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” ?

Page 1 of 10 Unit A-4 Speed of Sound

______Page 2 of 10 Unit A-4 Mach Number

aRT= 

Speed of Sound (Sea-Level Standard Value) SI Units: 340.3 m/s or 1,225.08 km/h U.S. Customary Units: 1,116.5 ft/s or 761.25 mph or 661.508 knots

______Page 3 of 10 Unit A-4 Measurement of Airspeed

V Pitot Tube: senses total pressure

− (subtract) Static Pressure Orifice: senses static pressure  Pitot-Static Probe

AIRSPEED MEASUREMENT DEVICE

Pitot-static probe measures both stagnation (or total) pressure and static pressure: provides pressure difference between them ( pp0 −  )

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 ( pp0 −  ) 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. Page 4 of 10 Unit A-4 Incompressible Flow (1) (Subsonic: M < 0.3)

1  V 2 p 2  (pp) (p0) (  )

VV pp0 −  p V = 2 

BERNOULLI’S EQUATION

For incompressible flow, we can employ Bernoulli’s equation. 1 Along a streamline: pV+= 2 constant 2 1 Let us define a dynamic pressure: qV  2 2 Then, the Bernoulli’s equation becomes: pq+=constant

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): 11 pVpV+=+22  22000

(Note 1) Assuming incompressible flow: ===0 constant )

(Note 2) At the stagnation point, velocity is zero: V0 = 0

pp0 −  Therefore: V = 2 (Airspeed Equation: M < 0.3, incompressible subsonic flow)  Page 5 of 10 Unit A-4 Incompressible Flow (2) (Subsonic: M < 0.3)

Pitot-static probe

pp0 −  Vtrue = 2 

pp0 −  Ve = 2 s

AIRSPEED CALIBRATIONS

pp0 −  Based on the airspeed equation: V = 2 , 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: (Vi ) .

(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: (VVVcip=+  ) .

(3) (e)quivalent airspeed = next, the pilot must multiply calibrated airspeed by the "f-factor" (called the f-factor correction) to determine equivalent airspeed: (VfVec= ) . Equivalent airspeed is basically, "if the air density is equal to the standard sea-level value," what would be the airspeed . . . the lower case ICeT/ICeTG ()pp0 −  (e) shows that this is usually the "lowest" airspeed value: Ve = 2 (Equivalent Airspeed) s

(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

s ()pp0 −  density ratio: VVtrue ==e 2 (True Airspeed, requires OAT information)  Page 6 of 10 Unit A-4 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.

______Page 7 of 10 Unit A-4 Compressible Flow (1) (Subsonic: 1 > M > 0.3)

( −1) p0  −1 2 =+1 M  p 2 T  −1  0 2 11( − ) =+1 M    −1 T 2 0 2  =+1 M   2

______Page 8 of 10 Unit A-4 Compressible Flow (2) (Subsonic: 1 > M > 0.3)

1 −1 1 2 ( ) 1 pp− 3.5 2a pp− 3.5 ( 0  ) 2 11 01( pp0 −  ) Vp=+− 711 VpV =+−=+− 711 11  p 1e    −1sp1 p 

(−1) 1  2 pp− 3.5 2 2appss 1 01− ( 0  ) VpV=+−=+−711 11 cal p  −1s ps  

AIRSPEED EQUATION (1 > M > 0.3): COMPRESSIBLE SUBSONIC FLOW

______Page 9 of 10 Unit A-4 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.

______Page 10 of 10 Unit A-4 What’s “Incompressible” ?

1  −1 0  −1 2 =+1 M   2

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: 11( − ) 0  −1 2 =+1 M   2

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.