Gliding, Climbing, and Turning Flight Performance

Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2012!

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!

•  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 by available 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 altitudes and airspeeds at which an aircraft can fly "–  in steady, level flight "–  at fixed weight"

Additional Factors Define the Flight Envelope"

•  Maximum Mach number"•  Maximum allowable

aerodynamic heating"•  Maximum thrust"•  Maximum dynamic

pressure"•  Performance ceiling"•  Wing stall"•  Flow-separation buffet"

–  Angle of attack"–  Local shock waves"

Piper Dakota Stall Buffet"http://www.youtube.com/watch?v=mCCjGAtbZ4g!

Boeing 787 Flight Envelope (HW #5, 2008)"

Best Cruise Region"

Gliding Flight�

D = CD12ρV 2S = −W sinγ

CL12ρV 2S =W cosγ

h = V sinγr = V cosγ

Equilibrium Gliding Flight" Gliding Flight"•  Thrust = 0"•  Flight path angle < 0 in gliding flight"•  Altitude is decreasing"•  Airspeed ~ constant"•  Air density ~ constant "

tanγ = −D

L= −

CD

CL

=hr=dhdr; γ = − tan−1 D

L#

$%

&

'(= −cot−1

LD#

$%

&

'(

•  Gliding flight path angle "

•  Corresponding airspeed "

Vglide =2W

ρS CD2 + CL

2

Maximum Steady Gliding Range"

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

•  This occurs at (L/D)max "

Maximum Steady Gliding Range"

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

•  This occurs at (L/D)max "

tanγ =hr= negative constant =

h−ho( )r − ro( )

Δr = Δhtanγ

=−Δh− tanγ

= maximum when LD= maximum

γmax = − tan−1 D

L#

$%

&

'(min

= −cot−1 LD#

$%

&

'(max

Sink Rate "•  Lift and drag define γ and V in gliding equilibrium"

D = CD12ρV 2S = −W sinγ

sinγ = − DW

L = CL12ρV 2S =W cosγ

V =2W cosγCLρS

h =V sinγ

= −2W cosγCLρS

DW$

%&

'

()= −

2W cosγCLρS

LW$

%&

'

()DL

$

%&

'

()

= −2W cosγCLρS

cosγ 1L D$

%&

'

()

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

•  Minimum sink rate provides maximum endurance"•  Minimize sink rate by setting ∂(dh/dt)/dCL = 0 (cos γ ~1)"

Conditions for Minimum Steady Sink Rate"

h = − 2W cosγCLρS

cosγ CD

CL

$

%&

'

()

= −2W cos3γ

ρSCD

CL3/2

$

%&

'

() ≈ −

2ρWS

$

%&

'

()CD

CL3/2

$

%&

'

()

CLME=

3CDo

εand CDME

= 4CDo

L/D and VME for Minimum Sink Rate"

VME =2W

ρS CDME

2 + CLME2≈

2 W S( )ρ

ε3CDo

≈ 0.76VL Dmax

LD( )

ME=14

3εCDo

=32

LD( )

max≈ 0.86 L

D( )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 L

D( )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 εCDo

=16.31

γMR = −cot−1 L

D$

%&

'

()max

= −cot−1(16.31)= −3.51°

CD( )L/Dmax = 2CDo= 0.0326

CL( )L/Dmax =CDo

ε= 0.531

VL/Dmax =76.49ρ

m / s

hL/Dmax =V sinγ = −4.68ρm / 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

ε=

3 0.0163( )0.0576

= 0.921

L D( )ME =14.13

hME = −2ρWS

$

%&

'

()CDME

CLME3/2

$

%&&

'

())= −

4.11ρm / s

γME = −4.05°

VME =58.12ρ

m / s

Maximum Endurance Glide"

Climbing Flight�

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

Climbing Flight"

V = 0 =T −D−W sinγ( )

m

sinγ =T −D( )W

; γ = sin−1T −D( )W

γ = 0 =L −W cosγ( )

mVL =W cosγ

h = V sinγ = VT − D( )W

=Pthrust − Pdrag( )

W

Specific Excess Power (SEP) = Excess PowerUnit Weight

≡Pthrust − Pdrag( )

W

•  Flight path angle " •  Required lift"

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

Steady Rate of Climb"

h =V sinγ =V TW"

#$

%

&'−

CDo+εCL

2( )qW S( )

*

+,,

-

.//

€

L = CLq S = W cosγ

CL =WS

#

$ %

&

' ( cosγ

q

V = 2 WS

#

$ %

&

' ( cosγCLρ

h =V TW!

"#

$

%&−

CDoq

W S( )−ε W S( )cos2 γ

q

*

+,

-

./

=VT h( )W

!

"#

$

%&−

CDoρ h( )V 3

2 W S( )−2ε W S( )cos2 γ

ρ h( )V

•  Climb rate "

•  Necessary condition for a maximum with respect to airspeed"

Condition for Maximum Steady Rate of Climb"

h =V TW!

"#

$

%&−

CDoρV 3

2 W S( )−2ε W S( )cos2 γ

ρV

∂ h∂V

= 0 = TW"

#$

%

&'+V

∂T /∂VW

"

#$

%

&'

(

)*

+

,-−3CDo

ρV 2

2 W S( )+2ε W S( )cos2 γ

ρV 2

Maximum Steady "Rate of Climb: "

Propeller-Driven Aircraft"

∂Pthrust∂V

= 0 = TW"

#$

%

&'+V

∂T /∂VW

"

#$

%

&'

(

)*

+

,-•  At constant power"

∂ h∂V

= 0 = −3CDo

ρV 2

2 W S( )+2ε W S( )ρV 2

•  With cos2γ ~ 1, optimality condition reduces to"

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

V 4 =43!

"#$

%&ε W S( )2

CDoρ2

; V = 2W S( )ρ

ε3CDo

=VME

Maximum Steady Rate of Climb: "

Jet-Driven Aircraft"•  Condition for a maximum at constant thrust and cos2γ ~ 1"

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

∂ h∂V

= 0

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

= −3CDo

ρ

2 W S( )V 4 +

TW#

$%

&

'(V 2 +

2ε W S( )ρ

= −3CDo

ρ

2 W S( )V 2( )2 + T

W#

$%

&

'( V 2( )+ 2ε W S( )

ρ

Optimal Climbing Flight�

What is the Fastest Way to Climb from One 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 energy height if thrust and drag were zero"

Total EnergyUnit Weight

≡ Specific Energy =mgh + mV 2 2

mg= h +

V 2

2g≡ Energy Height, Eh , ft or m

Specific Excess Power"

dEh

dt=ddt

h+V2

2g!

"#

$

%&=

dhdt+Vg!

"#

$

%&dVdt

•  Rate of change of Specific Energy "

=V sinγ + Vg"

#$

%

&'T −D−mgsinγ

m"

#$

%

&'=V

T −D( )W

=VCT −CD( ) 1

2ρ(h)V 2S

W

= Specific Excess Power (SEP)= Excess PowerUnit Weight

≡Pthrust −Pdrag( )

W

Contours of Constant Specific Excess Power"

•  Specific Excess Power is a function of altitude and airspeed"•  SEP is maximized at each altitude, h, when" d SEP(h)[ ]

dV= 0

Subsonic Energy Climb"•  Objective: Minimize time or fuel to climb to desired altitude

and airspeed"

Supersonic Energy Climb"•  Objective: Minimize time or fuel to climb to desired altitude

and airspeed"

The Maneuvering Envelope�

•  Maneuvering envelope: limits on normal load factor and allowable equivalent airspeed"–  Structural factors"–  Maximum and minimum

achievable lift coefficients"–  Maximum and minimum

airspeeds"–  Protection against

overstressing due to gusts"–  Corner Velocity: Intersection

of maximum lift coefficient and maximum load 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 envelope"http://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 = L mg = secµ,"g"s

•  Thrust required to maintain level flight"

Treq = CDo+ εCL

2( ) 12 ρV2S = Do +

2ερV 2S

Wcosµ

#

$%&

'(

2

= Do +2ερV 2S

nW( )2

µ :Bank Angle

•  Level flight = constant altitude"•  Sideslip angle = 0"

•  Bank angle"

Maximum Bank Angle in Level Flight"

cosµ = WCLqS

=1n=W 2ε

Treq −Do( )ρV 2S

µ = cos−1 WCLqS

$

%&

'

()= cos−1

1n$

%&'

()= cos−1 W

2εTreq −Do( )ρV 2S

*

+,,

-

.//

•  Bank angle is limited by "

€

µ :Bank Angle

CLmaxor Tmax or nmax

•  Turning rate"

Turning Rate and Radius in Level Flight"

ξ =CLqS sinµ

mV=W tanµmV

=g tanµV

=L2 −W 2

mV

=W n2 −1

mV=

Treq − Do( )ρV 2S 2ε −W 2

mV

•  Turning rate is limited by "

CLmaxor Tmax or nmax

•  Turning radius "

Rturn =

Vξ=

V 2

g n2 −1

Maximum Turn Rates"

“Wind-up turns”"

•  Corner velocity"

Corner Velocity Turn"

•  Turning radius "

Rturn =V 2 cos2 γ

g nmax2 − cos2 γ

Vcorner =2nmaxWCLmas

ρS

•  For steady climbing or diving flight"sinγ = Tmax − D

W

Corner Velocity Turn"

•  Time to complete a full circle "

t2π =V cosγ

g nmax2 − cos2 γ

•  Altitude gain/loss "Δh2π = t2πV sinγ

•  Turning rate "

ξ =g nmax

2 − cos2 γ( )V cosγ

�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"

Next Time:�Aircraft Equations of Motion�

�

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