Induced Drag and High-Speed Aerodynamics Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2016 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 • 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 Reading: Flight Dynamics Aerodynamic Coefficients, 85-96 Airplane Stability and Control Chapter 1 1 Early Developments in Stability and Control Chapter 1, Airplane Stability and Control, Abzug and Larrabee • What are the principal subject and scope of the chapter? • What technical ideas are needed to understand the chapter? • During what time period did the events covered in the chapter take place? • What are the three main "takeaway" points or conclusions from the reading? • What are the three most surprising or remarkable facts that you found in the reading? 2
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Induced Drag and High-Speed Aerodynamics!
Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2016
Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html
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
Reading:!Flight Dynamics !
Aerodynamic Coefficients, 85-96!Airplane Stability and Control!
Chapter 1!
1
Early Developments in Stability and Control!
Chapter 1, Airplane Stability and Control, Abzug and Larrabee!
•! What are the principal subject and scope of the chapter?!
•! What technical ideas are needed to understand the chapter?!
•! During what time period did the events covered in the chapter take place?!
•! What are the three main "takeaway" points or conclusions from the reading?!
•! What are the three most surprising or remarkable facts that you found in the reading?!
2
Review Questions!!! What is the relationship between circulation and
aerodynamic lift?!!! What causes aerodynamic “stall”?!!! What is the difference between leading-edge and
trailing-edge flap effects?!!! Linear angle of attack variation causes __ lift
variation…!!! Linear angle of attack variation causes __ drag
variation…!!! How does wing aspect ratio affect the lift slope?!!! What is control flap “carryover effect”?!
3
Induced Drag!
4
Aerodynamic Drag
Drag = CD12!V 2S " CD0
+ #CL2( ) 12 !V
2S
" CD0+ # CLo
+CL$$( )2%
&'()*12!V 2S
5
Induced Drag of a Wing, !CL2
!! Lift produces downwash (angle proportional to lift)!! Downwash rotates local velocity vector CW in figure!! Lift is perpendicular to velocity vector!! Axial component of rotated lift induces drag
6
!! But what is the proportionality factor, !?
Three Expressions for Induced Drag of a WingCDi
= CL sin! i " CL0+CL!
!( )sin! i
where! i = CL #eAR, Induced angle of attack
CDi! CL0
+CL""( )" i =
CL2
#eAR
!CL
2 1+$( )#AR
! %CL2
Spitfire
7
wheree = Oswald efficiency factor = 1 for elliptical distribution! = departure from ideal elliptical lift distribution
" = 1#eAR
=1+!( )#AR
Spanwise Lift Distribution of 3-D (Trapezoidal) Wings
Straight Wings (@ 1/4 chord), McCormick
TR = taper ratio, !!
For some taper ratio between 0.35 and 1, lift distribution is nearly elliptical 8
Maximum Lift-to-Drag RatioMaximize L/D by proper choice of CL
LD
= CL
CD
= CL
CDo+ !CL
2!(L /D)!CL
= 0
!(L /D)!CL
= 0 =CDo
+ "CL2( )#CL 2"CL( )
CDo+ "CL
2( )2=
CDo# "CL
2( )CDo
+ "CL2( )2
CL( ) L/D( )max=
CDo
!11
L /D( )max =1
2 !CDo
Large Angle Variations in Subsonic Drag Coefficient (0° < # < 90°)
All wing drag coefficients converge to Newtonian-like values at very high angle of attack
Low-AR wing has less drag than high-AR wing at given # 12
Lift vs. Drag for Large Variation in Angle-of-Attack (0° < # < 90°)
Subsonic Lift-Drag Polar
Low-AR wing has less drag than high-AR wing, but less lift as wellHigh-AR wing has the best overall subsonic L/D
13
Lift-to-Drag Ratio vs. Angle of Attack
•! L/D is an important performance metric for aircraft•! High-AR wing has best overall subsonic L/D•! Low-AR wing has best L/D at high angle of attack
LD
=CLq SCDq S
=CL
CD
14
Conversions from Propellers to JetsDouglas XB-43
Douglas XB-42 Mixmaster
Convair B-36 Convair YB-60
Northrop YB-35
Northrop XB-49
!!iiss""rriiccaall FFaacc""iidd
15
Jets at an Awkward Age•! Performance of the first jet aircraft
outstripped stability and control technology–! Lacked satisfactory actuators,
sensors, and control electronics–! Transistor: 1947, integrated circuit:
1958•! Dramatic dynamic variations over
larger flight envelope–! Control mechanisms designed to
lighten pilot loads were subject to instability
•! Reluctance of designers to embrace change, fearing decreased reliability, increased cost, and higher weight
North American B-45
Lockheed P-80
Douglas F3D
Convair XP-81
!!iiss""rriiccaall FFaacc""iidd
16
Mach Number Effects!
Ernst Mach1838-1916
Mach Number = True AirspeedSpeed of Sound
Supersonic Bullet, 1888
17
Drag Due to Pressure Differential
CDbase= Cpressurebase
SbaseS ! 0.029
CfrictionSwetSbase
SbaseS
M <1( ) Hoerner[ ]
< 2" M 2
SbaseS
#$%
&'( M > 2, " = specific heat ratio( )
“The Sonic Barrier”
Blunt base pressure drag
CDwave!CDincompressible
1"M 2M <1( )
!CDcompressible
M 2 "1M >1( )
!CDM! 2
M 2 "1M >1( )
Prandtl factor
18
Shock Waves inSupersonic Flow
•! Drag rises due to pressure increase across a shock wave
•! Subsonic flow–! Local airspeed is less than sonic
(i.e., speed of sound) everywhere
•! Transonic flow–! Airspeed is less than sonic at
some points, greater than sonic elsewhere
•! Supersonic flow–! Local airspeed is greater than
sonic virtually everywhere
•! Critical Mach number–! Mach number at which local
flow first becomes sonic–! Onset of drag-divergence–! Mcrit ~ 0.7 to 0.85
Air Compressibility Effect
19
Effect of Chord Thickness on Wing
Pressure Drag
•! Thinner chord sections lead to higher Mcrit, or drag-divergence Mach number
Lockheed P-38Lockheed F-104
20
Air Compressibility Effect on Wing Drag
Subsonic
SupersonicTransonic
Incompressible
Sonic Boomshttp://www.youtube.com/watch?v=gWGLAAYdbbc
21
Pressure Drag on Wing Depends on Sweep Angle
Mcritswept=Mcritunswept
cos!
Talay, NASA SP-367 22
From Straight to Swept Wings•! Straight-wing models were redesigned with swept wings to
reduce compressibility effects on drag and increase speed•! Dramatic change in stability, control, and flying qualitiesNorth American FJ-1 and
FJ-4 FuryRepublic F-84B Thunderbird and
F-84F ThunderstreakGrumman F9F-2 Panther and
F9F-6 Cougar
!!iiss""rriiccaall FFaacc""iidd
23
Supercritical Wing
•! Richard Whitcomb s supercritical airfoil –! Wing upper surface flattened to increase Mcrit–! Wing thickness can be restored
•! Important for structural efficiency, fuel storage, etc.
Pressure Distribution on Supercritical Airfoil ~ Section Lift
(–)
(+)
NASA Supercritical Wing F-8
Airbus A320
24
Subsonic Air Compressibility and Sweep Effects on 3-D Wing Lift Slope
•! Subsonic 3-D wing, with sweep effect
CL!=
"AR
1+ 1+ AR2cos#1 4
$
%&&
'
())
2
1*M 2 cos#1 4( )+
,
---
.
/
000
"1 4 = sweep angle of quarter chord25
Subsonic Air Compressibility Effects on 3-D Wing Lift Slope•! Subsonic 3-D wing, sweep = 0
plot(pi A / (1+sqrt(1 + ((A / 2)^2) (1 - M^2))), A=1 to 20, M = 0 to 0.9)
26
Subsonic Air Compressibility Effects on 3-D Wing Lift Slope•! Subsonic 3-D wing, sweep = 60°
plot(pi A / (1+sqrt(1 + (A ^2) (1 – 0.5 M^2))), A=1 to 20, M = 0 to 0.9)
27
Lift-Drag Polar for a Typical Bizjet
•! L/D equals slope of line drawn from the origin–! Single maximum for a given polar–! Two solutions for lower L/D (high and low airspeed)–! Available L/D decreases with Mach number
•! Intercept for L/Dmax depends only on !! and zero-lift drag
Note different scales for lift and drag
28
Wing Lift Slope at M = 1Approximation for all wing planforms
CL!="AR2
= 2" AR4
#
$%
&
'(
29
Supersonic Compressibility Effects on Triangular Wing Lift Slope
•! Supersonic delta (triangular) wing
CL!=
4M 2 "1
Supersonic leading edge
CL!= 2"
2 cot#" + $( )
where
$ = m 0.38 + 2.26m % 0.86m2( )m = cot#LE cot&
Subsonic leading edge
"LE = sweep angle of leading edge
30
Supersonic Effects on Arbitrary Wing and Wing-Body Lift Slope
•! Impinging shock waves•! Discrete areas with differing M and
local pressure coefficients, cp•! Areas change with #•! No simple equations for lift slope
Schlicting & Truckenbrodt, 197931
Fighter Jets of the 1950s: “Century Series”•! Emphasis on supersonic speed
Republic F-105
Lockheed F-104
Convair F-102
McDonnell F-101North American F-100
!!iiss""rriiccaall FFaacc""iidd
32
What Happened to the F-103?
Republic F-105(833 built)
!!iiss""rriiccaall FFaacc""iidd
Republic XF-103
33
Transonic Drag Rise and the Area Rule•! Richard Whitcomb (NASA Langley) and Wallace Hayes (Princeton)•! YF-102A (left) could not break the speed of sound in level flight;
F-102A (right) could
34
Area Rulehttps://en.wikipedia.org/wiki/Area_rule
Transonic Drag Rise and the Area Rule
Talay, NASA SP-367
Cross-sectional area of the total configuration should gradually increase and decrease to minimize transonic drag