Gliding, Climbing, and TurningFlight Performance

Robert Stengel, Aircraft Flight Dynamics

MAE 331, 2010

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

• Flight envelope

• Minimum glide angle/rate

• Maximum climb angle/rate

• V-n diagram

• Energy climb

• Corner velocity turn

• Herbst maneuver

The Flight Envelope

Flight Envelope Determined

by Available Thrust

• Flight ceiling defined byavailable climb rate– Absolute: 0 ft/min

– Service: 100 ft/min

– Performance: 200 ft/min• Excess thrust provides the

ability to accelerate or climb

• Flight Envelope: Encompasses all altitudesand airspeeds at which an aircraft can fly

– in steady, level flight

– at fixed weight

Additional Factors Define

the Flight Envelope• Maximum Mach number

• Maximum allowableaerodynamic heating

• Maximum thrust

• Maximum dynamicpressure

• Performance ceiling

• Wing stall

• Flow-separation buffet– Angle of attack

– Local shock waves

Piper Dakota Stall Buffethttp://www.youtube.com/watch?v=mCCjGAtbZ4g

Boeing 787 FlightEnvelope (HW #5, 2008)

Best

Cruise

Region

Gliding Flight

D = CD

1

2!V 2

S = "W sin#

CL

1

2!V 2

S =W cos#

!h = V sin#

!r = V cos#

Equilibrium Gliding Flight Gliding Flight

• Thrust = 0

• Flight path angle < 0 in gliding flight

• Altitude is decreasing

• Airspeed ~ constant

• Air density ~ constant

tan! = "D

L= "

CD

CL

=

!h

!r=dh

dr

! = " tan"1 D

L

#$%

&'(= " cot"1

L

D

#$%

&'(

• Gliding flight path angle

• Corresponding airspeed

Vglide =2W

!S CD

2+ CL

2

Maximum Steady Gliding Range

• Glide range is maximum when ! is least negative, i.e.,most positive

• This occurs at (L/D)max

Maximum SteadyGliding Range

• Glide range is maximum when ! is least negative, i.e.,most positive

• This occurs at (L/D)max

tan! =!h

!r= negative constant =

h " ho( )r " ro( )

#r =#h

tan!=

"#h

" tan!= maximum when

L

D= maximum

!max

= " tan"1 D

L

#$%

&'(min

= " cot"1L

D

#$%

&'(max

Sink Rate• Lift and drag define ! and V in gliding equilibrium

D = CD

1

2!V 2

S = "W sin#

sin# = "D

W

L = CL

1

2!V 2

S =W cos"

V =2W cos"

CL!S

!h = V sin!

= "2W cos!CL#S

D

W

$%&

'()= "

2W cos!CL#S

L

W

$%&

'()

D

L

$%&

'()

= "2W cos!CL#S

cos!1

L D

$

%&'

()

• Sink rate = altitude rate, dh/dt (negative)

• Minimum sink rate provides maximum endurance

• Minimize sink rate by setting ! (dh/dt)/dCL = 0 (cos ! ~1)

• See Mathematica performance calculations in BlackboardCourse Materials

Conditions for MinimumSteady Sink Rate

!h = !2W cos"CL#S

cos"CD

CL

$

%&'

()

= !2W cos

3 "#S

CD

CL

3/2

$

%&'

()* !

2

#W

S

$%&

'()

CD

CL

3/2

$

%&'

()

CLME

=3C

Do

!and C

DME

= 4CDo

L/D and VME for

Minimum Sink Rate

VME

=2W

!S CDME

2+ C

LME

2

"2 W S( )

!

#

3CDo

" 0.76VL Dmax

LD( )

ME

=1

4

3

!CDo

=3

2

LD( )

max

" 0.86 LD( )

max

L/D for Minimum Sink Rate

• For L/D < L/Dmax, there are two solutions

• Which one produces minimum sink rate?

LD( )

ME

! 0.86 LD( )

max

VME

! 0.76VL Dmax

Gliding Flight of

the P-51 Mustang

Loaded Weight = 9,200 lb (3, 465 kg)

L / D( )max

=1

2 !CDo

= 16.31

" MR = # cot#1L

D

$%&

'()max

= # cot#1(16.31) = #3.51°

CD( )L /Dmax

= 2CDo= 0.0326

CL( )L /Dmax

=CDo

!= 0.531

VL /Dmax =76.49

*m / s

!hL /Dmax = V sin" = #4.68

*m / s

Rho =10km = 16.31( ) 10( ) = 163.1 km

Maximum Range Glide

Loaded Weight = 9,200 lb (3, 465 kg)

S = 21.83m2

CDME= 4CDo

= 4 0.0163( ) = 0.0652

CLME=

3CDo

!=

3 0.0163( )0.0576

= 0.921

L D( )ME

= 14.13

!hME = "2

#W

S

$%&

'()

CDME

CLME

3/2

$

%&

'

() = "

4.11

#m / s

* ME = "4.05°

VME =58.12

#m / s

Maximum Endurance Glide

Climbing Flight

• Rate of climb, dh/dt = Specific Excess Power

Climbing

Flight

!V = 0 =T ! D !W sin"( )

m

sin" =T ! D( )

W; " = sin

!1 T ! D( )

W

!! = 0 =L "W cos!( )

mV

L =W cos!

!h = V sin! = VT " D( )

W=

Pthrust " Pdrag( )W

Specific Excess Power (SEP) =Excess Power

Unit Weight#

Pthrust " Pdrag( )W

• Note significance of thrust-to-weight ratio and wing loading

Steady Rate of Climb

!h = V sin! = VT

W

"#$

%&'(CDo

+ )CL

2( )qW S( )

*

+,,

-

.//

L = CLq S = W cos!

CL =W

S

"

# $

%

& ' cos!

q

V = 2W

S

"

# $

%

& ' cos!CL(

!h = VT

W

!"#

$%&'CDo

q

W S( )'( W S( )cos2 )

q

*

+,

-

./

= VT

W

!"#

$%&'CDo

0V 3

2 W S( )'2( W S( )cos2 )

0V

• Necessary condition for a maximum with respectto airspeed

Condition for Maximum

Steady Rate of Climb

!h = VT

W

!"#

$%&'CDo(V 3

2 W S( )'2) W S( )cos2 *

(V

! !h!V

= 0 =T

W

"#$

%&'+V

!T / !VW

"#$

%&'

(

)*

+

,- .

3CDo/V 2

2 W S( )+20 W S( )cos2 1

/V 2

Maximum SteadyRate of Climb:

Propeller-Driven Aircraft

!Pthrust

!V=

T

W

"#$

%&'+V

!T / !VW

"#$

%&'

(

)*

+

,- = 0

• At constantpower

! !h

!V= 0 = "

3CDo#V 2

2 W S( )+2$ W S( )

#V 2

• With cos2! ~ 1

• Airspeed for maximum rate of climb at maximum power, Pmax

V4=4

3

!"#

$%&' W S( )

2

CDo(2

; V = 2W S( )(

'3C

Do

= VME

Maximum Steady

Rate of Climb:Jet-Driven Aircraft

• Condition for a maximum at constant thrust and cos2! ~ 1

• Airspeed for maximum rate of climb at maximum thrust, Tmax

! !h!V

= 0

0 = "3C

Do#

2 W S( )V4+

T

W

$%&

'()V2+2* W S( )

#

= "3C

Do#

2 W S( )V2( )2

+T

W

$%&

'()V2( ) +

2* W S( )#

0 = ax2+ bx + c and V = + x

Optimal Climbing Flight

What is the Fastest Way to Climb fromOne Flight Condition to Another? • Specific Energy

• = (Potential + Kinetic Energy) per Unit Weight

• = Energy Height

Energy Height

• Could trade altitude with airspeed with no change in energyheight if thrust and drag were zero

Total Energy

Unit Weight! Specific Energy =

mgh + mV22

mg= h +

V2

2g

! Energy Height, Eh , ft or m

Specific Excess Power

dEh

dt=d

dth +

V2

2g

!"#

$%&=dh

dt+

V

g

!"#

$%&dV

dt

= V sin' +V

g

!"#

$%&T ( D ( mgsin'

m

!"#

$%&= V

T ( D( )

W= V

CT ( CD( )1

2)(h)V 2

S

W

= Specific Excess Power (SEP) =Excess Power

Unit Weight*

Pthrust ( Pdrag( )W

Contours of ConstantSpecific Excess Power

• Specific Excess Power is a function of altitude and airspeed

• SEP is maximized at each altitude, h, whend SEP(h)[ ]

dV= 0

Subsonic Energy Climb

• Objective: Minimize time or fuel to climb to desired altitudeand airspeed

Supersonic Energy Climb

• Objective: Minimize time or fuel to climb to desired altitudeand airspeed

The Maneuvering Envelope

• Maneuvering envelopedescribes limits on normalload factor and allowableequivalent airspeed– Structural factors

– Maximum and minimumachievable lift coefficients

– Maximum and minimumairspeeds

– Protection againstoverstressing due to gusts

– Corner Velocity:Intersection of maximum liftcoefficient and maximumload factor

Typical Maneuvering Envelope:

V-n Diagram

• Typical positive load factor limits– Transport: > 2.5

– Utility: > 4.4

– Aerobatic: > 6.3

– Fighter: > 9

• Typical negative load factor limits– Transport: < –1

– Others: < –1 to –3

C-130 exceeds maneuvering envelopehttp://www.youtube.com/watch?v=4bDNCac2N1o&feature=related

Maneuvering Envelopes (V-n Diagrams)for Three Fighters of the Korean War Era

Republic F-84

North American F-86

Lockheed F-94

Turning Flight

• Vertical force equilibrium

Level Turning Flight

L cosµ =W

• Load factor

n = LW

= Lmg

= secµ,"g"s

• Thrust required to maintain level flight

Treq = CDo+ !CL

2( )1

2"V 2

S = Do +2!

"V 2S

W

cosµ

#$%

&'(

2

= Do +2!

"V 2SnW( )

2

µ :Bank Angle

• Level flight = constant altitude

• Sideslip angle = 0

• Bank angle

Maximum Bank

Angle in Level Flight

cosµ =W

CLqS=1

n=W

2!Treq " Do( )#V 2

S

µ = cos"1W

CLqS

$

%&'

()= cos"1

1

n

$%&

'()= cos"1 W

2!Treq " Do( )#V 2

S

*

+,,

-

.//

• Bank angle is limited by

µ :Bank Angle

CLmax

or Tmax

or nmax

• Turning rate

Turning Rate and Radius in Level Flight

!! =CLqS sinµ

mV=W tanµ

mV=g tanµ

V=

L2 "W 2

mV

=W n

2 "1

mV=

Treq " Do( )#V 2S 2$ "W 2

mV

• Turning rate is limited by

CLmax

or Tmax

or nmax

• Turning radius

Rturn

=V

!!=

V2

g n2 "1

Maximum Turn Rates

• Corner velocity

Corner Velocity Turn

• Turning radius

Rturn

=V2cos

2 !

g nmax

2" cos

2 !

Vcorner

=2n

maxW

CLmas

!S

• For steady climbing or diving flight

sin! =Tmax

" D

W

Corner Velocity Turn

• Time to complete a full circle

t2! =

V cos"

g nmax

2# cos

2 "

• Altitude gain/loss

!h2" = t

2"V sin#

• Turning rate

!! =g n

max

2 " cos2 #( )V cos#

“Not a turning rate comparison”http://www.youtube.com/watch?v=z5aUGum2EiM

Herbst Maneuver• Minimum-time reversal of direction

• Kinetic-/potential-energy exchange

• Yaw maneuver at low airspeed

• X-31 performing the maneuver

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