Aircraft Flight Dynamics Robert F. Stengel Lecture5 Configuration Aerodynamics 2.

– affects leading-edge and trailing-edge flow separation" notched or dog-toothed wing leading edges" – Boundary layer control" – Maintain attached flow with increasing α!. • Strakes

Configuration Aerodynamics - 2


Robert 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 = C D 1

2ρV

2

S ≈ C D0 + εC L2

( )12ρV2

S

≈ C D0 + ε(C L o + C Lαα)2

%

1

2ρV

2

S

Induced Drag

Induced Drag of a Wing"

–   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"

C D i = C L isin αi ≈ C( L0+ C Lαα)sin αi

≈ C( L0+ C Lαα)αi≡ εC L2

≡ C L

2

πeAR=

C L2 ( 1+ δ )

πAR where

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"

•   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 "

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 withα!

–   affects leading-edge and

trailing-edge flow separation"

notched or dog-toothed wing leading edges"

–   Boundary layer control"

–   Maintain attached flow with increasing α! –   Avoid tip stall "

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 "

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 "

occurs"

rudder should operate properly"

and loss of roll control before the stall"

•   Strakes for improved high-α flight"

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)"

C L 2− D (y)c(y)

C L

3− D c

•   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 "

C l

δA

Induced Drag Factor, δ!

•   Graph for δ

(McCormick, p 172)"

Lower AR!

C D

2

1 + δ

( )

π AR

Oswald Efficiency Factor, e!

•   Approximation for e (Pamadi, p 390)"

e ≈ 1.1C Lα

RC L

α + (1 − R)π AR

where

R = 0.0004κ3

− 0.008κ2+ 0.05κ + 0.86

κ = AR λ cos ΛLE

C D i = C L

2

πeAR

P-51 Mustang"

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

Wing Span = 37 ft (9.83 m)

)

Loaded Weight = 9,200 lb (3, 465 kg) Maximum Power = 1, 720 hp (1,282 kW )

o = 0.0163

AR = 5.83

λ = 0.5

P-51 Mustang Example"

C L

1 + 1 + AR

2

#

$%

&

'(

2

)

*

+ +

,

- .

= 4.49 per rad (wing only)

e = 0.947

δ = 0.0557

ε = 0.0576

C D i=εC L= C L

πeAR=

C L( 1 + δ )

π AR

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

Mach Number Effects

Drag Due to Pressure Differential"

C D base = C pressure base S base

S ≈

0.029

C friction S wet

S base

S base

S (M < 1) [Hoerner]

< 2

γ M2

S base S

#

$

% &

' ( (M > 2, γ = specific heat ratio)

“The Sonic Barrier”!

Blunt base pressure drag"

wave≈C D incompressible

≈C D compressible

≈ C D M ≈ 2

M2

−1 (M > 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"

–  Mach number at which local

flow first becomes sonic"

–  Onset of drag-divergence"

–  M ~ 0.7 to 0.85 "

Pressure Drag"

•   Thinner chord sections lead to higher M crit,

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!

M crit swept = M crit unswept

cos Λ

Talay, NASA SP-367!

Transonic Drag Rise and the Area Rule"

•   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!

increase and decrease to minimize transonic drag"

Sears-Haack Body!

Supercritical

Wing"

–  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"

(1935)"

one specific Mach number"

Tohoku U (PAS, 47, 2011, 53-87)"

3-D wings"

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

Supersonic Transport Concept"

MIT, AIAA-2011-1248"

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 =

C L q S

C D q S =

C L

C D

Typical Bizjet"

•   Lift-Drag Polar: Cross-plot of C L (α) vs. C D (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 rate

Unit area

!

"

%

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

Newtonian Flow"

= ρV2

α

= 2sin ( 2α )#12ρV2

$

% &

'

(S

≡ C N

1

2ρV 2

#

$

% &

'

(S = C N qS

Lift = N cosα

C L= 2sin( 2α)cos α

Drag = N sinα

C D= 2sin3α

Normal Force =

Mass flow rate Unit area

!

"

$

%

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

Lift and drag coefficients"

Newtonian Lift and Drag Coefficients"

C L= 2sin ( 2α ) cosα

C D= 2sin 3 α

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"

X B

Y B

Z B

!

"

#

#

#

$

%

&

&

&

L B

N B

!

"

#

#

#

$

%

&

&

&

Force Vector"

Moment Vector"

r × f =

i j k

= yf( z − zf y)i + zf( x − xf z)j + xf( y − yf x)k

m =

#

$

%

%

%

%

&

'

( ( ( (

= rf =

#

$

%

%

%

&

'

( ( (

#

$

%

%

%

%

&

'

( ( ( (

Incremental Moment Produced

By Force Distribution"

Aerodynamic Force and Moment Vectors

of the Airplane"

yf z − zf y

zf x − xf z

xf y − yf x

"

#

$

$

$

$

%

&

' ' ' '

dx dy dz =

Surface

∫

L B

M B

N B

"

#

$

$

$

%

&

' ' '

f x

f y

f z

!

"

#

#

#

#

$

%

&

&

&

&

dx dy dz

Surface

X B

Y B

Z B

!

"

#

#

#

$

%

&

&

&

•   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 "

C m

α,C m q ,C m

α,C n

β,C n r ,C n

β

•   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"

V H = S ht

S

l ht c

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"

S

l vt b

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 = M B

= − #$Δp z(x, y)+ Δs z(x, y)%& x − x( cm)dx dy

surface∫∫

+ #$Δp x(y, z)+ Δs x(y, z)%&Δp x(z − z cm)dy dz

surface∫∫

Pitching Moment"

M B≈ − Z i(x i − x cm)

i=1

I

∑

+ X i(z i − z cm)+ Interference Effects + Pure Couples i=1

I

∑

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"

x cp net =!"(x cp C n)wing + x( cp C n)fuselage + x( cp C n)tail+ #$

C N total

Static Margin"

Static Margin = SM = 100 x( cm − x cp net)B

≡ 100 h( cm − h cp net)%

•   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( − xcp net)

≡ 100 hcm( − hcp net) %

Pitch-Moment Coefficient

Sensitivity to Angle of Attack"

M B = C m q Sc ≈ C( m o + C mαα)q Sc

M B = C m q Sc ≈ C$ m o − C Nα(h cm − h cp net)α&'q Sc

≈$%C m o − C Lα(h cm − h cp net)α&'q Sc

•  For small angle of attack and no control deflection"

•  Typically, static margin is positive and ∂C m /∂α

is negative for static pitch stability"

Effect of Static Margin

on Pitching Moment"

= C m o+∂C m

∂α α

#

$

'

(q Sc = C( m o + C mαα)q Sc

= 0 in trimmed (equilibrium) flight

Pitch-Moment Coefficient

Sensitivity to Angle of Attack"

•  For small angle of attack and no control deflection"

C m

α ≈ −C N αnet(h cm − h cp net)≈−C L αnet(h cm − h cp net)= −C L αnet

x cm − x cp net

c

$

%

( )

αwing

x cm − x cp wing

c

$

%

(

) − C L αht

x cm − x cp ht

c

$

%

(

) = −C L αwing

l wing c

$

%

& ' (

) − C L αht

l ht c

$

%

& ' ( )

referenced to wing area, S!

= C m αwing + C m αht

= −C L αtotal

Static Margin (%) 100

#

$

' (

Horizontal Tail Lift Sensitivity

to Angle of Attack"

∂C L

∂α

#

$

% &

'

(

horizontail tail

)

*

+ +

,

- .

aircraft reference

= C( Lαht)aircraft = C( Lαht)ht 1−∂ε

∂α

#

$

% &

' (ηelas

S ht S

#

$

% & '

( V ht

V N

#

$

% & ' (

2

•  Downwash effect on aft horizontal tail"

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

ηelas: Correction for aeroelastic effect

Angle of Attack"

αht

#

$

% &

' (

2

∂α

#

$

' (ηelas

S

#

$

% &

'

( l ht

c

#

$

% &

' (

αht

#

$

% &

' (

2

∂α

#

$

' (ηelasVHT

VHT = S ht l ht

Effects of Static Margin and Elevator Deflection on Pitching Coefficient "

of attack , i.e., sum of moments = 0"

stability"

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

∂C m ∂α

αTrim = − 1

α

o + C m

δE δE

αα + C m δ E δ E

Subsonic Pitching Coefficient

vs Angle of Attack (0° < α < 90°)"

Lateral-Directional Effects

of Sideslip Angle

Rolling and Yawing Moments

of the Airplane"

L B≈ Z i(y i − y cm)

i=1

I

∑

− Y i(z i − z cm)+ Interference Effects + Pure Couples

i=1

I

∑

Distributed effects can be aggregated to local

centers of pressure "

N B≈ Y i(x i − x cm)

i=1

I

∑

− X i(y i − y cm)+ Interference Effects + Pure Couples

i=1

I

∑

Rolling Moment!

Yawing Moment!

Sideslip Angle Produces Side Force,

!   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 ≈ ∂C Y

∂β qS • β = C YβqS • β

C Y

β ≈ C( )Yβ Fuselage + C( )Yβ Vertical Tail + C( )Yβ Wing

( )Vertical Tail≈ ∂C Y

∂ β

$

%

& '

(

)

vt

ηvt S Vertical Tail

S

β

( )Fuselage≈ −2S Base

πd Base2 4

( )Wing ≈ −C D Parasite, Wing − kΓ2

ηvt= Vertical tail efficiency

1+ 1+ AR2

Γ = Wing dihedral angle, rad

N ≈ ∂C n

∂β

ρV2

2

%

&

)

*Sb • β = C nβ

ρV2

2

%

&

)

*Sb • β

!  Side force contributions times respective moment arms"

–  Non-dimensional stability derivative"

( )C nβ Vertical Tail ≈ −C Y βvtηvt

S vt l vt

Sb  −C Y βvtηvtV VT

Vertical tail contribution"

V VT =S vt l vt

Sb = Vertical Tail Volume Ratio

ηvt= ηelas(1+∂σ∂β) V vt

2

2

%

&

)

*

l vt Vertical tail length (+)

= distance from center of mass to tail center of pressure

= x cm − x cp vt [x is positive forward; both are negative numbers]

C nβ

( )Fuselage=−2K Volume Fuselage

Sb

"

#

%

&

1.3

Fuselage contribution"

C nβ

( )Wing = 0.75C L N Γ + fcn Λ, AR,λ( )C L2N

Wing (differential lift and induced drag) contribution"