifting line theory
INFINITE VERSUS FINITE WINGS
High AR
Aspect Ratio
b: wingspan
S: wing area
2
b
AR 
S
b
Low AR
HOW TO ESTIMATE INDUCED DRAG
•
•
•
Calculation of angle ai is not trivial (MAE 3241)
Value of ai depends on distribution of downwash along span of wing
Downwash is governed by distribution of lift over span of wing
HOW TO ESTIMATE INDUCED DRAG
•
•
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Special Case: Elliptical Lift Distribution (produced by elliptical wing)
Lift/unit span varies elliptically along span
This special case produces a uniform downwash
CL
ai 
AR
Key Results:
Elliptical Lift Distribution
CL
C L2
Di  La i  L
 q S
AR
AR
Di
C L2

q S AR
C D ,i
C L2

AR
ELLIPTICAL LIFT DISTRIBUTION
• For a wing with same airfoil shape across span and no twist, an elliptical lift
distribution is characteristic of an elliptical wing plan form
• Example: Supermarine Spitfire
Key Results:
Elliptical Lift Distribution
CL
ai 
AR
C L2
C D ,i 
AR
HOW TO ESTIMATE INDUCED DRAG
• For all wings in general
• Define a span efficiency factor, e (also called span efficiency factor)
• Elliptical planforms, e = 1
– The word planform means shape as view by looking down on the wing
• For all other planforms, e < 1
• 0.85 < e < 0.99
C D ,i
2
L
C

eAR
Span Efficiency Factor
Goes with square of CL
Inversely related to AR
Drag due to lift
DRAG POLAR
2
L
C
C D  cd 
eAR
Total Drag = Profile Drag + Induced Drag
cd
{
EXAMPLE: U2 VS. F-15
1
2
L  W   V SC L
2
U2
• Cruise at 70,000 ft
– Air density highly reduced
• Flies at slow speeds, low q∞ →
high angle of attack, high CL
• U2 AR ~ 14.3 (WHY?)
C L2
C D  cd 
eAR
F-15
• Flies at high speed (and lower
altitudes), so high q∞ → low
angle of attack, low CL
• F-15 AR ~ 3 (WHY?)
EXAMPLE: U2 SPYPLANE
2
L
C
C D  cd 
eAR
• Cruise at 70,000 ft
– Out of USSR missile range
– Air density, ∞, highly
reduced
• In steady-level flight, L = W
1
2
L  W   V SC L
2
• As ∞ reduced, CL must
increase (angle of attack must
increase)
• AR ↑ CD ↓
• U2 AR ~ 14.3
U2 stall speed at altitude is only ten knots (18 km/h) less than its maximum speed
EXAMPLE: F-15 EAGLE
2
L
C
C D  cd 
eAR
• Flies at high speed at low angle
of attack → low CL
• Induced drag < Profile Drag
• Low AR, Low S
1
2
L  W   V SC L
2
U2 CRASH DETAILS
•
•
•
http://www.eisenhower.archives.gov/dl/U2Incident/u2documents.html
NASA issued a very detailed press release noting that an aircraft had “gone missing” north
of Turkey
I must tell you a secret. When I made my first report I deliberately did not say that the pilot
was alive and well… and now just look how many silly things [the Americans] have said.”
NASA U2
MYASISHCHEV M-55 "MYSTIC" HIGH ALTITUDE
RECONNAISANCE AIRCRAFT
WHY HIGH AR ON PREDATOR?
AIRBUS A380 / BOEING 747 COMPARISON
•
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•
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Wingspan: 79.8 m
AR: 7.53
GTOW: 560 T
Wing Loading: GTOW/b2: 87.94
•
•
•
•
Wingspan: 68.5 m
AR: 7.98
GTOW: 440 T
Wing Loading: GTOW/b2: 93.77
AIRPORT ACCOMODATIONS
• Airplanes must fit into 80 x 80 m box
Proposed changes to JFK
WINGLETS, FENCES, OR NO WINGLETS?
• Quote from ‘Airborne with the Captain’ website
– http://www.askcaptainlim.com/blog/index.php?catid=19
• “Now, to go back on your question on why the Airbus A380 did not follow the
Airbus A330/340 winglet design but rather more or less imitate the old design
“wingtip fences” of the Airbus A320. Basically winglets help to reduce induced
drag and improve performance (also increases aspect ratio slightly). However, the
Airbus A380 has very large wing area due to the large wingspan that gives it a
high aspect ratio. So, it need not have to worry about aspect ratio but needs only
to tackle the induced drag problem. Therefore, it does not require the winglets, but
merely “wingtip fences” similar to those of the Airbus A320.”
• What do you think of this answer?
• What are other trade-offs for winglets vs. no winglets?
– Consider Boeing 777 does not have winglets
WHY A LIFT DISTRIBUTION?
CHORD MAY VARY IN LENGTH
Thinner wing near tip
delay onset of high-speed
compressibility effects
Retain aileron control
WHY A LIFT DISTRIBUTION?
SHAPE OF AIRFOIL MAY VARY ALONG WING
F-111
NACA 64A209
NACA 64A210
WHY A LIFT DISTRIBUTION?
WING MAY BE TWISTED
P&W / G.E. GP7000 FAMILY
THINK ABOUT WING LOADING: W/b2
1
2
L  W   V SC L
2
U2
• Cruise at 70,000 ft
– Air density highly reduced
• Flies at slow speeds, low q∞ →
high angle of attack, high CL
• U2 AR ~ 14.3
C L2
C D  cd 
eAR
F-15
• Flies at high speed (and lower
altitudes), so high q∞ → low
angle of attack, low CL
• F-15 AR ~ 3
REALLY HIGH ASPECT RATIO
•
•
•
•
L/D ratios can be over 50!
Aspect ratio can be over 40
All out attempt to reduce induced drag
What we learned from the '24 regarding boundary layer control led us to believe
that we could move the trip line back onto the control surfaces and still be
"practical".
EXAMPLE: NASA HELIOS
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Helios: Proof-of-concept solar-electric flying wing, designed to operate at extremely high
altitudes for long duration, remotely piloted aircraft, AR = 31:1
Helios Prototype designed to fly at altitudes of up to 100,000 feet on single-day atmospheric
science and imaging missions, as well as perform multi-day telecommunications relay
missions at altitudes from 50,000 to 65,000 feet.
Helios Prototype set world altitude record for winged aircraft, 96,863 feet, during a flight in
August 2001
Flight at 100,000 ft. is quite similar to that expected in the Martian atmosphere, so data
obtained from the record altitude flight will also help to build NASA's technical and
operational data base for future Mars aircraft designs and missions
NOMINAL FLIGHT
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•
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Wingspan: 247 ft, Chord: 8ft, Wing Thickness: 12% of Chord, Wing Area: 1,976 ft2
Airspeed: 19 to 27 MPH cruise at low altitudes, up to 170 MPH at extreme altitude
Altitude: Up to 100,000 ft., typical endurance mission at 50,000 to 70,000 ft.
Aspect Ratio: 30.9
EXAMPLE: NASA HELIOS
This view of the Helios Prototype from a chase helicopter shows abnormally high wing
dihedral of more than 30 feet from wingtip to the center of the aircraft that resulted after
the Helios entered moderate air turbulence on its last test flight. The extreme dihedral
caused aerodynamic instability that led to an uncontrollable series of pitch oscillations and
over-speed conditions, resulting in structural failures and partial breakup of the aircraft.
EXAMPLE: NASA HELIOS, JUNE 26, 2003
FINITE WING CHANGE IN LIFT SLOPE
Infinite Wing
• In a wind tunnel, the easiest thing to
measure is the geometric angle of attack
• For infinite wings, there is no induced
angle of attack
– The angle you see = the angle the
infinite wing ‘sees’
ageom= aeff + ai = aeff
Finite Wing
• With finite wings, there is an induced
angle of attack
– The angle you see ≠ the angle the
finite wing ‘sees’
a geom  a eff  a i
ageom= aeff + ai
FINITE WING CHANGE IN LIFT SLOPE
a geom  a eff  a i
Infinite Wing
•
Finite Wing
Lift curve for a finite wing has a smaller
slope than corresponding curve for an
infinite wing with same airfoil cross-section
– Figure (a) shows infinite wing, ai = 0, so
plot is CL vs. ageom or aeff and slope is a0
– Figure (b) shows finite wing, ai ≠ 0
• Plot CL vs. what we see, ageom, (or
what would be easy to measure in a
wind tunnel), not what wing sees, aeff
1. Effect of finite wing is to reduce lift curve slope
– Finite wing lift slope = a = dCL/da
2. At CL = 0, ai = 0, so aL=0 same for infinite or
finite wings
SUMMARY: INFINITE VS. FINITE WINGS
Properties of a finite wing differ in two major respects from infinite wings:
1. Addition of induced drag
2. Lift curve for a finite wing has smaller slope than corresponding lift curve for
infinite wing with same airfoil cross section
SUMMARY
• Induced drag is price you pay for generation of lift
• CD,i proportional to CL2
– Airplane on take-off or landing, induced drag major component
– Significant at cruise (15-25% of total drag)
• CD,i inversely proportional to AR
– Desire high AR to reduce induced drag
– Compromise between structures and aerodynamics
– AR important tool as designer (more control than span efficiency, e)
• For an elliptic lift distribution, chord must vary elliptically along span
– Wing planform is elliptical
– Elliptical lift distribution gives good approximation for arbitrary finite wing
through use of span efficiency factor, e