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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