Configuration AerodynamicsRobert Stengel, Aircraft Flight Dynamics
MAE 331, 2008
• Configuration variables
• Longitudinal aerodynamicforce and momentcoefficients– Effects of configuration
variables
– Angle of Attack
–Mach number
Copyright 2008 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html
http://www.princeton.edu/~stengel/FlightDynamics.html
Reference Characteristics
Configuration
Variables
!
c =1
Sc2dy
"b 2
b 2
#
=2
3
$
% & '
( ) 1+ * + *2
1+ *croot
• Aspect Ratio
• Taper Ratio
• Mean Aerodynamic Chord
!
" =ctip
croot!
AR =b
crectangular wing
=b " b
c " b=b2
Sany wing
from Raymer
Medium to High Aspect Ratio Configurations
Cessna 337 DeLaurier Ornithopter Schweizer 2-32
• Typical for subsonic aircraft
Boeing 777-300
Low Aspect Ratio Configurations
A-5A Vigilante
• Typical for supersonic aircraft F-104 Starfighter
Variable Aspect Ratio ConfigurationsF-111 B-1
• Aerodynamic efficiency at high and low speeds
Reconnaissance AircraftU-2 (ER-2) SR-71
• Subsonic, high-altitude flight • Supersonic, high-altitude flight
Biplane
• Compared to monoplane
– Structurally stiff (guy wires)
– Twice the wing area for the samespan
– Lower aspect ratio than a singlewing 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
Longitudinal Aerodynamic
Forces and Moment
!
Lift = CLq S
Drag = CDq S
Pitching Moment = Cmq Sc
• Non-dimensional forcecoefficients aredimensionalized by dynamicpressure and reference area
• Non-dimensional momentcoefficients aredimensionalized by dynamicpressure, reference area, andreference length
Typical subsonic lift, drag, and pitchingmoment variations with angle of attack
Circulation and Lift
• Bernoulli!s equation (inviscid, incompressible flow)
!
pstatic +1
2"V 2
= constant along streamline = pstagnation
• Vorticity
!
Vupper(x) =V" + #V (x) 2
Vlower(x) =V" $#V (x) 2
!
"(x) =#V (x)
#z(x)
• Circulation
!
" = #(x)dx0
c
$
• 2-D Lift (inviscid, incompressible flow)
!
Lift = "#V#$
=1
2"#V#
2c 2%&( ) thin, symmetric airfoil[ ]
+ "#V#$camber
For Small Angles, Lift is
Proportional to Angle of Attack
!
Lift = CL
1
2"V 2
S # CL0
+$CL
$%%
&
' ( )
* + 1
2"V 2
S , CL0
+ CL%%[ ]1
2"V 2
S
where CL%= lift slope coefficient
• 2-D lift slope coefficient: inviscid,incompressible flow, unswept wing(referenced to chord length rather thanwing area)
!
CL"
= 2#
• 2-D lift slope coefficient: inviscid,incompressible flow, swept wing
!
CL"
= 2# cos$
Effect of Aspect
Ratio on Wing Lift
Slope Coefficient
• High Aspect Ratio (> 5) Wing
!
CL"
=2#AR
AR + 2
• Low Aspect Ratio (< 2) Wing
!
CL"
=#AR
2
• All Aspect Ratios (Helmboldequation)
!
CL"
=#AR
1+ 1+AR
2
$
% &
'
( ) 2*
+
, ,
-
.
/ /
– All wings at M = 1
Air Compressibility Effects on
Wing Lift Slope Coefficient
• Subsonic, 3-D wing, with sweep effect
!
CL"
=#AR
1+ 1+AR
2cos$1 4
%
& ' '
(
) * *
2
1+M 2cos$
1 4( ),
-
.
.
/
0
1 1
• Supersonic delta (triangular) wing
!
CL"
=4
M2#1
Supersonic leading edge
!
CL"
=2# 2
cot$
# + %( )
where % = m 0.38 + 2.26m & 0.86m2( )
m = cot$LEcot'
Subsonic leading edge
!
"1 4
= sweep angle of quarter chord
!
"LE = sweep angle of leading edge
Flow Separation Produces Stall
• Decreased lift
• Increased drag
Large Angle Variations in Subsonic
Lift Coefficient (0° < ! < 90°)
!
Lift = CL
1
2"V 2
S
• All lift coefficientshave at least onemaximum (stallcondition)
• All lift coefficientsare essentiallyNewtonian at high !
• Newtonian flow:TBD
Flap Effects on
Aerodynamic Lift
• Camber modification
• Trailing-edge flap deflectionshifts CL up and down
• Leading-edge flap deflectionincreases stall !
• Same effect applies forother control surfaces
– Elevator (horizontal tail)
– Ailerons (wing)
– Rudder (vertical tail)
Wing-Fuselage Interference Effects
• Wing lift induces
– Upwash in front of the wing
– Downwash behind the wing
– 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
Aerodynamic Drag
!
Drag = CD
1
2"V 2
S # CD0
+ $CL
2[ ]1
2"V 2
S
Parasitic Drag
!
Parasitic Drag = CD0
1
2"V 2
S
• Pressure differential,viscous shear stress,and separation
Reynolds Number, Skin Friction,
and Boundary Layer
!
Reynolds Number = Re ="Vl
µ=Vl
#
where
" = air density
V = true airspeed
l = characteristic length
µ = absolute (dynamic) viscosity
# = kinematic viscosity
• Skin friction coefficient
!
C f =Friction Drag
q Swet
where Swet = wetted area
!
Cf "1.33Re#1/ 2
laminar flow[ ]
" 0.46 log10 Re( )#2.58
turbulent flow[ ]
Typical Effect of Reynolds Number
on Drag
• Flow may stay attachedfarther at high Re, reducingthe drag
from Werle*
* See Van Dyke, M., An Album of Fluid Motion,Parabolic Press, Stanford, 1982
Effect of Streamlining on Drag
Induced Drag
• Lift produces downwash (angle proportional to lift)
• Downwash rotates velocity vector
• Lift is perpendicular to velocity vector
• Axial component of lift induces drag
!
CDi= CLi
sin" i # CL0
+ CL""( )sin" i # CL
0
+ CL""( )" i $ %CL
2
$CL
2
&eAR=CL
2
1+ '( )&AR
where
e =Oswald efficiency factor
' = departure from ideal elliptical lift distribution
Spitfire
Spanwise Lift Distribution
of 3-D Wings
• Wing does not have to have an elliptical planformto have a nearly elliptical lift distribution
Straight Wings (@ 1/4 chord)(McCormick)
Straight and Swept Wings(NASA SP-367)
TR = taper ratio, "
Oswald Efficiency and
Induced Drag Factors• Approximation for e
(Pamadi, p. 390)
!
e "1.1C
L#
RCL#
+ (1$ R)%AR
where
R = 0.0004& 3+$ 0.008& 2 + 0.05& + 0.86
& =AR '
cos(LE
• Graph for #(McCormick, p. 172)
Higher AR
!
CDi
=CL
2
"eAR
!
CDi
=CL
2
1+ "( )#AR
P-51 Mustang
http://en.wikipedia.org/wiki/P-51_Mustang
!
Wing Span = 37 ft (9.83m)
Wing Area = 235 ft (21.83m2)
LoadedWeight = 9,200 lb (3,465 kg)
Maximum Power =1,720 hp (1,282 kW )
CDo= 0.0163
AR = 5.83
" = 0.5
P-51 Mustang Example
!
CL"=
#AR
1+ 1+AR
2
$
% &
'
( )
2*
+
, ,
-
.
/ /
= 4.49 per rad (wing only)
e = 0.947
0 = 0.0557
1 = 0.0576
!
CDi
= "CL
2
=CL
2
#eAR=CL
2
1+ $( )#AR
Drag Due to
Pressure Differential
• Blunt base pressure drag
!
CDbase= Cpressurebase
SbaseS" 0.29Sbase
C frictionSwetM <1( ) Hoerner[ ]
<2
#M 2
Sbase
S
$
% &
'
( ) M > 2, # = specific heat ratio( )
• Prandtl factor
!
CDwave"CDincompressible
1#M2
M <1( )
"CDcompressible
M2#1
M >1( )
"CD
M " 2
M2#1
M >1( )
Air Compressibility Effect
Subsonic
Supersonic
Transonic
Incompressible
Shock Waves inSupersonic Flow
• Drag rises due to pressureincrease 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 atsome points, greater than sonicelsewhere
• Supersonic flow
– Local airspeed is greater thansonic virtually everywhere
Drag Coefficient
vs. Mach
Number• Critical Mach number
– Mach number at which local flow
first becomes sonic
– Onset of drag-divergence
– Mcrit ~ 0.7 to 0.85
Sweep AngleEffect on Wing Drag
!
Mcritswept=Mcritunswept
cos"
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
Supercritical
Wing
• Thinner chord sections lead to higher Mcrit
• 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 distributionon Supercritical Airfoil
(–)
(+)
Large Angle Variations in Subsonic
Drag Coefficient (0° < ! < 90°)
• All drag coefficients converge to Newtonian-likevalues at high angle of attack
• Low-AR wing has less drag than high-AR wing
Newtonian Flow
• No circulation
• “Cookie-cutter” flow
• Equal pressure acrossbottom of the flat plate
!
Normal Force =Mass flow rate
Unit area
"
# $
%
& ' Change in velocity( ) Projected Area( ) Angle between plate and velocity( )
!
N = "V( ) V( ) S sin#( ) sin#( )
= "V2( ) S sin
2#( )
= 2sin2#( )
1
2"V
2$
% &
'
( ) S
* CN
1
2"V
2$
% &
'
( ) S = CNq S
!
Lift = N cos"
CL = 2sin2"( )cos"
!
Drag = N sin"
CD = 2sin3"
Application of Newtonian Flow
• Hypersonic flow (M ~> 5)– Shock wave close to surface
(thin shock layer), merging withthe boundary layer
– Flow is ~ parallel to the surface
– Separated upper surface flow
Space Shuttle inSupersonic Flow
High-Angle-of-AttackResearch Vehicle (F-18)
• All Mach numbers athigh angle of attack– Separated flow on upper
(leeward) surfaces
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 well
• High-AR wing has the best overall L/D
Lift/Drag vs. Angle of Attack• L/D is an important performance metric for aircraft
!
L
D=
CLq S
CDq S=
CL
CD
• Low-AR wing has best L/D at intermediateangle of attack
Stagger Effect on Biplane
CL vs. C
D and C
L vs. L/D
NACA TN-70
• Biplane wing with no stagger has the best L/D
Pitching Moment
!
Body " Axis Pitching Moment = MB = " #pz + #sz( )xdxdysurface
$$ + #px + #sx( )#pxzdydzsurface
$$
• Pressure and shear stress differentials times moment armsintegrate over the surface to produce a net pitching moment
• Center of mass establishes the moment arm center
Pitching Moment
!
MB " # Zix1 +i=1
I
$ Xiz1 + Interference Effects+ Pure Couplesi=1
I
$
• Distributed effects canbe aggregated to localcenters of pressure
Pure Couple
• Net force = 0
• Net moment " 0
Rockets Cambered Lifting Surface
Fuselage
Net Center of Pressure
and Static Margin• Local centers of pressure can be aggregated at a net center of
pressure (or neutral point)
!
xcpnet =xcpCn( )
wing+ xcpCn( )
fuselage+ xcpCn( )
tail+ ...[ ]
CNtotal
!
Static Margin = SM =100 xcm " xcpnet
( )c
,%
#100 hcm " hcpnet( )%
• Static margin reflects the distance between the center of massand the net center of pressure
Effect of Static Margin on
Pitching Moment
!
MB = Cmq Sc " Cm o#CN$
hcm # hcpnet( )$[ ]q Sc " Cm o
#CL$hcm # hcpnet
( )$[ ]q Sc
" Cm o+%Cm
%$$
&
' (
)
* + q Sc = Cm o
+ Cm$$( )q Sc
= 0 in trimmed (equilibrium) flight
• For small angle of attack and no control deflection
• Typically, static margin is positive and !Cm/!! is negative forstatic pitch stability
Pitch-Moment Coefficient
Sensitivity to Angle of Attack
!
MB = Cmq Sc " Cm o+ Cm#
#( )q Sc
• For small angle of attack and no control deflection
!
Cm"# $CN"net
hcm $ hcpnet( ) # $CL"net
hcm $ hcpnet( ) = $CL"net
xcm $ xcpnet
c
%
& '
(
) *
= $CL"wing
xcm $ xcpwing
c
%
& '
(
) * $CL"ht
xcm $ xcpht
c
%
& '
(
) * = $CL"wing
lwing
c
%
& '
(
) * $CL"ht
lht
c
%
& '
(
) *
= Cm"wing
+ Cm"ht referenced to wing area, S
Horizontal Tail Lift Sensitivity
to Angle of Attack
!
CL"ht( )
aircraft=Vtail
VN
#
$ %
&
' (
2
1)*+
*"
#
$ %
&
' ( ,elas
Sht
S
#
$ %
&
' ( CL"ht( )
ht
$ = Wing downwash angle at
the tailVTail = Airspeed at vertical tail;
“scrubbing” lowers VTail,propeller slipstreamincreases VTail
%elas = Aeroelastic correction
factor
Effects of Static Margin and Elevator
Deflection on Pitching Coefficient
• Zero crossingdetermines trim angleof attack
• Negative sloperequired for staticstability
• Slope, !Cm/!!, varieswith static margin
• Control deflectionaffects Cmo and trimangle of attack
!
MB = Cm o+ Cm"
" + Cm#E#E( )q Sc
Subsonic Pitching Coefficient
vs. Angle of Attack (0° < ! < 90°)
“Pitch Up” and Deep Stall
• Possibility of 2 stableequilibrium (trim) points withsame control setting– Low !
– High !
• High-angle trim is calleddeep stall– Low lift
– High drag
• Large control momentrequired to regain low-angletrim
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