ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 1

AEROSPACE ENGINEERING 1
(ENG1002)
Aircraft Performance
Lecture 2: Performance
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 2

Aircraft Performance
 Lift-to-drag ratio (CL/CD);
 Gliding performance
 Minimum glide angle, maximum
range, glide speed and rate of
descent

 Best range airspeed for propeller and jet aircraft;

 Best endurance airspeed for propeller and jet aircraft;
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 3

Lift-to-drag ratio
Remember that lift and drag coefficient
are :
𝐿 𝐷
𝐶𝐿 = 1 𝐶𝐷 = 1
𝜌𝑈 2 𝑆 𝜌𝑈 2 𝑆
2 2

The ratio between CL and CD is called

the lift-to-drag ratio (L/D).

𝐿𝑖𝑓𝑡 𝐶𝐿
i.e. 𝐿Τ𝐷 = =
𝐷𝑟𝑎𝑔 𝐶𝐷

CL/CD has only one maximum value at

one angle of attack.
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 4

Lift-to-drag ratio
L/D ratio of range of “vehicles”
 High-performance
sailplane : 40
 Modern jet airliner : 18 - 25
(Boeing 787 ~21)
 Boeing 747 : 17.7
 Turboprop : 13
 Helicopter : 4
 Space shuttle : 2.5
 Modern F1 cars : 1 - 4
 Lorry < 1
 Reliant Robin Space Shuttle < 0.5.
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 5

Gliding flight
If all engines in an
aeroplane are turned off.
Then no thrust is
generated by the
engines.

The aircraft then starts

gliding.
Note:
a = angle of attack.
In gliding flight: q = pitch angle
 Thrust = 0  = glide angle (i.e. 𝛾 = 𝛼 + θ)
 Flight path angle < 0o in gliding flight
 Altitude is decreasing
 Airspeed ~ constant – Maximise the range,
Objective
which means need to minimise the
glide angle ()
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 6

Gliding flight
Objective – Maximise the range, which means need to
minimise the glide angle () …… HOW ?
L
Consider force equilibrium: 
D
Parallel to U 
−𝐷 + 𝑊 sin 𝛾 = 0
Perp to U
W = mg
𝐿 − 𝑊 cos 𝛾 = 0

⇒ 𝐷 = 𝑊 sin 𝛾 ∶ 1 Therefore,
⇒ 𝐿 = 𝑊 cos 𝛾 ∶ (2) to minimise the glide
Dividing (1) with (2), then: angle (),
𝐷 1 need to maximise the
⇒ tan 𝛾 = = ∶ 3 lift-to-drag ratio (L/D).
𝐿 𝐿ൗ
𝐷
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 7

Gliding flight
What is the maximum GLIDE RANGE, R, when aircraft in
gliding flight at the glide angle ()?

From diagram,
ℎ −ℎ
tan 𝛾 = 1 2 ∶ 4
𝑅

Remember
1
⇒ tan 𝛾 = 𝐿 ∶ (5)
ൗ𝐷
Combining (4) and (5) gives
𝐿
𝑅 = (ℎ1 − ℎ2 ) ∶ (6)
𝐷
From (6), the Range, R, is a maximum when the lift-drag ratio
𝐿
is maximum. i.e. 𝑅𝑚𝑎𝑥 = (ℎ1 − ℎ2 )
𝐷 𝑚𝑎𝑥
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 8

Gliding flight
Objective – What is the corresponding gliding speed at the
minimum glide angle ()?
Remember that 𝐷 = 𝑊 sin 𝛾 ∶ 1 and 𝐿 = 𝑊 cos 𝛾 ∶ (2)

ASSUME  IS SMALL, hence sin 𝛾 ≈ tan 𝛾 ≈ 𝛾 and cos 𝛾 ≈ 1

( in radians)

Hence, (1) and (2) become:

𝐷 = 𝑊 𝛾 ≈ 0 and 𝐿 = 𝑊 = 𝐶𝐿 . 12𝜌𝑈 2 𝑆 ∶ (7)

Re-arranging gives

𝑈𝑠𝑚𝑎𝑙𝑙 𝛾 =
2𝑊
∶ (8) If  is not small, can show
𝜌𝑆𝐶𝐿
2𝑊
𝑈𝛾 =
Note, (8) is only valid when 𝜌𝑆 𝐶𝐿2 + 𝐶𝐷2
glide angle is small.
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 9

Gliding flight
What is the corresponding rate of descent, 𝒉,ሶ when
aircraft in gliding flight at the minimum glide angle ()?

From diagram,
ℎሶ 𝒉ሶ
sin 𝛾 = ∶ 9 Ud
𝑈𝑑

where  is the glide angle, ℎሶ is the rate of descent and Ud is the

descent speed. Re-arranging (9)

⇒ ℎሶ = 𝑈𝑑 sin 𝛾 ∶ (10)
From previous slide, for small , the descent speed is

Subst (8) into (10), then

2𝑊
𝑈𝑠𝑚𝑎𝑙𝑙 𝛾 = ∶ (8) 2𝑊
𝜌𝑆𝐶𝐿 ℎሶ = sin 𝛾
𝜌𝑆𝐶𝐿
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 10

Example 2
An airliner can glide 120 km from an altitude of 9000 m.
Calculate:

(a) the value of CL/CD and the glide angle, .

(b) the rate of descent if a steady airspeed of 200 ms-1 is

maintained.
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 11

Total Aircraft Drag

How does aircraft drag vary with velocity?
Aircraft drag (CD)is composed of
two main drag components:

(i) Form drag or profile drag

(CD0) – a “pressure drag”
due to the shape of the
aircraft.

(ii) Induced drag or drag-due-

to-lift (𝑘𝐶𝐿2 ) – induced by the
lift generated by the aircraft.
Total drag = Profile drag + Induced drag
2 k is the lift dependent drag
𝐶𝐷 = 𝐶𝐷0 + 𝑘𝐶𝐿 ∶ (11) factor.

Important equation, termed the “DRAG POLAR”

 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 12

Total Aircraft Drag

Important equation, “DRAG POLAR”

Total drag = Profile drag + Induced drag

𝐶𝐷 = 𝐶𝐷0 + 𝑘𝐶𝐿2 +…
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 13

Minimum Drag Speed

What is the minimum drag speed of an aeroplane, UMD?

Using the drag polar equation

𝐶𝐷 = 𝐶𝐷0 + 𝑘𝐶𝐿2 ∶ (11)

Also, recalling that:

𝐷 𝐿 𝑊
𝐶𝐷 = 1 2 ; 𝐶𝐿 = 1 =1 𝑖𝑛 𝑐𝑟𝑢𝑖𝑠𝑒 ∶ (12)
𝜌𝑈 𝑆 𝜌𝑈 2 𝑆 𝜌𝑈 2 𝑆
2 2 2
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 14

Minimum Drag Speed

Minimum drag speed, UMD is

𝑘 0.25 2𝑊
𝑈𝑀𝐷 =
𝐶𝐷0 𝜌𝑆

Important :
𝑘
(i) The ratio is an important parameter in
𝐶𝐷0
determining the min. drag speed (UMD).

(i) The min. drag speed (UMD) decreases with

the wing area S.

Derivation of UMD presented in video on Moodle – also in supplementary notes.

 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 15

Speed for maximum L/D

What is the speed that gives maximum lift-to-
drag ratio, L/D?
Again, use the drag polar equation
𝐶𝐷 = 𝐶𝐷0 + 𝑘𝐶𝐿2

⇒ 𝐶𝐷0 = 𝑘𝐶𝐿2

Important:
Maximum L/D speed (or Minimum
D/L speed) occurs at the point
when the profile drag (𝐶𝐷0 ) is equal
to the induced drag (𝑘𝐶𝐿2 ).
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 16

Speed for the Minimum Drag to Speed Ratio

What is the speed, UD/U, that gives minimum Drag-to-Speed
Ratio (D/U)?
Use Drag Polar : 𝐶𝐷 = 𝐶𝐷0 + 𝑘𝐶𝐿2

Can show

𝑈𝐷/𝑈 = 1.316𝑈𝑀𝐷

Note : Minimum drag to speed ratio speed (UD/U) is 1.316 times of the
minimum drag speed (UMD).
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 17

Minimum Power Airspeed

The power required to propel an aeroplane is given by the thrust equation
(assuming steady constant speed flight) ;
𝑃 = 𝑇𝑈 = 𝐷𝑈
Where P = propulsive (Thrust) power, T = Engine Thrust, D = drag and U =
the airspeed

Again, using drag (Force) polar, can show

𝑈𝑀𝑃 = 0.76𝑈𝑀𝐷

Note : Minimum power speed (UMP) is 0.76 times of the minimum drag speed (UMD).
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 18

Speed for Minimum Power to Speed Ratio

Again using thrust equation for constant speed flight : 𝑃 = 𝑇𝑈 = 𝐷𝑈
Therefore, the corresponding power-to-speed ratio (P/U) is:
𝑃
=𝑇=𝐷
𝑈
Using Drag polar, and differentiating w.r.t U can show

4 4
4𝑘𝑊 2
𝑈𝑃/𝑈 = 𝑈𝑀𝐷 =
𝐶𝐷0 𝜌𝑆 2

Note : The speed for minimum power-to-speed ratio (UP/U) is

same as the minimum drag speed (UMD).
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 19

Summary : Performance-speed
relations
4 4𝑘𝑊 2
 Minimum Drag speed : 𝑈𝑀𝐷 =
𝐶𝐷0 𝜌𝑆 2

 Maximum L/D when 𝐶𝐷0 = 𝑘𝐶𝐿2

 UD/U (speed for minimum drag-to-velocity ratio) = 1.316 UMD

 UMP (minimum power speed) = 0.76 UMD

 UP/U (speed for minimum power-to-speed ratio) = UMD

Derivations of all performance speeds available on Moodle in supplementary notes.

 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 20

Performance : Jet aeroplanes

 For a jet aeroplane,
fuel flow 𝑚ሶ 𝑓 is
proportional to thrust.

 In steady flight,
thrust = drag (i.e. T = D),
so fuel flow is proportional to drag (i.e. 𝑚ሶ 𝑓 ∝ 𝐷).
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 21

Performance : Jet aeroplanes

 For a jet aeroplane,
fuel flow 𝑚ሶ 𝑓 is
proportional to thrust.

 In steady flight,
thrust = drag (i.e. T = D),
so fuel flow is proportional to drag (i.e. 𝑚ሶ 𝑓 ∝ 𝐷).
 Best range occurs at speed where the drag-to-speed ratio
(UD/U) is minimum (i.e. lowest fuel flow (or drag) at the
highest speed).
𝑈𝑏𝑒𝑠𝑡 𝑟𝑎𝑛𝑔𝑒,𝑗𝑒𝑡 = 𝑈𝐷/𝑈 = 1.316 𝑈𝑀𝐷
 Best endurance occurs at speed where drag is minimum
(lowest fuel flow). 𝑈𝑏𝑒𝑠𝑡 𝑒𝑛𝑑𝑢𝑟𝑎𝑛𝑐𝑒,𝑗𝑒𝑡 = 𝑈𝑀𝐷
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 22

Performance : Propeller aeroplanes

 For a propeller,
fuel flow 𝑚ሶ 𝑓 is
proportional to engine power

 Best range occurs at speed where

the power-to-speed ratio (UP/U) is
minimum (i.e. lowest fuel flow at the highest speed).
𝑈𝑏𝑒𝑠𝑡 𝑟𝑎𝑛𝑔𝑒,𝑝𝑟𝑜𝑝 = 𝑈𝑃/𝑈 = 𝑈𝑀𝐷

 Best endurance occurs at speed where power is minimum

(lowest fuel flow).
𝑈𝑏𝑒𝑠𝑡 𝑒𝑛𝑑𝑢𝑟𝑎𝑛𝑐𝑒,𝑝𝑟𝑜𝑝 = 𝑈𝑀𝑃 = 0.76 𝑈𝑀𝐷
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 23

Example 3
(a) A jet transport aeroplane has a best-range speed of 250
ms-1. Calculate the endurance speed.

(b) A propeller-driven aeroplane has an endurance airspeed

of 80 ms-1. Calculate the best-range speed.
 ENG1002 : Aerospace Engineering 1 – Aerodynamic Performance : Performance 24

Summary : Performance
 Lift-Drag Ratio, L/D
 Gliding Flight
 Glide angle, ; Gliding range; Gliding Speed
 Total Aircraft Drag
 Drag Polar; Induced Drag + Profile Drag
 Minimum Drag Speed, UMD
 Speed for Minimum Drag to Speed Ratio
 Minimum Power Air Speed
 Speed for Minimum Power to Speed Ratio
 Performance : Jets and Propeller Aircraft