Low-Speed Aerodynamics!
Robert Stengel, Aircraft Flight Dynamics, MAE
331, 2016
•!
•!
•!
•!
•!
Learning Objectives
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
Reading:!
Flight Dynamics !
Aerodynamic Coefficients, 65-84!
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
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
Non-dimensional force coefficients, CL and CD, are
dimensionalized by
dynamic pressure, q, N/m2 or lb/sq ft
reference area, S, m2 of ft2
"1
%
Lift = C L q S = C L $ !V 2 ' S
#2
&
"1
%
Drag = C D q S = C D $ !V 2 ' S
#2
&
5
Circulation of Incompressible Air Flow
About a 2-D Airfoil
Bernoullis equation (inviscid, incompressible flow)
(Motivational, but not the whole story of lift)
1
pstatic + !V 2 = constant along streamline = pstagnation
2
Vorticity at point x
Vupper (x) = V! + "V (x) 2
! 2"D (x) =
Vlower (x) = V! # "V (x) 2
#V (x)
#z(x)
Circulation about airfoil
! 2"D
%V (x)
= $ # 2"D (x)dx = $
dx
%z(x)
0
0
c
c
Lower pressure on upper surface
see “Lift: A History of Explanations – in Plain English – for How Airplanes Fly, 1910-1950,”
6
Princeton A.B. Thesis, Mackenzie Hawkins, 2015.
Relationship Between
Circulation and Lift
Differential pressure along chord section
2'
2'
1
1
$
$
!p ( x ) = & pstatic + " # (V# + !V ( x ) 2 ) ) * & pstatic + " # (V# * !V ( x ) 2 ) )
2
2
%
( %
(
2
2
1 $
= " # (V# + !V ( x ) 2 ) * (V# * !V ( x ) 2 ) '
(
2 %
= " #V# !V ( x ) = " #V# !z(x)+ 2*D (x)
2-D Lift (inviscid, incompressible flow)
c
c
0
0
( Lift )2!D = # "p ( x ) dx = $ %V% # & 2!D (x)dx = $ %V% ( ' )2!D
!
1
! "V" 2 c ( 2#$ )[ thin, symmetric airfoil] + ! "V" ( % camber )2&D
2
1
! ! "V" 2 c C L$
$ + ! "V" ( % camber )2&D
2&D
2
( )
7
Lift vs. Angle of Attack
2-D Lift (inviscid, incompressible flow)
1
(
! "V" 2 c C L#
# * + %& ! "V" ( + camber )2$D ()
2$D
&2
)
= [Lift due to angle of attack]
+ [Lift due to camber]
( Lift )2-D ! %'
( )
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 ratio
AR =
b
c
=
rectangular wing
b ! b b2
=
c!b S
any wing
Infinite-span wing –> infinite aspect ratio
10
What Do We Mean by !
2-Dimensional Aerodynamics?
Assuming constant chord section, the 2-D Lift
is the
same at any y station of the infinite-span wing
Lift 3!D = C L3!D
1
1
"V 2 S = C L3!D "V 2 ( bc ) [Rectangular wing]
2
2
1
# ( Lift 3!D ) = C L3!D "V 2 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
11
Effect of Sweep Angle on Lift
Unswept wing, 2-D lift slope coefficient
Inviscid, incompressible flow
Referenced to chord length, c, rather than wing area
$ #C '
C L2!D = " & L )
= " C L"
% #" ( 2!D
( )
2!D
= ( 2* )" [Thin Airfoil Theory]
Swept wing, 2-D lift slope coefficient
Inviscid, incompressible flow
( )
C L2!D = " C L"
2!D
= ( 2# cos $ )"
12
Thin Airfoil Theory
Downward velocity, w, at xo due to vortex at x
Integral
Differential
dw ( xo ) =
! ( x ) dx
2" ( xo # x )
w ( xo ) =
1
2!
1
" ( x)
dx
o # x)
$ (x
0
Boundary condition: flow tangent to mean camber line
w ( xo )
# dz &
=! "% (
$ dx ' xo
V
McCormick, 1995
13
Thin Airfoil Theory
Integral equation for ! ( x )
1
2! V
1
" ( x)
& dz )
dx = % # ( +
' dx * xo
o # x)
$ (x
0
Coordinate transformation
x=
1
(1! cos" )
2
Solution for ! ( x )
% 1+ cos" #
(
! = 2V ' A0
+ $ An sin n" *
sin "
&
)
n=1
McCormick, 1995
Coefficients
1 # dz
d$
# %0 dx
2 # dz
An = %
cos n$ d$
# 0 dx
A0 = ! "
14
Thin Airfoil Theory
Lift, from Kutta-Joukowski theorem
1
L = # !V " ( x ) dx = 2$ A0 + $ A1
0
For thin airfoil with circular arc
A0 = ! , A1 = 4zmax
C L2!D = 2"# + 4" zmax = C L# # + C Lo
=C L# # [Flat plate]
C L! =
[Circular arc]
"C L
= 2#
"!
McCormick, 1995
15
Classic Airfoil
Profiles
NACA Airfoils
http://en.wikipedia.org/wiki/NACA_airfoil
•!
NACA 4-digit Profiles (e.g., NACA 2412)
•!
Clark Y (1922): Flat lower surface, 11.7%
thickness
–! 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%
–!
–!
–!
–!
GA, WWII aircraft
Reasonable L/D
Benign theoretical stall characteristics
Experimental result is more abrupt
Fluent, Inc, 2007
Clark Y Airfoil
http://en.wikipedia.org/wiki/Clark_Y
16
Typical Airfoil Profiles
Positive camber
Neutral camber
Negative camber
Talay, NASA SP-367
17
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-ofpressure (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
CD
CL, 60° flap
Rough ~ Turbulent
Smooth ~ Laminar
“Drag Bucket”
CL, w/o flap
Cm
Cm, w/o flap
Cm, 60° flap
!
CL
19
!is"rical Fac"id
Measuring Lift and Drag with Whirling Arms
and Early Wind Tunnels
Whirling Arm Experimentalists
Otto Lillienthal
Hiram Maxim
Samuel Langley
Wind Tunnel Experimentalists
Frank Wenham
Gustave Eiffel
Hiram Maxim
Wright Brothers 20
!is"rical Fac"id
Wright Brothers Wind Tunnel
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
1 2
1
!V S " C D0 + # C L2 !V 2 S
2
2
2 1
+ # C Lo + C L$ $ (* !V 2 S
)2
(
Drag = C D
" %'C D0
&
(
)
)
23
Parasitic Drag, CDo
•! Pressure differential,
viscous shear stress,
and separation
Parasitic Drag = C D0
Talay, NASA SP-367
1 2
!V S
2
24
Reynolds Number and Boundary Layer
Reynolds Number = Re =
!Vl Vl
=
µ
"
where
! = air density, kg/m 2
V = true airspeed, m/s
l = characteristic length, m
µ = absolute (dynamic) viscosity = 1.725 " 10 #5 kg / m i s
$ = kinematic viscosity (SL) = 1.343 " 10 #5 m / s 2
25
Reynolds Number,
Skin Friction, and
Boundary Layer
Skin friction coefficient for a flat plate
Friction Drag
qSwet
where Swet = wetted area
Cf =
Wetted Area: Total surface
area of the wing or aircraft,
subject to skin friction
Boundary layer thickens in transition, then
thins in turbulent flow
C f ! 1.33Re "1/2
! 0.46 ( log10 Re )
"2.58
[laminar flow ]
[turbulent flow ]
26
Effect of Streamlining on Parasitic Drag
CD = 2.0
CD = 1.2
CD = 0.12
CD = 1.2
CD = 0.6
Talay, NASA SP-367
27
Subsonic CDo Estimate (Raymer)
28
!is"rical Fac"id
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
29
!is"rical Fac"id
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
30
Description of Aircraft
Configurations!
31
A Few Definitions
Republic F-84F
Thunderstreak
32
Wing Planform Variables
Aspect Ratio
b
AR =
rectangular wing
c
b ! b b2
=
any wing
=
c!b S
Taper Ratio
!=
ctip
tip chord
=
croot root chord
Delta Wing
Rectangular Wing
Swept 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-367
34
•!
Mean Aerodynamic Chord and
Wing Aerodynamic Center
Mean aerodynamic chord (m.a.c.) ~ mean geometric chord
b2
1
c = " c 2 ( y ) dy
S !b 2
# 2 & 1+ ) + )
=% (
croot
$ 3 ' 1+ )
2
•!
!=
[for trapezoidal wing]
Trapezoidal Wing
s subsonic
Axial location of the wing
aerodynamic center (a.c.)
–! Determine spanwise location of m.a.c.
–! Assume that aerodynamic center is at
25% m.a.c.
Elliptical Wing
ctip
tip chord
=
croot root chord
Midchord
line
from Raymer
from Sunderland
35
3-Dimensional Aerodynamic!
Lift and Drag!
https://www.youtube.com/watch?v=S7V0awkweZc
36
Wing Twist Effects
•! 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
•! separationburble
produces buffet at tail surface, warning of stall
–! improves aileron effectiveness at high !
Talay, NASA SP-367
37
Aerodynamic Strip Theory
•!
Airfoil section may vary from tip-to-tip
•!
3-D Wing Lift: Integrate 2-D lift coefficients
of airfoil sections across finite span
–!
–!
–!
–!
Chord length
Airfoil thickness
Airfoil profile
Airfoil twist
Incremental lift along span
dL = C L2!D ( y ) c ( y ) qdy
=
dC L3!D ( y )
dy
Aero L-39 Albatros
c ( y ) qdy
3-D wing lift
L3! D =
b /2
"
!b /2
C L2! D ( y ) c ( y ) q dy
38
Bombardier Dash 8
Effect of Aspect Ratio on
3-Dimensional Wing Lift
Slope Coefficient
Handley Page HP.115
(Incompressible Flow)
High Aspect Ratio (> 5) Wing
# "C L &
# AR &
2 * AR
C L! ! %
= 2* %
( =
(
$ "! '3)D AR + 2
$ AR + 2 '
Low Aspect Ratio (< 2) Wing
C L! =
# AR &
" AR
= 2" %
(
$ 4 '
2
39
Effect of Aspect Ratio on 3-D
Wing Lift Slope Coefficient
(Incompressible Flow)
All Aspect Ratios (Helmbold equation)
C L! =
" AR
2,
)
#
&
AR
+1+ 1+ %
( .
$ 2 ' .+*
40
Effect of Aspect Ratio on 3-D
Wing Lift Slope Coefficient
All Aspect Ratios (Helmbold equation)
Wolfram Alpha (https://www.wolframalpha.com/)
plot(pi A / (1+sqrt(1 + (A / 2)^2)), A=1 to 20)
41
!is"rical Fac"ids
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
•!
Flow around fuselage induces upwash on the wing, canard,
and tail
–! 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
from Etkin
43
Longitudinal Control Surfaces
Wing-Tail Configuration
Flap
Elevator
Delta-Wing Configuration
Elevator
44
Angle of Attack and
Control Surface Deflection
•!
Horizontal tail with
elevator control
surface
•!
Horizontal tail at
positive angle of attack
•!
Horizontal tail with
positive elevator
deflection
45
Control Flap Carryover Effect on
Lift Produced By Total Surface
from Schlichting & Truckenbrodt
C L! E
C L"
vs.
cf
xf + cf
cf
(x
f
+ cf )
46
Lift due to Elevator Deflection
Lift coefficient variation due to elevator deflection
C L! E !
(C )
L#
"C L
= # ht$ht C L%
"! E
&C L = C L! E ! E
( )
ht
Sht
S
! ht = Carryover effect
"ht = Tail efficiency factor
ht
= Horizontal tail lift-coefficient slope
Sht = Horizontal tail reference area
Lift variation due to elevator deflection
!L = C L" E qS" E
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!
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
49
Supplementary Material
50
•!
Typical Effect of Reynolds
Number on Parasitic Drag
Flow may stay attached
farther at high Re,
reducing the drag
from Werle*
51
* See Van Dyke, M., An Album of Fluid Motion,
Parabolic Press, Stanford, 1982
Effect of Aspect Ratio on 3-Dimensional
Wing Lift Slope Coefficient
•! High Aspect Ratio (> 5) Wing
•! Wolfram Alpha
plot(2 pi (a/(a+2)), a=5 to 20)
•! Low Aspect Ratio (< 2) Wing
•! Wolfram Alpha
plot(2 pi (a / 4), a=1 to 2)
52
Aerodynamic Stall, Theory and Experiment
•! Flow separation produces stall
•! Straight rectangular wing, AR = 5.536, NACA 0015
•! Hysteresis for increasing/decreasing !
Anderson et al, 1980
53
Maximum Lift of
Rectangular Wings
Maximum
Lift
Coefficient,
CL max
Schlicting & Truckenbrodt, 1979
Angle of
Attack for
CL max
Aspect Ratio
! : Sweep angle
" : Thickness ratio
54
Maximum Lift of Delta Wings with
Straight Trailing Edges
Maximum Lift
Coefficient, CL max
Angle of Attack
for CL max
Aspect Ratio
Aspect Ratio
! : Taper 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
surface
56
Multi-Engine Aircraft of World War II
Consolidated B-24
Boeing B-17
Boeing B-29
Douglas A-26
•! Large W.W.II aircraft had
unpowered controls:
–! High foot-pedal force
–! Rudder stability problems
arising from balancing to
reduce pedal force
North American B-25
•! Severe engine-out problem
for twin-engine aircraft
Martin B-26
57
Medium to High Aspect Ratio Configurations
Cessna 337
DeLaurier Ornithopter
Schweizer 2-32
Vtakeoff = 82 km/h
hcruise = 15 ft
Vcruise = 144 mph
hcruise = 10 kft
Mtypical = 75 mph
hmax = 35 kft
Mcruise = 0.84
hcruise = 35 kft
•!
Boeing 777-300
Typical for subsonic aircraft
58
Uninhabited Air Vehicles
Northrop-Grumman/Ryan Global Hawk
General Atomics Predator
Vcruise = 70-90 kt
hcruise = 25 kft
Vcruise = 310 kt
hcruise = 50 kft
59
Stealth and Small UAVs
Lockheed-Martin RQ-170
Northrop-Grumman X-47B
http://en.wikipedia.org/wiki/Stealth_aircraft
General Atomics Predator-C (Avenger)
InSitu/Boeing ScanEagle
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 Videos
Flow over a narrow airfoil, with downstream vortices
http://www.youtube.com/watch?v=zsO5BQA_CZk
Flow over transverse flat plate, with downstream vortices
http://www.youtube.com/watch?v=0z_hFZx7qvE
Laminar vs. turbulent flow
http://www.youtube.com/watch?v=WG-YCpAGgQQ&feature=related
Smoke flow visualization, wing with flap
http://www.youtube.com/watch?feature=fvwp&NR=1&v=eBBZF_3DLCU/
1930s test in NACA wind tunnel
http://www.youtube.com/watch?v=3_WgkVQWtno&feature=related
62