Aircraft Performance lecture 1: Introduction
incompressible subsonic flow
Pressure p and Volume V (or speed v) are variable. Up till
now we assumed a frictionless, stationary,
incompressible and adiabatic flow. Continuity equation
and Bernoulli’s equation.
𝑣" ∙ 𝐴" = 𝑣& ∙ 𝐴&
1 1
𝑝" + 𝜌 ∙ 𝑔 ∙ ℎ" + ∙ 𝜌 ∙ 𝑣"& = 𝑝& + 𝜌 ∙ 𝑔 ∙ ℎ& + ∙ 𝜌 ∙ 𝑣&&
2 2
Figure 1 Transonic flow
Compressible transonic flow
Pressure p, Volume V, density r and temperature T are variable. Now we assumed a
frictionless, stationary and adiabatic flow. Continuity equation is still valid, but we have to
add the density r. It now changes from constant volume flow [m3/s] in constant mass flow
[kg/s]. (Bernoulli’s equation is not valid anymore!)
𝜌" ∙ 𝑣" ∙ 𝐴" = 𝜌& ∙ 𝑣& ∙ 𝐴&
An important dimensionless figure is the Mach number
By definition the Mach number M is equal to the True Airspeed TAS divided through the
local speed of sound:
𝑇𝐴𝑆
𝑀=
𝑎
Mach number Type of flow
< 0,3 Incompressible subsonic flow.
0,3 – 0,8 Compressible subsonic flow, the speed of all the airflow around the
aircraft is still below Mach 1.
0,8 – 1,2 Transonic flow, the speed of the airflow around the aircraft is both below
and above Mach 1.
> 1,2 Supersonic flow, the speed of all the airflow around the aircraft is above
Mach 1.
>5 Hypersonic flow, this is of course also a supersonic flow, but the friction
of air causes huge temperature problems.
Table 1: Mach number division
Local speed of sound a
Local speed of sound a is equal to 𝑎 = 2𝛾 ∙ 𝑅5 ∙ 𝑇 [m/s]
6
𝛾 = 7 1,4 for air.
68
𝑗
𝑅 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 287 [ ]
𝑘𝑔. 𝐾
T is temperature in Kelvin!
,Local speed of sound a at sea level
For air at sea level in ISA Conditions:
𝑚 3600
𝑎 = 21,4 ∙ 287 ∙ 288 = 340,2 N P = 340,2 ∙ = 661,3 [𝑘𝑡𝑠]
𝑠 1852
Speed of sound is only depending upon temperature in air and nothing else!
Also, the temperature changes in a transonic flow. First law of thermodynamics, the energy
equation:
𝑇𝐴𝑆 &
𝑆𝐴𝑇 + = 𝑇𝐴𝑇
2 ∙ 𝑐S
This temperature rise has nothing to do with friction. This formula is usually written with
Mach numbers.
𝑇𝐴𝑆
𝑀= → 𝑇𝐴𝑆 = 𝑀 ∙ 𝑎
𝑎
𝑎 = 2𝛾 ∙ 𝑅 ∙ 𝑇
𝑇𝐴𝑆 & 𝑀& ∙ 𝑎& 𝑀& ∙ 𝛾 ∙ 𝑅 ∙ 𝑆𝐴𝑇
𝑇𝐴𝑇 = 𝑆𝐴𝑇 + = 𝑆𝐴𝑇 + = 𝑆𝐴𝑇 + →
2 ∙ 𝑐S 2 ∙ 𝑐S 2 ∙ 𝑐S
𝑐S 1
𝑀& ∙ 𝑎& 𝑀& ∙ 𝑐 ∙ 𝑅 𝑀& ∙ 𝑐 ∙ 𝑅
X X
𝑇𝐴𝑇 = 𝑆𝐴𝑇 U1 + V = 𝑆𝐴𝑇 W1 + Y = 𝑆𝐴𝑇 W1 + Y→
2 ∙ 𝑐S 2 ∙ 𝑐S 2
1 1 𝑐S
𝑀& ∙ 𝑐 ∙ Z𝑐S − 𝑐X \ 𝑀& ∙ 𝑐 ∙ ]𝑐 − 1^
X X X
𝑇𝐴𝑆 = 𝑆𝐴𝑇 W1 + Y = 𝑆𝐴𝑇 W1 + Y→
2 2
𝑀& ∙ (𝛾 − 1)
𝑇𝐴𝑇 = 𝑆𝐴𝑇 U1 + V
2
For air 𝛾 = 1,4 so:
𝑀& ∙ (𝛾 − 1)
𝑇𝐴𝑇 = 𝑆𝐴𝑇 U1 + V = 𝑆𝐴𝑇(1 + 0,2 ∙ 𝑀& )
2
𝑆𝐴𝑇 + 0,2 ∙ 𝑀& ∙ 𝑆𝐴𝑇
- TAT: Total Air Temperature. Temperature in Kelvin after the air molecules are
suddenly brought to a stop. (analog to total pressure).
- SAT: Static Air Temperature. Temperature in Kelvin in still air. (Analog to static
pressure).
- Ram Rise: 0,2 ∙ 𝑀& ∙ 𝑆𝐴𝑇. Temperature rise in Kelvin due to the sudden stop of the
air molecules. (Analog to dynamic pressure).
, The critical Mach number is the flight
Mach number where somewhere on the
airplane construction, the speed of sound
is reached.
Figure 2: Normal shock wave
Figure 3: Static pressure distribution
Figure 4: consequences of exceeding the critical Mach number
Figure 5: consequences of exceeding the critical Mach number
Figure 6: consequences of exceeding the critical Mach number
, Low speed buffet Item Shock buffet
Large Angle of attack Small
Low Mach number High
Large amplitude Characteristic Small amplitude
Low frequency High frequency
Lower your angle of attack Action taken by pilot Decrease your Mach
and increase your speed number
Table 2: Difference between low speed and shock stall
+/- 25.000 ft
+/- 12.000 ft
Figure 7: The Buffet Onset Boundary Chart
"
𝐿 = 𝑛 ∙ 𝑊 = 𝐶d ∙ & ∙ 𝜌 ∙ 𝐸𝐴𝑆 & ∙ 𝑆
2∙𝑛∙𝑊
𝐸𝐴𝑆 & =
𝐶d ∙ 𝜌 ∙ 𝑆
2∙𝑛∙𝑊
𝐸𝐴𝑆 = f
𝐶d ∙ 𝜌 ∙ 𝑆
2∙𝑛∙𝑊
𝐸𝐴𝑆5ghii = f
Figure 8: Stall speed increases with altitude due to 𝐶djhk ∙ 𝜌 ∙ 𝑆
compressibility effects
De snelheid en dus ook de stallspeed is evenredig met de wortel uit de loadfactor. Een stall waarbij de
loadfactor groter is dan een wordt een accelerated of g-stall genoemd.
The area rules
Consequence no area ruling: Shock
stall starts at the wing root. Lift vector
shifts aft due to shock stall
𝑇 ∙ 𝑎 = 𝑁 ∙ 𝑏 (equilibrium around
the lateral axis).
Figure 8: displacement versus speed
incompressible subsonic flow
Pressure p and Volume V (or speed v) are variable. Up till
now we assumed a frictionless, stationary,
incompressible and adiabatic flow. Continuity equation
and Bernoulli’s equation.
𝑣" ∙ 𝐴" = 𝑣& ∙ 𝐴&
1 1
𝑝" + 𝜌 ∙ 𝑔 ∙ ℎ" + ∙ 𝜌 ∙ 𝑣"& = 𝑝& + 𝜌 ∙ 𝑔 ∙ ℎ& + ∙ 𝜌 ∙ 𝑣&&
2 2
Figure 1 Transonic flow
Compressible transonic flow
Pressure p, Volume V, density r and temperature T are variable. Now we assumed a
frictionless, stationary and adiabatic flow. Continuity equation is still valid, but we have to
add the density r. It now changes from constant volume flow [m3/s] in constant mass flow
[kg/s]. (Bernoulli’s equation is not valid anymore!)
𝜌" ∙ 𝑣" ∙ 𝐴" = 𝜌& ∙ 𝑣& ∙ 𝐴&
An important dimensionless figure is the Mach number
By definition the Mach number M is equal to the True Airspeed TAS divided through the
local speed of sound:
𝑇𝐴𝑆
𝑀=
𝑎
Mach number Type of flow
< 0,3 Incompressible subsonic flow.
0,3 – 0,8 Compressible subsonic flow, the speed of all the airflow around the
aircraft is still below Mach 1.
0,8 – 1,2 Transonic flow, the speed of the airflow around the aircraft is both below
and above Mach 1.
> 1,2 Supersonic flow, the speed of all the airflow around the aircraft is above
Mach 1.
>5 Hypersonic flow, this is of course also a supersonic flow, but the friction
of air causes huge temperature problems.
Table 1: Mach number division
Local speed of sound a
Local speed of sound a is equal to 𝑎 = 2𝛾 ∙ 𝑅5 ∙ 𝑇 [m/s]
6
𝛾 = 7 1,4 for air.
68
𝑗
𝑅 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 287 [ ]
𝑘𝑔. 𝐾
T is temperature in Kelvin!
,Local speed of sound a at sea level
For air at sea level in ISA Conditions:
𝑚 3600
𝑎 = 21,4 ∙ 287 ∙ 288 = 340,2 N P = 340,2 ∙ = 661,3 [𝑘𝑡𝑠]
𝑠 1852
Speed of sound is only depending upon temperature in air and nothing else!
Also, the temperature changes in a transonic flow. First law of thermodynamics, the energy
equation:
𝑇𝐴𝑆 &
𝑆𝐴𝑇 + = 𝑇𝐴𝑇
2 ∙ 𝑐S
This temperature rise has nothing to do with friction. This formula is usually written with
Mach numbers.
𝑇𝐴𝑆
𝑀= → 𝑇𝐴𝑆 = 𝑀 ∙ 𝑎
𝑎
𝑎 = 2𝛾 ∙ 𝑅 ∙ 𝑇
𝑇𝐴𝑆 & 𝑀& ∙ 𝑎& 𝑀& ∙ 𝛾 ∙ 𝑅 ∙ 𝑆𝐴𝑇
𝑇𝐴𝑇 = 𝑆𝐴𝑇 + = 𝑆𝐴𝑇 + = 𝑆𝐴𝑇 + →
2 ∙ 𝑐S 2 ∙ 𝑐S 2 ∙ 𝑐S
𝑐S 1
𝑀& ∙ 𝑎& 𝑀& ∙ 𝑐 ∙ 𝑅 𝑀& ∙ 𝑐 ∙ 𝑅
X X
𝑇𝐴𝑇 = 𝑆𝐴𝑇 U1 + V = 𝑆𝐴𝑇 W1 + Y = 𝑆𝐴𝑇 W1 + Y→
2 ∙ 𝑐S 2 ∙ 𝑐S 2
1 1 𝑐S
𝑀& ∙ 𝑐 ∙ Z𝑐S − 𝑐X \ 𝑀& ∙ 𝑐 ∙ ]𝑐 − 1^
X X X
𝑇𝐴𝑆 = 𝑆𝐴𝑇 W1 + Y = 𝑆𝐴𝑇 W1 + Y→
2 2
𝑀& ∙ (𝛾 − 1)
𝑇𝐴𝑇 = 𝑆𝐴𝑇 U1 + V
2
For air 𝛾 = 1,4 so:
𝑀& ∙ (𝛾 − 1)
𝑇𝐴𝑇 = 𝑆𝐴𝑇 U1 + V = 𝑆𝐴𝑇(1 + 0,2 ∙ 𝑀& )
2
𝑆𝐴𝑇 + 0,2 ∙ 𝑀& ∙ 𝑆𝐴𝑇
- TAT: Total Air Temperature. Temperature in Kelvin after the air molecules are
suddenly brought to a stop. (analog to total pressure).
- SAT: Static Air Temperature. Temperature in Kelvin in still air. (Analog to static
pressure).
- Ram Rise: 0,2 ∙ 𝑀& ∙ 𝑆𝐴𝑇. Temperature rise in Kelvin due to the sudden stop of the
air molecules. (Analog to dynamic pressure).
, The critical Mach number is the flight
Mach number where somewhere on the
airplane construction, the speed of sound
is reached.
Figure 2: Normal shock wave
Figure 3: Static pressure distribution
Figure 4: consequences of exceeding the critical Mach number
Figure 5: consequences of exceeding the critical Mach number
Figure 6: consequences of exceeding the critical Mach number
, Low speed buffet Item Shock buffet
Large Angle of attack Small
Low Mach number High
Large amplitude Characteristic Small amplitude
Low frequency High frequency
Lower your angle of attack Action taken by pilot Decrease your Mach
and increase your speed number
Table 2: Difference between low speed and shock stall
+/- 25.000 ft
+/- 12.000 ft
Figure 7: The Buffet Onset Boundary Chart
"
𝐿 = 𝑛 ∙ 𝑊 = 𝐶d ∙ & ∙ 𝜌 ∙ 𝐸𝐴𝑆 & ∙ 𝑆
2∙𝑛∙𝑊
𝐸𝐴𝑆 & =
𝐶d ∙ 𝜌 ∙ 𝑆
2∙𝑛∙𝑊
𝐸𝐴𝑆 = f
𝐶d ∙ 𝜌 ∙ 𝑆
2∙𝑛∙𝑊
𝐸𝐴𝑆5ghii = f
Figure 8: Stall speed increases with altitude due to 𝐶djhk ∙ 𝜌 ∙ 𝑆
compressibility effects
De snelheid en dus ook de stallspeed is evenredig met de wortel uit de loadfactor. Een stall waarbij de
loadfactor groter is dan een wordt een accelerated of g-stall genoemd.
The area rules
Consequence no area ruling: Shock
stall starts at the wing root. Lift vector
shifts aft due to shock stall
𝑇 ∙ 𝑎 = 𝑁 ∙ 𝑏 (equilibrium around
the lateral axis).
Figure 8: displacement versus speed
