Conguration Aerodynamics - 1
Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2010
Conguration Variables Lift
Effects of shape, angle, and Mach number Stall
Description of Aircraft Conguration
Parasitic Drag
Skin friction Base drag
Copyright 2010 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE331.html http://www.princeton.edu/~stengel/FlightDynamics.html
A Few Denitions
Wing Planform Variables
Aspect Ratio Taper Ratio
b AR = rectangular wing c b ! b b2 = any wing = c!b S
Rectangular Wing Delta Wing
!=
ctip croot
Swept Trapezoidal Wing
Republic F-84F
Mean Aerodynamic Chord and Wing Center of Pressure
Mean aerodynamic chord (m.a.c.) ~ mean geometric chord
c= 1 2 " c ( y ) dy S !b 2 # 2& 1+ ) + ) =% ( croot $ 3' 1 + )
2 b2
Medium to High Aspect Ratio Congurations
Cessna 337 DeLaurier Ornithopter Schweizer 2-32
[for trapezoidal wing]
Trapezoidal Wing
Axial location of the wing!s subsonic aerodynamic center (a.c.)
Determine spanwise location of m.a.c. Assume that aerodynamic center is at 25% m.a.c.
Midchord line
Typical for subsonic aircraft
Elliptical Wing
from Raymer from Sunderland
Boeing 777-300
Low Aspect Ratio Congurations
North American A-5A Vigilante
Variable Aspect Ratio Congurations
North American B-1 General Dynamics F-111
Typical for supersonic aircraft
Lockheed F-104 Starghter
Aerodynamic efciency at sub- and supersonic speeds
Reconnaissance Aircraft
Lockheed U-2 (ER-2) Lockheed SR-71 Trainer
Uninhabited Air Vehicles
Northrop-Grumman/Ryan Global Hawk General Atomics Predator
Subsonic, high-altitude ight
Supersonic, high-altitude ight
Stealth and Small UAVs
Northrop-Grumman X-47B General Atomics Predator-C (Avenger)
Re-entry Vehicles
Northrop HL-10 Martin Marietta X-24A
Northrop M2-F2
InSitu/Boeing ScanEagle
JAXA ALFLEX
NASA X-38
Martin Marietta X-24B
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)
Aerodynamic Lift and Drag
Interference effects of two wings
Gap Aspect ratio Relative areas and spans Stagger
Longitudinal Aerodynamic Forces and Moment of the Airplane
Non-dimensional force coefcients are dimensionalized by
dynamic pressure, q reference area, S Typical subsonic lift, drag, and pitching moment variations with angle of attack
Circulation of Incompressible Air Flow About a 2-D Airfoil
Bernoulli!s equation (inviscid, incompressible ow)
pstatic +
Vorticity
1 2 !V = constant along streamline = pstagnation 2
Vupper (x) = V! + "V (x) 2 Vlower (x) = V! # "V (x) 2
Non-dimensional moment coefcients also dimensionalized by
reference length, c
! 2 " D (x) =
Circulation
c
#V (x) #z(x)
Lower pressure on upper surface
Lift = C L q S Drag = C D q S Pitching Moment = Cm q Sc
! 2 " D = $ # 2 " D (x)dx
0
What Do We Mean by 2-Dimensional Aerodynamics?
Finite-span wing > nite aspect ratio
AR = b rectangular wing c b ! b b2 = any wing = c!b S
What Do We Mean by 2Dimensional Aerodynamics?
Assuming constant chord section, the 2-D Lift is the same at any y station of the innite-span wing
1 2 1 "V S = C L3! D "V 2 ( bc ) [Rectangular wing] 2 2 1 2 # ( Lift 3! D ) = C L3! D "V c#y 2 1 1 % ( lim # ( Lift 3! D ) = lim ' C L3! D "V 2 c#y* + "2-D Lift" = C L2! D "V 2 c #y$0 #y$0 & ) 2 2 Lift 3! D = C L3! D
Innite-span wing > innite aspect ratio
For Small Angles, Lift is Proportional to Angle of Attack
Unswept wing, 2-D lift slope coefcient
Inviscid, incompressible ow Referenced to chord length, c, rather than wing area
Classic Airfoil Proles
NACA 4-digit Proles (e.g., NACA 2412)
http://en.wikipedia.org/wiki/NACA_airfoil Maximum camber as percentage of chord (2) Distance of maximum camber from leading edge, 10s of percent (4) Maximum thickness as percentage of chord (12) See NACA Report No. 460, 1935, for lift and drag characteristics of 78 airfoils Airfoils used on various aircraft: The Incomplete Guide to Airfoil Usage http://www.ae.illinois.edu/m-selig/ads/aircraft.html NACA Airfoils
C L2! D = C L"
( )
2! D
" = ( 2# )" [Lifting-line Theory]
Swept wing, 2-D lift slope coefcient
Inviscid, incompressible ow
C L2! D = C L"
( )
2! D
" = ( 2# cos $ )"
Clark Y (1922): Flat lower surface, 11.7% thickness
GA, WWII aircraft Clark Y Airfoil http://en.wikipedia.org/wiki/Clark_Y Reasonable L/D Benign computed stall characteristics, but experimental result is more abrupt
Fluent, Inc, 2007
Relationship Between Circulation and Lift
2-D Lift (inviscid, incompressible ow)
Aerodynamic Strip Theory
Airfoil section may vary from tip-to-tip
Chord length Airfoil thickness Airfoil prole Airfoil twist
( Lift )2 ! D = "#V# ( $ )2 ! D
1 2 "#V# c ( 2%& ) [ thin, symmetric airfoil ] + "#V# ( $ camber )2 ! D 2 1 2 ! "#V# c C L& & + "#V# ( $ camber )2 ! D 2! D 2 !
( )
Lift of a 3-D wing is found by integrating 2-D lift coefcients of airfoil sections across the nite span Incremental lift
dL = C L2! D ( y ) c ( y ) qdy
3-D wing lift
b /2
Aero L-39 Albatros
L3! D =
!b /2
"
C L2! D ( y ) c ( y ) q dy
Effect of Aspect Ratio on Wing Lift Slope Coefcient
(Incompressible Flow)
Airfoil section lift coefcients and lift slopes near wingtips are lower than their estimated 2-D values
Bombardier Dash 8
Effect of Aspect Ratio on 3-Dimensional Wing Lift Slope Coefcient
(Incompressible Flow) High Aspect Ratio (> 5) Wing
Handley Page HP.115
C L! =
2" AR # AR & = 2" % $ AR + 2 ( ' AR + 2
Low Aspect Ratio (< 2) Wing
C L! =
" AR # AR & = 2" % $ 4 ( ' 2
All wings at M = 1
For Small Angles, Lift is Proportional to Angle of Attack
Lift = C L 1 2 #C ( 1 1 % !V S " 'C L0 + L $ * !V 2 S + %C L0 + C L$ $ ( !V 2 S & )2 2 #$ ) 2 & where C L$ = lift slope coefficient
Maximum Lift Coefcient
Maximum Lift of Rectangular Wings
Schlicting & Truckenbrodt, 1979
At higher angles,
ow separates wing loses lift
Angle of Attack
Flow separation produces stall
http://www.youtube.com/watch?v=RgUtFm93Jfo
Aspect Ratio
! : Sweep angle " : Thickness ratio
Maximum Lift of Delta Wings with Straight Trailing Edges
Maximum Lift Coefcient Angle of Attack Schlicting & Truckenbrodt, 1979
Large Angle Variations in Subsonic Lift Coefcient (0 < ! < 90)
Lift = C L
All lift coefcients have at least one maximum (stall condition) All lift coefcients are essentially Newtonian at high ! Newtonian ow: TBD
1 2 !V S 2
Aspect Ratio Aspect Ratio
! : Taper ratio
Flap Effects on Aerodynamic Lift
Effect of Aspect Ratio on 3-D Wing Lift Slope Coefcient
(Incompressible Flow)
All Aspect Ratios (Helmbold equation)
Camber modication Trailing-edge ap deection shifts CL up and down Leading-edge ap (slat) deection increases stall ! Same effect applies for other control surfaces
Elevator (horizontal tail) Ailerons (wing) Rudder (vertical tail)
C L! =
" AR
2 ) # AR & , . +1 + 1 + % $ 2 ( . ' + *
Air Compressibility and Sweep Effects on 3-D Wing Lift Slope Coefcient
Subsonic 3-D wing, with sweep effect
C L! =
Air Compressibility Effects on 3-D Wing Lift Slope Coefcient
Supersonic delta (triangular) wing
Supersonic leading edge Subsonic leading edge
" AR
2 + $ AR ' 2 -1 + 1 + & 2 cos # ) 1 * M cos #1 4 % 14( ,
. 0 0 0 /
C L! =
4 M2 "1
C L! =
2" 2 cot # (" + $ )
where $ = m 0.38 + 2.26m % 0.86m 2 m = cot # LE cot &
! LE = sweep angle of leading edge
!1 4 = sweep angle of quarter chord
Wing-Fuselage Interference Effects
Wing lift induces
Upwash in front of the wing Downwash behind the wing, having major effect on the tail Local angles of attack over canard and tail surface are modied, affecting net lift and pitching moment
Aerodynamic Drag
Drag = C D " %C D0 ' & 1 2 2 1 !V S " C D0 + # C L !V 2 S 2 2 2 1 + # C Lo + C L$ $ ( !V 2 S *2 )
Flow around fuselage induces upwash on the wing, canard, and tail
from Etkin
Parasitic Drag
Pressure differential, viscous shear stress, and separation
Parasitic Drag = C D0 1 2 !V S 2
Reynolds Number and Boundary Layer
Reynolds Number = Re =
!Vl Vl = "
where ! = air density V = true airspeed l = characteristic length = absolute (dynamic) viscosity " = kinematic viscosity
Reynolds Number, Skin Friction, and Boundary Layer
Skin friction coefcient for a at plate
Cf = Friction Drag qSwet where Swet = wetted area
Typical Effect of Reynolds Number on Parasitic Drag
Flow may stay attached farther at high Re, reducing the drag
from Werle*
C f ! 1.33Re "1/2 ! 0.46 ( log10
[laminar flow ] Re ) [turbulent
"2.58
flow ]
* See Van Dyke, M., An Album of Fluid Motion, Parabolic Press, Stanford, 1982
Effect of Streamlining on Parasitic Drag
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