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Low-Speed Aerodynamics!Robert Stengel, Aircraft Flight Dynamics, MAE
331, 2016
•! 2D lift and drag•! Appreciate effects of Reynolds number•! Relationships between airplane shape
and aerodynamic characteristics•! Distinguish between 2D and 3D lift and
drag•! How to compute static and dynamic
effects of aerodynamic control surfaces
Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html
http://www.princeton.edu/~stengel/FlightDynamics.html
Learning Objectives
Reading:!Flight Dynamics !
Aerodynamic Coefficients, 65-84!
1
Review Questions!!! What are Newton’s three laws of motion?!!! Why are non-dimensional aerodynamic
coefficients useful?!!! What are longitudinal motion variables?!!! What are lateral-directional motion variables?!!! Why are 2-D and 3-D aerodynamics
different?!!! Describe some different kinds of aircraft
engines.!!! Why aren’t rockets used for cruising aircraft
propulsion?!
2
2-Dimensional Aerodynamic !Lift and Drag!
3
Lift and Drag•! Lift is perpendicular to the free-stream
airflow direction•! Drag is parallel to the free-stream airflow
direction
4
Longitudinal Aerodynamic Forces
Lift = CLq S = CL12!V 2"
#$%&' S
Drag = CDq S = CD12!V 2"
#$%&' S
Non-dimensional force coefficients, CL and CD, are dimensionalized by
dynamic pressure, q, N/m2 or lb/sq ftreference area, S, m2 of ft2
5
Circulation of Incompressible Air Flow About a 2-D Airfoil
Bernoulli s equation (inviscid, incompressible flow)(Motivational, but not the whole story of lift)
pstatic +12!V 2 = constant along streamline = pstagnation
Vorticity at point x
Vupper (x) = V! + "V (x) 2Vlower (x) = V! # "V (x) 2
! 2"D (x) =#V (x)#z(x)
Circulation about airfoil
!2"D = # 2"D (x)dx0
c
$ = %V (x)%z(x)
dx0
c
$Lower pressure on upper surface
6see “Lift: A History of Explanations – in Plain English – for How Airplanes Fly, 1910-1950,”Princeton A.B. Thesis, Mackenzie Hawkins, 2015.
Relationship Between Circulation and Lift
2-D Lift (inviscid, incompressible flow)
Lift( )2!D = "p x( )dx0
c
# = $%V% & 2!D (x)dx0
c
# = $%V% '( )2!D
!p x( ) = pstatic +12"# V# + !V x( ) 2( )2$
%&'()* pstatic +
12"# V#* !V x( ) 2( )2$
%&'()
= 12"# V# + !V x( ) 2( )2 * V# * !V x( ) 2( )2$
%'(
= "#V#!V x( ) = "#V#!z(x)+ 2*D (x)
Differential pressure along chord section
!12!"V"
2c 2#$( ) thin, symmetric airfoil[ ]+ !"V" %camber( )2&D!12!"V"
2c CL$( )2&D
$ + !"V" %camber( )2&D 7
Lift vs. Angle of Attack2-D Lift (inviscid, incompressible flow)
Lift( )2-D !12!"V"
2c CL#( )2$D
#%&'
()*+ !"V" +camber( )2$D%& ()
= [Lift due to angle of attack] + [Lift due to camber]
8
Typical Flow Variation with Angle of Attack
•! At higher angles, –! flow separates–! wing loses lift
•! Flow separation produces stall
9
What Do We Mean by !2-Dimensional Aerodynamics?
Finite-span wing –> finite aspect ratioAR = b
crectangular wing
=b ! bc ! b
=b2
Sany wing
Infinite-span wing –> infinite aspect ratio
10
What Do We Mean by !2-Dimensional Aerodynamics?
lim!y"0
! Lift3#D( ) = lim!y"0
CL3#D
12$V 2 c!y%
&'()* + "2-D Lift" = CL2#D
12$V 2 c
Assuming constant chord section, the 2-D Lift is the same at any y station of the infinite-span wing
Lift3!D = CL3!D
12"V 2 S = CL3!D
12"V 2 bc( ) [Rectangular wing]
# Lift3!D( ) = CL3!D
12"V 2 c#y
11
Effect of Sweep Angle on LiftUnswept wing, 2-D lift slope coefficient
Inviscid, incompressible flowReferenced to chord length, c, rather than wing area
CL2!D=" #CL
#"$%&
'() 2!D
=" CL"( )2!D
= 2*( )" [Thin Airfoil Theory]
Swept wing, 2-D lift slope coefficientInviscid, incompressible flow
CL2!D=" CL"( )
2!D= 2# cos$( )"
12
13McCormick, 1995
Thin Airfoil Theory
Downward velocity, w, at xo due to vortex at x
dw xo( ) = ! x( )dx2" xo # x( ) w xo( ) = 1
2!" x( )xo # x( ) dx0
1
$
Differential Integral
Boundary condition: flow tangent to mean camber line
w xo( )V
=! " dzdx
#$%
&'( xo
14McCormick, 1995
Thin Airfoil Theory
Coordinate transformation
Integral equation for
x = 121! cos"( )
12!V
" x( )xo # x( ) dx0
1
$ =% # dzdx
&'(
)*+ xo
! x( )
Solution for ! x( )
! = 2V A01+ cos"sin"
+ An sinn"n=1
#
$%&'
()*
Coefficients
A0 =! " 1#
dzdxd$
0
#
%An =
2#
dzdxcosn$ d$
0
#
%
15McCormick, 1995
Thin Airfoil Theory
For thin airfoil with circular arc
L = !V" x( )dx0
1
# = 2$A0 +$A1
Lift, from Kutta-Joukowski theorem
A0 =! , A1 = 4zmaxCL2!D
= 2"# + 4"zmax = CL## +CLo
[Circular arc]=CL#
# [Flat plate]
CL!= "CL
"!= 2#
Classic Airfoil Profiles
•! NACA 4-digit Profiles (e.g., NACA 2412)–! Maximum camber as percentage of chord
(2) = 2%–! Distance of maximum camber from leading
edge, (4) = 40%–! Maximum thickness as percentage of chord
(12) = 12%
NACA Airfoilshttp://en.wikipedia.org/wiki/NACA_airfoil
•! Clark Y (1922): Flat lower surface, 11.7% thickness
–! GA, WWII aircraft–! Reasonable L/D–! Benign theoretical stall characteristics–! Experimental result is more abrupt
Fluent, Inc, 2007
Clark Y Airfoilhttp://en.wikipedia.org/wiki/Clark_Y16
Typical Airfoil ProfilesPositive camber
Neutral camber
Negative camber
Talay, NASA SP-36717
Airfoil Effects•! Camber increases zero-! lift
coefficient•! Thickness
–! increases ! for stall and softens the stall break
–! reduces subsonic drag –! increases transonic drag –! causes abrupt pitching
moment variation
•! Profile design –! can reduce center-of-
pressure (static margin, TBD) variation with !
–! affects leading-edge and trailing-edge flow separation
Talay, NASA SP-367 18
NACA 641-012 Chord Section Lift, Drag, and Moment (NACA TR-824)
CL, 60° flap
CL, w/o flap
CL
Cm, w/o flap
CD
Cm, 60° flap
“Drag Bucket”
!
Smooth ~ Laminar
Rough ~ Turbulent
19
CL
Cm
Measuring Lift and Drag with Whirling Arms and Early Wind Tunnels
Whirling Arm Experimentalists
!!iiss""rriiccaall FFaacc""iidd
Otto Lillienthal Hiram Maxim Samuel Langley
Wind Tunnel Experimentalists
Hiram MaximFrank Wenham Wright BrothersGustave Eiffel 20
Wright Brothers Wind Tunnel!!iiss""rriiccaall FFaacc""iidd
21
Flap Effects on Aerodynamic Lift
•! Camber modification•! Trailing-edge flap deflection
shifts CL up and down•! Leading-edge flap (slat)
deflection increases stall !•! Same effect applies for
other control surfaces–! Elevator (horizontal tail)–! Ailerons (wing)–! Rudder (vertical tail)
22
Aerodynamic Drag
Drag = CD12!V 2S " CD0
+ #CL2( ) 12 !V
2S
" CD0+ # CLo
+ CL$$( )2%
&'()*12!V 2S
23
Parasitic Drag, CDo
•! Pressure differential, viscous shear stress, and separation
Parasitic Drag = CD0
12!V 2S
Talay, NASA SP-367 24
Reynolds Number and Boundary LayerReynolds Number = Re = !Vl
µ= Vl"
25
where! = air density, kg/m2
V = true airspeed, m/sl = characteristic length, m
µ = absolute (dynamic) viscosity = 1.725 "10#5 kg /m i s$ = kinematic viscosity (SL) = 1.343"10#5m / s2
Reynolds Number, Skin Friction, and Boundary Layer
Skin friction coefficient for a flat plate
Cf =Friction Drag
qSwetwhere Swet = wetted area
Cf !1.33Re"1/2 laminar flow[ ]
! 0.46 log10 Re( )"2.58 turbulent flow[ ]
Boundary layer thickens in transition, then thins in turbulent flow
Wetted Area: Total surface area of the wing or aircraft, subject to skin friction
26
Effect of Streamlining on Parasitic Drag
Talay, NASA SP-367
CD = 2.0
CD = 1.2
CD = 0.12
CD = 1.2
CD = 0.627
Subsonic CDo Estimate (Raymer)
28
Samuel Pierpoint Langley (1834-1906)
•! Astronomer supported by Smithsonian Institution•! Whirling-arm experiments•! 1896: Langley's steam-powered Aerodrome model
flies 3/4 mile•! Oct 7 & Dec 8, 1903: Manned aircraft flights end in
failure
!!iiss""rriiccaall FFaacc""iidd
29
Wilbur (1867-1912) and Orville (1871-1948)
Wright•! Bicycle mechanics from Dayton, OH•! Self-taught, empirical approach to flight•! Wind-tunnel, kite, and glider
experiments•! Dec 17, 1903: Powered, manned
aircraft flight ends in success
!!iiss""rriiccaall FFaacc""iidd
30
Description of Aircraft Configurations!
31
Republic F-84FThunderstreak
A Few Definitions
32
Wing Planform VariablesAspect Ratio
Taper Ratio
! =ctipcroot
=tip chord
root chordAR = b
crectangular wing
= b ! bc ! b
= b2
Sany wing
Rectangular Wing
Delta WingSwept Trapezoidal Wing
33
Wing Design Parameters•! Planform
–! Aspect ratio–! Sweep–! Taper–! Complex geometries–! Shapes at root and tip
•! Chord section–! Airfoils–! Twist
•! Movable surfaces–! Leading- and trailing-edge devices–! Ailerons–! Spoilers
•! Interfaces–! Fuselage–! Powerplants–! Dihedral angle
Talay, NASA SP-36734
Mean Aerodynamic Chord and Wing Aerodynamic Center
c = 1S
c2 y( )dy!b 2
b 2
"
= 23
#$%
&'(
1+ ) + ) 2
1+ )croot [for trapezoidal wing]
from Raymer
•! Mean aerodynamic chord (m.a.c.) ~ mean geometric chord
•! 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.
from Sunderland
Trapezoidal Wing
Elliptical Wing
Mid-chord line
! =ctipcroot
=tip chord
root chord
35
3-Dimensional Aerodynamic !Lift and Drag!
36
https://www.youtube.com/watch?v=S7V0awkweZc
Wing Twist Effects
Talay, NASA SP-367
•! Washout twist–! reduces tip angle of attack–! typical value: 2° - 4°–! changes lift distribution (interplay with taper ratio)–! reduces likelihood of tip stall–! allows stall to begin at the wing root
•! separation burble produces buffet at tail surface, warning of stall–! improves aileron effectiveness at high !
37
Aerodynamic Strip Theory•! Airfoil section may vary from tip-to-tip
–! Chord length–! Airfoil thickness–! Airfoil profile–! Airfoil twist
•! 3-D Wing Lift: Integrate 2-D lift coefficients of airfoil sections across finite span
Incremental lift along span
Aero L-39 AlbatrosdL = CL2!D
y( )c y( )qdy
=dCL3!D
y( )dy
c y( )qdy
3-D wing lift
L3!D = CL2!Dy( )c y( )q dy
!b /2
b /2
"38
Effect of Aspect Ratio on 3-Dimensional Wing Lift
Slope Coefficient (Incompressible Flow)
High Aspect Ratio (> 5) Wing
CL!!
"CL
"!
#
$%
&
'(3)D
=2*ARAR+2
= 2* ARAR+2#
$%
&
'(
Low Aspect Ratio (< 2) Wing
CL!="AR2
= 2" AR4
#
$%
&
'(
Bombardier Dash 8 Handley Page HP.115
39
Effect of Aspect Ratio on 3-D Wing Lift Slope Coefficient
(Incompressible Flow)All Aspect Ratios (Helmbold equation)
CL!=
"AR
1+ 1+ AR2
#
$%
&
'(2)
*++
,
-..
40
Effect of Aspect Ratio on 3-D Wing Lift Slope Coefficient
All Aspect Ratios (Helmbold equation)
plot(pi A / (1+sqrt(1 + (A / 2)^2)), A=1 to 20)
41
Wolfram Alpha (https://www.wolframalpha.com/)
!!iiss""rriiccaall FFaacc""iiddss•! 1906: 2nd successful aviator: Alberto
Santos-Dumont, standing!–! High dihedral, forward control surface
•! Wrights secretive about results until 1908; few further technical contributions
•! 1908: Glenn Curtiss et al incorporate ailerons
–! Wright brothers sue for infringement of 1906 US patent (and win)
•! 1909: Louis Bleriot's flight across the English Channel
42
Wing-Fuselage Interference Effects•! Wing lift induces
–! Upwash in front of the wing–! Downwash behind the wing, having major effect on the tail–! Local angles of attack over canard and tail surface are modified,
affecting net lift and pitching moment•! Flow around fuselage induces upwash on the wing, canard,
and tail
from Etkin
43
Longitudinal Control Surfaces
Flap Elevator
Elevator
Wing-Tail Configuration
Delta-Wing Configuration
44
Angle of Attack and Control Surface Deflection
•! Horizontal tail at positive angle of attack
•! Horizontal tail with elevator control surface
•! Horizontal tail with positive elevator deflection
45
Control Flap Carryover Effect on Lift Produced By Total Surface
from Schlichting & Truckenbrodt
CL!E
CL"
vs.cf
x f + cf
c f x f + c f( ) 46
Lift due to Elevator Deflection
CL!E!"CL
"!E= # ht$ht CL%( )
ht
ShtS
&CL = CL!E!E
Lift coefficient variation due to elevator deflection
!L = CL"EqS"E
Lift variation due to elevator deflection
! ht = Carryover effect"ht = Tail efficiency factor
CL#( )ht= Horizontal tail lift-coefficient slope
Sht = Horizontal tail reference area
47
Example of Configuration and Flap Effects
48
Next Time:!Induced Drag and High-Speed !
Aerodynamics!Reading:!
Flight Dynamics !Aerodynamic Coefficients, 85-96!Airplane Stability and Control!
Chapter 1!
49
Learning Objectives•! Understand drag-due-to-lift and effects of
wing planform •! Recognize effect of angle of attack on lift
and drag coefficients •! How to estimate Mach number (i.e., air
compressibility) effects on aerodynamics•! Be able to use Newtonian approximation to
estimate lift and drag
Supplementary Material
50
Typical Effect of Reynolds Number on Parasitic Drag
from Werle*
* See Van Dyke, M., An Album of Fluid Motion, Parabolic Press, Stanford, 1982
•! Flow may stay attached farther at high Re, reducing the drag
51
Effect of Aspect Ratio on 3-Dimensional Wing Lift Slope Coefficient
•! High Aspect Ratio (> 5) Wing•! Wolfram Alpha
•! Low Aspect Ratio (< 2) Wing•! Wolfram Alpha
plot(2 pi (a / 4), a=1 to 2)
plot(2 pi (a/(a+2)), a=5 to 20)
52
Aerodynamic Stall, Theory and Experiment
Anderson et al, 1980
•! Flow separation produces stall•! Straight rectangular wing, AR = 5.536, NACA 0015•! Hysteresis for increasing/decreasing !
53
Angle of Attack for
CL max
Maximum Lift of Rectangular Wings
Schlicting & Truckenbrodt, 1979
Aspect Ratio
MaximumLift
Coefficient,CL max
! : Sweep angle" : Thickness ratio 54
Maximum Lift of Delta Wings with Straight Trailing Edges
!: Taper ratioAspect Ratio
Angle of Attack for CL max
Maximum Lift Coefficient, CL max
Aspect Ratio
Schlicting & Truckenbrodt, 1979 55
Aft Flap vs. All-Moving Control Surface
•! Carryover effect of aft flap–! Aft-flap deflection can be almost as effective as
full surface deflection at subsonic speeds–! Negligible at supersonic speed
•! Aft flap –! Mass and inertia lower, reducing likelihood of
mechanical instability–! Aerodynamic hinge moment is lower–! Can be mounted on structurally rigid main
surface56
Multi-Engine Aircraft of World War II
•! Large W.W.II aircraft had unpowered controls:–! High foot-pedal force–! Rudder stability problems
arising from balancing to reduce pedal force
•! Severe engine-out problem for twin-engine aircraft
Boeing B-17 Boeing B-29Consolidated B-24
Douglas A-26
North American B-25
Martin B-26
57
Medium to High Aspect Ratio ConfigurationsCessna 337 DeLaurier Ornithopter Schweizer 2-32
•! Typical for subsonic aircraft
Boeing 777-300
Mtypical = 75 mphhmax = 35 kft
Mcruise = 0.84hcruise = 35 kft
Vtakeoff = 82 km/hhcruise = 15 ft
Vcruise = 144 mphhcruise = 10 kft
58
Uninhabited Air VehiclesNorthrop-Grumman/Ryan Global Hawk General Atomics Predator
Vcruise = 70-90 kthcruise = 25 kft
Vcruise = 310 kthcruise = 50 kft
59
Stealth and Small UAVsLockheed-Martin RQ-170 General Atomics Predator-C (Avenger)
InSitu/Boeing ScanEagle
http://en.wikipedia.org/wiki/Stealth_aircraft
Northrop-Grumman X-47B
60
Subsonic 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)
•! Interference effects of two wings–! Gap–! Aspect ratio–! Relative areas and spans–! Stagger
61
Some VideosFlow over a narrow airfoil, with downstream vortices
http://www.youtube.com/watch?v=zsO5BQA_CZk
http://www.youtube.com/watch?v=0z_hFZx7qvE
Flow over transverse flat plate, with downstream vortices
http://www.youtube.com/watch?v=WG-YCpAGgQQ&feature=related
Laminar vs. turbulent flow
Smoke flow visualization, wing with flap
1930s test in NACA wind tunnelhttp://www.youtube.com/watch?feature=fvwp&NR=1&v=eBBZF_3DLCU/
http://www.youtube.com/watch?v=3_WgkVQWtno&feature=related62