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 Low Pressure Low Pressure 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 Low Pressure Low Pressure High Pressure Upper surface (upper side of wing): low pressure Lower surface (underside of wing): high pressure

AIRFOILS VERSUS WINGS Low Pressure Low Pressure High Pressure 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] 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 Chord line Local relative wind 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: Increase in drag, called induced drag (drag due to lift) Angle of attack is effectively reduced, aeff as compared with V∞

ANGLE OF ATTACK DEFINITIONS Chord line 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 Chord line aeffective Local Relative Wind, V∞ 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 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 Relative wind gets tilted downward under the airfoil LIFT is still always perpendicular to the RELATIVE WIND

FINITE WING DESCRIPTION Finite Wing Case 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 Local relative wind is canted downward, lift vector is tilted back so a component of L acts in direction normal to incoming relative wind Wing tip vortices alter surface pressure distributions in direction of increased drag Vortices contain rotational energy put into flow by propulsion system to overcome induced drag

INDUCED DRAG: IMPLICATIONS FOR WINGS V∞ Infinite Wing (Appendix D) Finite Wing

HOW TO ESTIMATE INDUCED DRAG 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 May be calculated from Inviscid theory: Lifting line theory Look up

INFINITE VERSUS FINITE WINGS High AR Aspect Ratio b: wingspan S: wing area Low AR b

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 Special Case: Elliptical Lift Distribution (produced by elliptical wing) Lift/unit span varies elliptically along span This special case produces a uniform downwash Key Results: Elliptical Lift Distribution

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

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 Goes with square of CL Inversely related to AR Drag due to lift Span Efficiency Factor

DRAG POLAR Total Drag = Profile Drag + Induced Drag { cd

EXAMPLE: U2 VS. F-15 U2 F-15 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?) Flies at high speed (and lower altitudes), so high q∞ → low angle of attack, low CL F-15 AR ~ 3 (WHY?)

EXAMPLE: U2 SPYPLANE Cruise at 70,000 ft Out of USSR missile range Air density, r∞, highly reduced In steady-level flight, L = W As r∞ 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 Flies at high speed at low angle of attack → low CL Induced drag < Profile Drag Low AR, Low S

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 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 NACA 64A209 NACA 64A210

WHY A LIFT DISTRIBUTION? WING MAY BE TWISTED

P&W / G.E. GP7000 FAMILY

THINK ABOUT WING LOADING: W/b2 F-15 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 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 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 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’ With finite wings, there is an induced angle of attack The angle you see ≠ the angle the finite wing ‘sees’ ageom= aeff + ai = aeff Finite Wing ageom= aeff + ai

FINITE WING CHANGE IN LIFT SLOPE Infinite 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 Effect of finite wing is to reduce lift curve slope Finite wing lift slope = a = dCL/da At CL = 0, ai = 0, so aL=0 same for infinite or finite wings Finite Wing

SUMMARY: INFINITE VS. FINITE WINGS Properties of a finite wing differ in two major respects from infinite wings: Addition of induced drag 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