4 Incompressible Flow Measurement of Airspeed A- 5...

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” ?

APPLICATION OF CONTINUITY ON A SOUND WAVE

Let us consider a coordinate system attached to (and thus, moving with the same speed with) the sound

wave. Continuity equation (1 2m m ) yields:

( ) ( )Aa d A a da

( )( )a d a da a ad da d da

Therefore, da

ad

(eqn. 1)

SPEED OF SOUND (1)

Recall, the Euler’s equation (in terms of speed of sound): dp ada => dp

daa

(eqn. 2)

Substituting eqn. 2 into eqn. 1 yields: dp

ad a

=> 2 dp

ad

The flow through a sound wave involves no heat addition, and the effect of friction is negligible: means,

it is isentropic flow).

Therefore: isentropic

dpa

d

Unit A-4Page 1 of 10

Speed of Sound

0

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ISENTROPIC FLOW

For isentropic flow: 2 2

1 1

p

p

=> 2 1

2 1

constantp p

c , or, simply:

pc

(eqn. 1)

SPEED OF SOUND (2)

Starting from isentropic

dpa

d

From eqn. 1, the pressure of isentropic flow can be expressed as: p c

Therefore, 1

isentropic

( )dp d

c cd d

(eqn. 2)

Substituting eqn. 1 back into eqn. 2, and simplifying: 1

isentropic

dp p p

d

Therefore, the speed of sound is: isentropic

dp pa

d

For an ideal gas: p RT => p

RT ; therefore, the speed of sound becomes: a RT


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:

stagnation pressure total pressure.


Measurement of Airspeed

Pitot-Static Probe

Pitot Tube: senses total pressure

Static Pressure Orifice: senses static pressure

(subtract)

BERNOULLI’S EQUATION

For incompressible flow, we can employ Bernoulli’s equation.

Along a streamline: 21

constant2

p V = p0

Let us define a dynamic pressure: 21

2q V

Then, the Bernoulli’s equation becomes: constantp q = p0

AIRSPEED MEASUREMENT FOR SUBSONIC INCOMPRESSIBLE FLOW (M < 0.3)

Let us define: location ‘1’ 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 (‘1’) and the tip of the Pitot-Static tube (‘tip’):

2 2

1 1 tip tip

1 1

2 2p V p V

At the freestream: p1 = p, because V1 = V (this is freestream).

At the tip: ptip = p0, because Vtip = 0 (this is stagnation point).

Therefore, 2

0

1

2p V p =>

0 12p p

V


Incompressible Flow (1) (Subsonic: M < 0.3)

(p0)(p)

V

TRUE AND EQUIVALENT AIRSPEEDS

The air density is difficult to measure. For small (low subsonic and low altitude cruise) airplanes, often

the equivalent airspeed is indicated on its airspeed indicator:

Equivalent (or indicated) airspeed is the airspeed that uses the standard sea-level air density value for the

airspeed calculation:

0( )2e

s

p pV

(Equivalent Airspeed)

As long as the altitude is low (close to the sea-level), the equivalent (or indicated) airspeed is fairly

accurate.

The true airspeed is the airspeed that uses the actual air density value for a given flight altitude for the

airspeed calculation:

0true

( )2

p pV

(True Airspeed)

Have you heard about “KEAS” = “Knots in Equivalent AirSpeed”?

(1 knot = 1.15 mph)


Incompressible Flow (2) (Subsonic: M < 0.3)

Pitot-static probe

The equivalent airspeed can be calculated (by using the air density at standard sea-level).

Using s = 0.0023769 slugs/ft3 and 2

0 53.3 lb/ftp p :

02( ) 2(53.3)

0.0023769e

s

p pV

211.774 ft/s

If the temperature is known, it is possible to calculate the true air density:

Using 21,896.7 lb/ftp (pressure altitude 3,000 ft):

1,896.7

(1,716)(50 460)

p

RT

= 0.00216726 slugs/ft3

Using this true air density:

true

02( ) 2(53.3)

0.00216726

p pV

221.780 ft/s

The error of equivalent airspeed: 221.780 211.774

100221.780

4.5 %


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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ENERGY EQUATION

For compressible flow, we can no longer use Bernoulli’s equation. Let us look at the energy equation

one more time.

Recall, the energy equation:

2

constant2

p

Vc T (along the streamline)

AIRSPEED MEASUREMENT FOR SUBSONIC COMPRESSIBLE FLOW (M > 0.3)

Applying the energy equation for a Pitot tube (freestream ‘1’ and stagnation point ‘0’): 2

11 0

2p p

Vc T c T =>

2

0 1

1 1

12 p

T V

T c T (eqn. 1)

Also, the definition of specific heat can be given by: 1

p

Rc

Substituting this into eqn. 1:

2 2

0 1 1

1 1 1

11 1

2[ / ( 1)] 2

T V V

T R T RT

Note that from speed of sound, 2

1 1a RT : thus the equation becomes,

2

20 112

1 1

1 11 1

2 2

T VM

T a

=>

201

1

11

2

TM

T

Using the isentropic relationship: 1

0 0 0

1 1 1

p T

p T

120

1

1

11

2

pM

p

and

1

120

1

1

11

2M


Compressible Flow (1)(Subsonic: 1 > M > 0.3)

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TRUE AND CALIBRATED AIRSPEEDS (1)

Starting from:1

201

1

11

2

pM

p

Solving this equation for M1:

1

2 01

1

21

1

pM

p

(Note: V

Ma

)

1 1

2 22 0 0 11 1

1

1 1

2 21 1 1

1 1

p p pa aV

p p

(True Airspeed)

TRUE AND CALIBRATED AIRSPEEDS (2)

True airspeed requires information of a1 (i.e., T1) and p1. The static temperature and pressure in the air

surrounding the airplane is often very difficult to measure (in high-speed flight).

Therefore, all high-speed airspeed indicators are “calibrated.” For example, assuming that a1 and p1 are

both equal to the standard sea-level value (as = 340.3 m/s = 1,116.5 ft/s and ps = 1.013×105 N/m2 =

2,116.2 lb/ft2), the calibrated airspeed (based on the standard sea-level condition) becomes:

1

22 0 1

cal

21 1

1

s

s

a p pV

p

(Calibrated Airspeed)


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

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Pitot-static probe measurement:0 1p p = 5.5 103 N/m2

At standard sea-level: ps = 1.013 105 N/m2 and as = 340.3 m/s

1 1.4 1

2 2 3 1.42 0 1

cal 5

2 2(340.3) 5.5 101 1 1 1

1 1.4 1 1.013 10

s

s

a p pV

p

= > Calibrated airspeed: calV 93.878 m/s cal

s

93.8780.276

340.3

VM

a

Pressure altitude of 10 km: p1 = 2.65 104 N/m2

Measured temperature is: T1 = 45 C = 45+273 = 228 K

Speed of sound is: 1 1 (1.4)(287)(228) 302.672 m/sa RT

1 1.4 1

2 2 3 1.42 0 11

1 4

1

2 2(302.672) 5.5 101 1 1 1

1 1.4 1 2.65 10

p paV

p

= > True airspeed: 1V 159.243 m/s 1

1

159.2430.526

302.672

VM

a

Error of calibrated airspeed 159.243 93.878

100159.243

41.05 %


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 ‘1’

(freestream) and location ‘0’ (stagnation point) can be given as: 1 1

201

1

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.


What’s “Incompressible” ?

4 Incompressible Flow Measurement of Airspeed A- 5...

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