Configuration Aerodynamics - 2Robert Stengel, Aircraft Flight Dynamics, MAE 331,

2010

Drag

Induced drag

Compressibility effects

P-51 example

Newtonian Flow

Pitching Moment

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

Aerodynamic Drag

Drag = CD1

2!V 2S " CD

0

+ #CL2( )1

2!V 2S

" CD0

+ # CLo + CL$$( )2%

&'()*1

2!V 2S

Induced Drag

Induced Drag

Lift produces downwash (angle proportional to lift)

Downwash rotates velocity vector

Lift is perpendicular to velocity vector

Axial component of rotated lift induces drag

Induced Drag

CDi = CLi sin! i " CL0 + CL!!( )sin! i" CL0 + CL!!( )! i # $CL

2

#CL2

%eAR=CL21+ &( )

%AR

where

e = Oswald efficiency factor = 1 for elliptical distribution

& = departure from ideal elliptical lift distribution

Spitfire Spanwise Lift Distribution

of 3-D (Trapezoidal) WingsStraight Wings (@ 1/4 chord)

(McCormick)

TR = taper ratio, !

For some taper ratio between 0.35 and 1,lift distribution is nearly elliptical

Spanwise Lift Distributionof 3-D Wings

Wing does nothave to have anellipticalplanform to havea nearly ellipticallift distribution

Sweep moves liftdistributiontoward tips

Straight and Swept Wings(NASA SP-367)

CL2!D (y)c(y)

CL3!D c

Induced Drag Factor Graph for " (McCormick, p. 172)

Lower AR

CDi=CL

21+ !( )

"AR

Oswald Efficiency Factor 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

CDi=

CL

2

!eAR

P-51 Mustang

http://en.wikipedia.org/wiki/P-51_Mustang

Wing Span = 37 ft (9.83m)

Wing Area = 235 ft (21.83m2)

Loaded Weight = 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

/ = 0.05570 = 0.0576

CDi= !C

L

2=

CL

2

"eAR=CL

21+ #( )

"AR

Mach Number Effects

Drag Due toPressure Differential

Blunt base pressure drag

CDbase = CpressurebaseSbase

S! 0.29Sbase

C frictionSwetM < 1( ) Hoerner[ ]

1( )

!

CDM ! 2

M2"1

M > 1( )

Air Compressibility EffectShock Waves in

Supersonic 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

Critical Mach number

Mach number at which localflow first becomes sonic

Onset of drag-divergence

Mcrit ~ 0.7 to 0.85

Effect of Chord

Thickness on Wing

Pressure Drag

Thinner chord sections lead to higher Mcrit

Lockheed P-38

Lockheed F-104

Pressure Drag on WingDepends on Sweep Angle

Sweep AngleEffect on Wing Drag

Mcritswept

=

Mcritunswept

cos!

Air CompressibilityEffect

Subsonic

SupersonicTransonic

Incompressible

Sonic Boomshttp://www.youtube.com/watch?v=gWGLAAYdbbc

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

Transonic Drag Rise and the Area Rule 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

()

(+)

NASA Supercritical

Wing F-8

Airbus A320

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 andHigh-Angle-of-Attack

Lift and Drag

Newtonian Flow

No circulation

Cookie-cutter

flow

Equal pressureacross bottom of

the flat plate

Normal Force =

Mass flow rate

Unit area

!

"#$

%&Change in velocity( ) Projected Area( ) Angle between plate and velocity( )

Newtonian Flow

N = !V( ) V( ) S sin"( ) sin"( )

= !V 2( ) S sin2"( )

= 2sin2"( )

1

2!V 2#

$%&'(S

) CN1

2!V 2#

$%&'(S = CNqS

Lift = N cos!

CL = 2sin2!( )cos!

Drag = N sin!

CD = 2sin3!

Normal Force =

Mass flow rate

Unit area

!

"#$

%&Change in velocity( ) Projected Area( ) Angle between plate and velocity( )

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 in

Supersonic Flow

High-Angle-of-Attack Research

Vehicle (F-18) All Mach numbers at

high 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-to-Drag Ratio vs.Angle of Attack

L/D is an important performance metric for aircraft

High-AR wing has best overall L/D

Low-AR wing has best L/D at intermediate angle of attack

L

D=

CLq S

CDq S=

CL

CD

Pitching Moment

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 can be aggregated tolocal centers of pressure

Pure Couple

Net force = 0

Net moment " 0

Rockets Cambered Lifting Surface

Fuselage

Net Center of Pressure

Local centers of pressure can be aggregatedat a net center of pressure (or neutral point)

xcpnet =xcpCn( )wing + xcpCn( ) fuselage + xcpCn( )tail + ...

!"

#$

CNtotal

Static Margin

Static Margin = SM =100 xcm ! xcpnet( )

c, %

" 100 hcm ! hcpnet( )%

Static margin reflects the distance between thecenter of mass and the net center of pressure

Static Margin

Static Margin = SM =100 xcm ! xcpnet( )

c, %

" 100 hcm ! hcpnet( )%

Effect of Static Margin onPitching Moment

MB = Cmq Sc ! Cmo " CN# hcm " hcpnet( )#$% &'q Sc ! Cmo " CL# hcm " hcpnet( )#$% &'q Sc

! Cmo +(Cm(#

#)*+

,-.q Sc = Cmo + 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 ! Cmo + Cm""( )q Sc

For small angle of attack and no controldeflection

Pitch-Moment Coefficient

Sensitivity to Angle of Attack For small angle of attack and no control deflection

Cm! " #CN!net hcm # hcpnet( ) " #CL!net hcm # hcpnet( ) = #CL!netxcm # xcpnet

c

$%&

'()

= #CL!wingxcm # xcpwing

c

$

%&'

()# CL!ht

xcm # xcphtc

$%&

'()= #CL!wing

lwing

c

$%&

'()# CL!ht

lht

c

$%&

'()

= Cm!wing+ Cm!ht

= #CL!totalStatic Margin (%)

100

$%&

'()

referenced to wing area, S

Horizontal Tail Lift Sensitivity

to Angle of Attack

!CL!"

#$%

&'( horizontailtail

)

*

++

,

-

.

.aircraftreference

= CL"ht( )aircraft =Vtail

VN

#

$%&

'(

2

1/!0!"

#$%

&'(1elas

Sht

S

#$%

&'(CL"ht( )ht

Upwash effect on acanard (i.e., forward)surface

Vtail

: Airspeed at the horizontal tail [Flow over body (), Scrubbing (), Propeller slipstream(+)]

! : Downwash angle due to wing lift at the horizontal tail

"! "# : Sensitivity of downwash angle to angle of attack

$elas

: Correction for aeroelastic effect

Horizontal Tail MomentSensitivity to Angle of Attack

Cm!ht

= "Vtail

VN

#

$%&

'(

2

1")*)!

#$%

&'(+elas

Sht

S

#$%

&'(CL!ht

( )ht

lht

c

#$%

&'(

Normal force (~lift) x moment arm

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 = Cmo + Cm!! + Cm"E"E( )q Sc

Subsonic Pitching Coefficient

vs. Angle of Attack (0 < # < 90) Pitch Up and Deep Stall

Possibility of 2 stable equilibrium(trim) points with same control setting

Low #

High #

High-angle trim is called deep stall

Low lift

High drag

Large control moment required toregain low-angle trim

Pitch Up andDeep Stall

Possibility of 2stable equilibrium(trim) points withsame controlsetting Low #

High #

High-angle trim iscalled deep stall Low lift

High drag

Large controlmoment requiredto regain low-angle trim

Air Midwest 5481, Beechcraft 1900Dhttp://www.youtube.com/watch?v=2pVBN9cLVuc

Colgan Flight 3407, Bombardier, Dash 8 Q400http://www.youtube.com/watch?v=lxywEE1kK6I&feature=fvw

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