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AE301 Aerodynamics I
UNIT A: Fundamental Concepts
ROAD MAP . . .
A-1: Engineering Fundamentals Review
A-2: Standard Atmosphere
A-3: Governing Equations of Aerodynamics
A-4: Airspeed Measurements
A-5: Aerodynamic Forces and Moments
AE301 Aerodynamics I
Unit A-4: List of Subjects
Speed of Sound
Mach number
Measurement of Airspeed
Incompressible Flow
Compressible Flow
What’s “Incompressible” ?
Unit A-4Page 1 of 10
Speed of Sound
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Unit A-4Page 2 of 10
Mach Number
Speed of Sound (Sea-Level Standard Value)
SI Units: 340.3 m/s or 1,225.08 km/hU.S. Customary Units: 1,116.5 ft/s or 761.25 mphor 661.508 knots
a RT=
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AIRSPEED MEASUREMENT DEVICE
Pitot-static probe measures both stagnation (or total) pressure and static pressure: provides pressure
difference between them ( 0p p− )
STATIC, DYNAMIC, AND TOTAL (OR STAGNATION) PRESSURES
Static pressure (p) at a given point is the pressure we would feel if we were moving along with the
flow at that point.
Total pressure (p0) at a given point in a flow is the pressure that would exist if the flow was slowed
down “isentropically” to zero velocity: therefore, p < p0 (for a stagnant air: p = p0).
Dynamic pressure is a pressure due to the added energy into the moving fluid (air). The difference
between total and static pressures ( 0p p− ) is dynamic pressure. Dynamic pressure is zero for a
stagnant air (p = p0).
Stagnation point is where V = 0: so at stagnation point, the pressure becomes close to the total pressure
(p0): stagnation pressure total pressure.
Unit A-4Page 3 of 10
Measurement of Airspeed
Pitot-Static Probe
Pitot Tube: senses total pressure
Static Pressure Orifice: senses static pressure
− (subtract)
V
BERNOULLI’S EQUATION
For incompressible flow, we can employ Bernoulli’s equation.
Along a streamline: 21
constant2
p V+ =
Let us define a dynamic pressure: 21
2q V
Then, the Bernoulli’s equation becomes: constantp q+ =
AIRSPEED MEASUREMENT FOR SUBSONIC INCOMPRESSIBLE FLOW (M < 0.3)
Let us define: location ‘’ being the flow far upstream (called, the “freestream”) and location ‘0’ being
the location of zero velocity, the ‘tip’ of the Pitot-Static tube (called, the “stagnation point”).
Applying Bernoulli’s equation between freestream () and the tip of the Pitot-Static tube (0):
2 2
0 0 0
1 1
2 2p V p V + = +
(Note 1) Assuming incompressible flow: 0 constant = = = )
(Note 2) At the stagnation point, velocity is zero: 0 0V =
Therefore: 02
p pV
−= (Airspeed Equation: M < 0.3, incompressible subsonic flow)
Unit A-4Page 4 of 10
Incompressible Flow (1) (Subsonic: M < 0.3)
(p0)(p)
V
( )p
02p p
V
−=
V
p
21
2V
p
AIRSPEED CALIBRATIONS
Based on the airspeed equation: 02p p
V
−= , the aircraft airspeed will be determined. However, in
order to accurately determine airspeed, it is required to go through a series of stages of error corrections.
(1) (I)ndicated airspeed = the airspeed that the needle on the airspeed indicator points at for a given set
of flight condition is called (simply) the indicated airspeed: ( )iV .
(2) (C)alibrated airspeed = first, errors in total pressure measurements for certain conditions (also the
indicator itself) will need to be calibrated from indicated airspeed: ( )c i pV V V= + .
(3) (e)quivalent airspeed = next, the pilot must multiply calibrated airspeed by the "f-factor" (called the
f-factor correction) to determine equivalent airspeed: ( )e cV fV= . Equivalent airspeed is basically, "if
the air density is equal to the standard sea-level value," what would be the airspeed . . . the lower case
(e) shows that this is usually the "lowest" airspeed value: 0( )
2e
s
p pV
−
= (Equivalent Airspeed)
(4) (T)rue airspeed = the true airspeed is the airspeed that uses the actual air density value for a given
flight altitude for the airspeed calculation. This can be done by multiplying the square root of the air
density ratio: 0
true
( )2s
e
p pV V
−= = (True Airspeed)
Unit A-4Page 5 of 10
Incompressible Flow (2) (Subsonic: M < 0.3)
Pitot-static probe
ICeT/ICeTG
0true 2
p pV
−=
02e
s
p pV
−
=
Unit A-4Page 6 of 10
Class Example Problem A-4-1
Related Subjects . . . “Airspeed Measurement: M < 0.3”
The altimeter on a low-speed private aircraft (M < 0.3) reads 3,000
ft. If a Pitot-static probe (as shown in the figure) measures a
pressure of 53.3 lb/ft2, what is the equivalent airspeed of the
airplane? Suppose, if you know the outside air temperature
(through an independent measurement) is 50 ºF, what is the true
airspeed? Calculate the error of equivalent airspeed.
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Unit A-4Page 7 of 10
Compressible Flow (1)(Subsonic: 1 > M > 0.3)
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20 11
2
TM
T
−= +
( )
( )
1
20
1 1
20
11
2
11
2
pM
p
M
−
−
− = +
− = +
AIRSPEED EQUATION (1 > M > 0.3): COMPRESSIBLE SUBSONIC FLOW
Unit A-4Page 8 of 10
Compressible Flow (2) (Subsonic: 1 > M > 0.3)
( )12
2 0 111
1
21 1
1
p paV
p
− − = + −
−
( )12
2 0 1cal
21 1
1
s
s
a p pV
p
− − = + −
−
( )1
3.501
7 1 1s
s
p pV p
p
−
= + −
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( )1
3.501
7 1 1p p
V pp
−
= + −
( )1
3.501
7 1 1e
s
p pV p
p
−
= + −
Unit A-4Page 9 of 10
Class Example Problem A-4-2
Related Subjects . . . “Airspeed Measurement: M > 0.3”
A jet aircraft is cruising high speed (high subsonic:
M > 0.3) at 10 km cruising altitude. If a Pitot-static
probe (as shown in the figure) measures a pressure
of 5.5 103 N/m2, what is the calibrated airspeed
(and associated Mach number) of the airplane?
Suppose, if you know that the outside air
temperature (through an independent measurement)
is −45 ºC, what is the true airspeed (and associated
Mach number)? Calculate the error of calibrated
airspeed.
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DEFINITION OF INCOMPRESSIBLE FLOW
So far, we employed the rule of thumb (M∞ < 0.3) as an indicator of incompressible flow.
But, why this is valid?
Recall, for isentropic flow, with calorically perfect ideal gas, the ratio of density between location ‘∞’
(freestream) and location ‘0’ (stagnation point) can be given as: ( )1 1
20 11
2M
−
− = +
Note that the “freestream” is the location where the density is “lowest” within the flow field, while
“stagnation point” is the location where the density is “highest” (most compressed). Hence, this
equation is the “density variation” within the given flow field (from lowest to highest density).
For isentropic flows with Mach numbers less than about 0.3, the density variation within the flow field
is less than 5 percent. The variation is small, and thus the flow can be treated as incompressible.
Unit A-4Page 10 of 10
What’s “Incompressible” ?
1
120 1
12
M
−
− = +