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

2012!!  Drag"

!  Induced drag"!  Compressibility effects"

!  P-51 example"!  Newtonian Flow"!  Moments"!  Effects of Sideslip

Angle"Copyright 2012 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 = CD12ρV 2S ≈ CD0

+ εCL2( ) 12 ρV

2S

≈ CD0+ ε CLo

+ CLαα( )2%

&'()*12ρV 2S

Induced Drag�

Induced Drag of a Wing"

•  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"

Induced Drag of a Wing"CDi

=CLisinαi ≈ CL0

+CLαα( )sinαi

≈ CL0+CLα

α( )αi ≡εCL2

≡CL2

πeAR=CL2 1+δ( )πAR

wheree =Oswald efficiency factor =1 for elliptical distributionδ = departure from ideal elliptical lift distribution

Spitfire!Straight, Swept, and

Tapered Wings"

•  Straight at the quarter chord"

•  Swept at the quarter chord"

•  Progression of separated flow from trailing edge with increasing angle of attack"

Wing Design Parameters"•  Planform"

–  Aspect ratio"–  Sweep"–  Taper"–  Complex geometries"–  Shape at root"–  Shape at tip"

•  Chord section"–  Airfoils"–  Twist"

•  Movable surfaces"–  Leading- and trailing-edge devices"–  Ailerons"–  Spoilers"

•  Interfaces"–  Fuselage"–  Powerplants"–  Dihedral angle"

Taper Ratio Effects"

•  Taper makes lift distribution more elliptical"

–  λ ~ 0.45 is best"–  L/D effect (phugoid)"

•  Tip stall (pitch up)"•  Bending stress"•  Roll Damping"

Airfoil Effects"•  Camber increases zero-α lift

coefficient"•  Thickness"

–  increases α for stall and softens the stall break"

–  reduces subsonic drag "–  increases transonic drag "–  causes abrupt pitching

moment variation (more to follow)"

•  Profile design "–  can reduce c.p. (static

margin) variation with α!–  affects leading-edge and

trailing-edge flow separation"

•  Vortex generators, fences, vortilons, notched or dog-toothed wing leading edges"–  Boundary layer control"–  Maintain attached flow with increasing α!–  Avoid tip stall"

Secondary Wing Structures"

McDonnell-Douglas F-4!

Sukhoi Su-22!

LTV F-8!

•  Strakes or leading edge extensions"–  Maintain lift at high α!–  Reduce c.p. shift at high Mach number"

Leading-Edge Extensions"

McDonnell Douglas F-18! General Dynamics F-16!

•  Winglets, rake, and Hoerner tip reduce induced drag by controlling the tip vortices"

•  End plate, wingtip fence straightens flow, increasing apparent aspect ratio (L/D)"

•  Chamfer produces favorable roll w/ sideslip"

Wingtip Design"

Yankee AA-1!

Boeing 747-400! Boeing P-8A!

Airbus A319!

•  Marked by noticeable, uncommanded changes in pitch, yaw, or roll and/or by a marked increase in buffet"

•  Stall must be detectable"•  Aircraft must pitch down when it

occurs"•  Up to the stall break, ailerons and

rudder should operate properly"•  Inboard stall strips to prevent tip stall

and loss of roll control before the stall"•  Strakes for improved high-α flight"

Design for Satisfactory Stalls" Spanwise Lift Distribution of 3-D (Trapezoidal) Wings"

Straight Wings (@ 1/4 chord)"(McCormick)"

TR = taper ratio, λ!

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

Spanwise Lift Distribution of 3-D Wings"

•  Wing does not have to have a geometrically elliptical planform to have a nearly elliptical lift distribution"

•  Sweep moves lift distribution toward tips"Straight and Swept Wings"

(NASA SP-367)"

CL2−D(y)c(y)

CL3−Dc

•  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"

•  separation�burble� produces buffet at tail surface, warning of stall"

–  improves aileron effectiveness at high α"

Wing Twist Effects"

ClδA

Induced Drag Factor, δ!

•  Graph for δ (McCormick, p. 172)"

Lower AR!

CDi=CL2 1+ δ( )πAR

Oswald Efficiency Factor, e!

•  Approximation for e (Pamadi, p. 390)"

e ≈1.1CLα

RCLα+ (1− R)πAR

whereR = 0.0004κ 3 − 0.008κ 2 + 0.05κ + 0.86

κ =AR λcosΛLE

CDi=

CL2

πeAR

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

= 0.0163AR = 5.83λ = 0.5

P-51 Mustang Example"

CLα=

πAR

1+ 1+ AR2

#$%

&'(

2)

*++

,

-..

= 4.49 per rad (wing only)

e = 0.947δ = 0.0557ε = 0.0576

CDi= εCL

2 =CL2

πeAR=CL2 1+ δ( )πAR

http://www.youtube.com/watch?v=WE0sr4vmZtU!

Mach Number Effects�

Drag Due to Pressure Differential"

CDbase=Cpressurebase

SbaseS ≈

0.029

CfrictionSwetSbase

SbaseS

M <1( ) Hoerner[ ]

<2

γM 2SbaseS

#

$%

&

'( M > 2, γ = 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"

Shock Waves in!Supersonic 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" Effect of Chord Thickness on Wing

Pressure Drag"

•  Thinner chord sections lead to higher Mcrit, or drag-divergence Mach number"

Lockheed P-38!Lockheed F-104!

Air Compressibility Effect on Wing Drag"

Subsonic!

Supersonic!Transonic!

Incompressible!

Sonic Booms"http://www.youtube.com/watch?v=gWGLAAYdbbc"

Pressure Drag on Wing Depends on Sweep Angle "

Sweep Angle!Effect on Wing Drag!

Mcritswept=Mcritunswept

cosΛ

Talay, NASA SP-367!

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"

Talay, NASA SP-367!

•  Cross-sectional area of the total configuration should gradually increase and decrease to minimize transonic drag"

Sears-Haack Body!http://en.wikipedia.org/wiki/Sears-Haack_body!

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! Supersonic Biplane"•  Concept of Adolf Busemann

(1935)"•  Shock wave cancellation at

one specific Mach number"•  2-D wing"

•  Kazuhiro Kusunose et al , Tohoku U (PAS, 47, 2011, 53-87)"•  Adjustable flaps"•  Tapered, variably spaced

3-D wings"•  Fuselage added"

http://en.wikipedia.org/wiki/Adolf_Busemann!

Supersonic Transport Concept"

•  Rui Hu, Qiqi Wang, Antony Jameson, Stanford, MIT, AIAA-2011-1248"•  Optimization of biplane aerodynamics"•  Sketch of possible configuration"

Large Angle Variations in Subsonic Drag Coefficient (0° < α < 90°)"

•  All wing drag coefficients converge to Newtonian-like values at high angle of attack"

•  Low-AR wing has less drag than high-AR wing at given α!

Lift vs. Drag for Large Variation in Angle-of-Attack (0° < α < 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"

€

LD

=CLq SCDq S

=CL

CD

Lift-Drag Polar for a Typical Bizjet"

•  Lift-Drag Polar: Cross-plot of CL(α) vs. CD(α)"

Note different scaling for lift and drag!

•  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)"

Newtonian Flow and High-Angle-of-Attack

Lift and Drag�

Newtonian Flow"

•  No circulation"•  �Cookie-cutter�

flow"•  Equal pressure

across bottom of the flat plate"

Normal Force =

Mass flow rateUnit area

!

"#

$

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

Newtonian Flow"

N = ρV( ) V( ) S sinα( ) sinα( )= ρV 2( ) S sin2α( )

= 2sin2α( ) 12 ρV2#

$%

&

'(S

≡CN12ρV 2#

$%

&

'(S =CNqS

Lift = N cosα

CL = 2sin2α( )cosα

Drag = N sinαCD = 2sin

3α

Normal Force =

Mass flow rateUnit area

!

"#

$

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

Lift and drag coefficients"

Newtonian Lift and Drag Coefficients"

CL = 2sin2α( )cosα

CD = 2sin3α

Application of Newtonian Flow"

•  Hypersonic flow (M ~> 5)"–  Shock wave close to surface

(thin shock layer), merging with the 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"

Moments of the Airplane �

Airplane Forces and Moments Resolved into Body Axes"

fB =XB

YBZB

!

"

###

$

%

&&&

mB =

LBMB

NB

!

"

###

$

%

&&&

Force Vector"

Moment Vector"

r× f =i j kx y zfx fy fz

= yfz − zfy( ) i+ zfx − xfz( ) j + xfy − yfx( )k

m =

yfz − zfy( )zfx − xfz( )xfy − yfx( )

#

$

%%%%

&

'

((((

= rf =0 −z yz 0 −x−y x 0

#

$

%%%

&

'

(((

fxfyfz

#

$

%%%%

&

'

((((

Incremental Moment Produced

By Force Distribution"

Aerodynamic Force and Moment Vectors

of the Airplane"

mB =

yfz − zfy( )zfx − xfz( )xfy − yfx( )

"

#

$$$$

%

&

''''

dxdydz =Surface∫

LBMB

NB

"

#

$$$

%

&

'''

fB =fxfyfz

!

"

####

$

%

&&&&

dxdydzSurface∫ =

XB

YBZB

!

"

###

$

%

&&&

•  Aerodynamics analogous to those of the wing"

•  Longitudinal stability"–  Horizontal stabilizer"–  Short period natural

frequency and damping"•  Directional stability"

–  Vertical stabilizer (fin)"•  Ventral fins"•  Strakes"•  Leading-edge extensions"•  Multiple surfaces"•  Butterfly (V) tail"

–  Dutch roll natural frequency and damping"

•  Stall or spin prevention/ recovery"

•  Avoid rudder lock (TBD)!

Tail Design Effects"

Cmα

,Cmq,Cm α

,Cnβ,Cnr

,Cn β

Horizontal Tail Location and Size!•  15-30% of wing area"•  ~ wing semi-span behind the c.m."•  Must trim neutrally stable airplane at maximum lift in ground effect"•  Effect on short period mode"•  Horizontal Tail Volume: Typical value = 0.48"

VH =ShtSlhtc

North American F-86!Lockheed Martin F-35!

•  Analogous to horizontal tail volume"•  Effect on Dutch roll mode"•  Powerful rudder for spin recovery"

–  Full-length rudder located behind the elevator"–  High horizontal tail so as not to block the flow over the rudder"

•  Vertical Tail Volume: Typical value = 0.18"

VV =SvtSlvtb

Vertical Tail Location and Size!

Curtiss SB2C! Piper Tomahawk!

Pitching Moment of the Airplane �

Pitching Moment"•  Pressure and shear stress differentials times moment arms integrate

over the airplane surface to produce a net pitching moment"•  Center of mass establishes the moment arm center"

Body - Axis Pitching Moment =MB

= − Δpz x, y( )+Δsz x, y( )#$ %& x − xcm( )dxdysurface∫∫

+ Δpx y, z( )+Δsx y, z( )#$ %&Δpx z− zcm( )dydzsurface∫∫

Pitching Moment"

MB ≈ − Zi xi − x cm( )i=1

I

∑

+ Xi zi − zcm( )+ Interference Effects+Pure Couplesi=1

I

∑

•  Distributed effects can be aggregated to local centers of pressure"

Pure Couple"•  Net force = 0"

Rockets! Cambered Lifting Surface!

Fuselage!•  Cross-sectional area, A!•  x positive to the right"•  At small α!

–  Positive lift with dA/dx > 0"–  Negative lift with dA/dx < 0"

•  Net moment ≠ 0"

Net Center of Pressure "•  Local centers of pressure can be aggregated

at a net center of pressure (or neutral point) along the body x axis"

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

!"

#$

CNtotal

Static Margin"

Static Margin = SM =100 xcm − xcpnet( )B

c, %

≡100 hcm −hcpnet( )%

•  Static margin reflects the distance between the center of mass and the net center of pressure"•  Body axes"•  Normalized by mean aerodynamic chord"•  Does not reflect z position of c.p.!

Static Margin"

Static Margin = SM =100 xcm − xcpnet( )

c, %

≡ 100 hcm − hcpnet( )%

Pitch-Moment Coefficient Sensitivity to Angle of Attack"

MB = Cmq Sc ≈ Cmo+ Cmα

α( )q Sc•  For small angle of attack and no control deflection"

MB =Cmq Sc ≈ Cmo−CNα

hcm −hcpnet( )α$%

&'q Sc

≈ Cmo−CLα

hcm −hcpnet( )α$%

&'q Sc

•  For small angle of attack and no control deflection"

•  Typically, static margin is positive and ∂Cm/∂α is negative for static pitch stability"

Effect of Static Margin on Pitching Moment"

= Cmo+∂Cm

∂αα

#

$%

&

'(q Sc = Cmo

+Cmαα( )q Sc

= 0 in trimmed (equilibrium) flight

Pitch-Moment Coefficient Sensitivity to Angle of Attack"

•  For small angle of attack and no control deflection"

Cmα≈ −CNαnet

hcm −hcpnet( ) ≈ −CLαnethcm −hcpnet( ) = −CLαnet

xcm − xcpnetc

$

%&

'

()

≈ −CLαwing

xcm − xcpwingc

$

%&

'

()−CLαht

xcm − xcphtc

$

%&

'

()= −CLαwing

lwingc

$

%&

'

()−CLαht

lhtc

$

%&

'

()

referenced to wing area, S!=Cmαwing

+Cmαht

= −CLαtotal

Static Margin (%)100

#

$%

&

'(

Horizontal Tail Lift Sensitivity to Angle of Attack"

∂CL

∂α

#

$%

&

'(horizontailtail

)

*

++

,

-

.

.aircraftreference

= CLαht( )aircraft = CLαht( )ht 1−∂ε∂α

#

$%

&

'(ηelas

ShtS

#

$%

&

'(VhtVN

#

$%

&

'(

2

•  Downwash effect on aft horizontal tail"

•  Upwash effect on a canard (i.e., forward) surface"

Vht : Airspeed at the horizontal tail [Flow over body (±), Scrubbing (–), Propeller slipstream (+)]ε : Downwash angle due to wing lift at the horizontal tail∂ε ∂α : Sensitivity of downwash angle to angle of attackηelas : Correction for aeroelastic effect

Tail Moment Sensitivity to Angle of Attack"

Cmαht= − CLαht( )ht

VhtVN

#

$%

&

'(

2

1− ∂ε∂α

#

$%

&

'(ηelas

ShtS

#

$%

&

'(lhtc

#

$%

&

'(

= − CLαht( )htVhtVN

#

$%

&

'(

2

1− ∂ε∂α

#

$%

&

'(ηelasVHT

VHT =

ShtlhtSc

= Horizontal Tail Volume Ratio

Effects of Static Margin and Elevator Deflection on Pitching Coefficient "

•  Zero crossing determines trim angle of attack, i.e., sum of moments = 0"

•  Negative slope required for static stability"

•  Slope, ∂Cm/∂α, varies with static margin"

•  Control deflection shift curve up and down, affecting trim angle of attack"

∂Cm ∂α

αTrim = −1Cmα

Cmo+CmδE

δE( )

MB = Cmo+ Cmα

α + CmδEδE( )q Sc

Subsonic Pitching Coefficient vs. Angle of Attack (0° < α < 90°)"

Lateral-Directional Effects of Sideslip Angle�

Rolling and Yawing Moments of the Airplane"

LB ≈ Zi yi − y cm( )i=1

I

∑

− Yi zi − zcm( )+ Interference Effects+Pure Couplesi=1

I

∑

Distributed effects can be aggregated to local centers of pressure"

NB ≈ Yi xi − x cm( )i=1

I

∑

− Xi yi − ycm( )+ Interference Effects+Pure Couplesi=1

I

∑

Rolling Moment!

Yawing Moment!

Sideslip Angle Produces Side Force, Yawing Moment, and Rolling Moment!

!  Sideslip usually a small angle ( ±5 deg)"

!  Side force generally not a significant effect"

!  Yawing and rolling moments are principal effects"

Side Force due to Sideslip Angle!

Y ≈∂CY

∂βqS•β =CYβ

qS•β

CYβ≈ CYβ( )Fuselage + CYβ( )Vertical Tail + CYβ( )Wing

CYβ( )Vertical Tail ≈∂CY

∂β

$

%&

'

()vt

ηvt

SVerticalTailS

CYβ( )Fuselage ≈ −2SBaseS; SB =

πdBase2

4

CYβ( )Wing ≈ −CDParasite,Wing− kΓ2

ηvt = Vertical tail efficiency

k = πAR1+ 1+ AR2

Γ = Wing dihedral angle, rad

•  Fuselage, vertical tail, and wing are main contributors"

Yawing Moment due to Sideslip Angle!

N ≈∂Cn

∂βρV 2

2%

&'

(

)*Sb•β =Cnβ

ρV 2

2%

&'

(

)*Sb•β

!  Side force contributions times respective moment arms"–  Non-dimensional stability

derivative"

Cnβ≈ Cnβ( )

Vertical Tail+ Cnβ( )

Fuselage+ Cnβ( )

Wing+ Cnβ( )

Propeller

Cnβ( )

Vertical Tail≈ −CYβvt

ηvtSvtlvtSb −CYβvt

ηvtVVT

Vertical tail contribution"

VVT =

SvtlvtSb

=Vertical Tail Volume Ratio

ηvt =ηelas 1+∂σ ∂β( ) Vvt2

VN2

%

&'

(

)*

Yawing Moment due to Sideslip Angle!

lvt Vertical tail length (+)= distance from center of mass to tail center of pressure= xcm − xcpvt [x is positive forward; both are negative numbers]

Cnβ( )Fuselage =−2K VolumeFuselage

Sb

K = 1−dmax Lengthfuselage"

#$

%

&'1.3

Fuselage contribution"

Cnβ( )Wing = 0.75CLNΓ+ fcn Λ,AR,λ( )CLN

2

Wing (differential lift and induced drag) contribution"

Yawing Moment due to Sideslip Angle!

L ≈ ClβqSb•β

Clβ≈ Clβ( )

Wing+ Clβ( )

Wing−Fuselage+ Clβ( )

Vertical Tail

Rolling Moment due to Sideslip Angle!

•  Dihedral effect"

•  Vertical tail effect"

Rolling Moment due to Sideslip Angle!•  Crossflow effects depend on

vertical location of the wing"

Example of Configuration and Flap Effects!

NACA 641-012 Chord Section Lift, Drag, and Moment (NACA TR-824)!

CL, 60° flap!

CL, w/o flap!

CL!

Cm, w/o flap!

CD!

Cm, 60° flap!

“Drag Bucket”!

α!

Smooth ~ Laminar!

Rough ~ Turbulent!

CDo Estimate (Raymer)!

Next Time:�Aircraft Performance�

�Reading�

Flight Dynamics, 107-115, 118-130 �Virtual Textbook, Parts 6,7 �

Supplemental Material

Downwash and Elasticity Also Effect Elevator Sensitivity"

∂CL

∂δE#

$%

&

'(horizontailtail

)

*

++

,

-

.

.aircraftreference

= CLδE( )aircraft =VtailVN

#

$%

&

'(

2

1− ∂ε∂α

#

$%

&

'(ηelas

ShtS

#

$%

&

'( CLδE( )ht

�Pitch Up� and Deep Stall"•  Possibility of 2 stable equilibrium

(trim) points with same control setting"–  Low α"–  High α!

•  High-angle trim is called deep stall"–  Low lift"–  High drag"

•  Large control moment required to regain low-angle trim"

http://www.youtube.com/watch?v=IpZ8YukAwwI&feature=related !

Anatomy of a Cirrus Stall Accident"

http://www.youtube.com/watch?v=7nm_hoHhbFo!

Some Videos"

http://www.youtube.com/watch?v=saeejPWQTHw!

http://www.youtube.com/watch?v=WmseXJ7DV4c&feature=related!

!  First flight of B-58 Hustler, 1956"

!  Century series fighters, bombers, 1959"

http://www.youtube.com/watch?v=BMcuVhzCrX8&feature=related!

!  Bird of Prey, 1990s, and X-45, 2000s "

http://www.youtube.com/watch?v=hVjaiMXvCTQ!

!  XF-92A, 1948"