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4 : Climb and Turn Performance

or how nimble is the aircraft ?

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The rate of climb (ROC)

The rate of climb (ROC) is the rate at which an aircraft increases its altitude

Climb records

h (km) time (s) ROC (m/s)

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Climb model

γ : flight path (climb) angle

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Resolving forces parallel and perpendicular to flight path (velocity V)

T – D – W sinγ = m dV/dt

L – W cosγ = m V dγ/dt

The climb rate is then

dh/dt = V sinγ

=

or(T – D) V =

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The climb equation

(T – D) V = dh/dt + d [V2 / (2g) ] / dtW

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Specific energy

The quantity on the rhs of the climb equation can be written as :

dh/dt + d [V2 / (2g) ] / dt

=

=

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Energy perspective

Hence the climb equation is actually an ‘energy” equation

Ps = dhe/dt

The time to transit from one energy state to another is given by :

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Example : F104 Ps curves

How to fly from energy state A to state B as fast as possible ?

A

B

Source : NASA TN D6398

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Comparison F4H-1 vs F15A

F4H-1 2 x J79-GE-2A turbojets

thrust = 2 x 10,350 lb ( 2 x 4,693 kg)

MTOW = 54,600 lb (24,761 kg)

T/W =

F15A 2 x F100-PW-100 turbofan

thrust = 2 x 25,000 lb (2 x 11,250 kg)

MTOW = 56,000 lb (25,200 kg)

T/W =

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Energy-Maneuverability Theory

• Developed by USAF Col John Boyd at Georgia Tech in 1962

• Thesis :

• Proved that the F-4 could not out turn a MiG-21 except at low altitude and high speed

• E-M theory was applied to the design of the F15 and F16

USAF Col John Boyd 1927-1997

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Sustained level turn

Conventional aircraft change heading by banking

i.e. tilt the lift force so that the horizontal component provides the centripedal force for circular motion.

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Sustained level turn model

φ

L cosφ =

L sinφ =

Resolving forces in the vertical and horizontal directionsL

φ: : bank angle

dχ/dt : turn rateW = mg

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Define the load factor n = L/W

= 1 / cosφ ( n > 1 )

Dividing the 2 force equations

V/g dχ/dt = tan φ

=

=

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The turn equation is :

dχ/dt = (g/V) √ ( n2 - 1 )

The turn radius is :

R = V / dχ/dt

= V2/ ( g √ ( n2 - 1 ) )

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Example : Turn envelope

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Exercise : Performance

estimation

Assuming A has a corner velocity of 450 kts at a load factor of 8, what’s the turn radius and turn rate ?

Assuming B is flying at the same speed, what’s the load factor and turn rate for B ?

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Turn at maximum load factor

Question : What’s the bank angle ?

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load factor n = ⇒ bank angle =

airspeed V = 450 kts = 450*0.5151 = 232 m/s

turn rate =

=

=

=

turn radius = V / turn rate

=

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RA = V2 / (g √ (nA2 – 1 ) )

RB = V2 / (g √ (nB2 – 1 ) )

load factor nB =

=

turn rate =

=

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Comparison of turn rates

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Bottom Line

High turn rates require high load factors

Notes :

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V-n Diagram