AIRSPEED / PITOT TUBE

We have seen in the previous post that velocity in the test section is obtained with the help of pressure difference, this pressure difference is measured with the help of U shape manometer.
How do we calculate pressure difference in real life, hence how do we calculate the speed of an aircraft?   

Pitot tube, this is an aerodynamic instrument that actually measures the total pressure at a point of the flow.

TOTAL CONDITIONS 

Stagnation condition is when the fluid is brought to rest adiabatically.

• Total Temperature:  The value of the temperature of the fluid element after it has been brought to rest adiabatically is defined as the total temperature, denoted by T0.
• Total enthalpy: The value of the enthalpy of the fluid element after it has brought to rest adiabatically is defined as the total enthalpy denotes by h0.
• Total pressure: The value of the pressure of the fluid element is brought to rest isentropically, is called total pressure denoted by p0.
• Total density: The value of the density of the fluid element is brought to rest isentropically, called as total density denoted by ρ 0

TOTAL CONDITIONS  are imaginary quantities that would exist at a point in flow if the fluid element passing through that point was brought into rest adiabatically.

• Static temperature: Static temperature is measured of purely random motion of the molecules in a gas. It is the temperature u feel when you u are ride along with the gas at the local flow velocity.
•  Static pressure: Static pressure is measured of purely random motion of the molecules in a gas. It is the pressure u feel when you u are ride along with the gas at the local flow velocity.
• Static density: Static density is measured of purely random motion of the molecules in a gas. It is the density u feel when you u are ride along with the gas at the local flow velocity. 
  

It is an important equation in aerodynamic, it states that at any point in the flow, the total enthalpy is given by the sum of the static enthalpy and kinetic energy, all per unit mass.

EXAMPLE OF STATIC AND TOTAL PRESSURE

 Think that u are sitting in a car which is moving with the velocity of 100m/s, windows are closed and there is a fly inside the car moving with u in a random motion. your speed is 100m/s and in that mean, so is that fly. This fly hits u on your skin with this random motion, you will feel a slight impact .this slight effect is analogous to static pressure in flowing gas.
Now think there is a person standing along the side of the road and u open the window and the fly hits the skin of the person, the impact will be more. The strength of the impact is analogous to the total pressure.

PITOT TUBE

It consists of a tube placed parallel to the flow and open to the flow at the endpoint(A). the other end of the tube (point B) is closed, as shown in the figure1.

1

Now, imagine flow is started and some amount of gas is pile up inside the tube as the tube is closed from pressure gauge at point (B). There is no place for the gas to go hence, gas will pile up and stagnate everywhere inside the tube, including the open face of the tube at point (A). The gas is brought into reset isentropiclly as friction is negligible and no heat exchange. Hence, from the above discussion, we can say that pressure at point A is total pressure. point A is a stagnation point, according to the definition of stagnation point, any point of the flow where V=0.

CAN WE MEASURE BOTH THE PRESSURE (STATIC AND TOTAL PRESSURE ) IN ONE INSTRUMENT? 

Yes
As pitot tube only measure total pressure
 PITOT-STATIC PROBE is the instrument which measures both the pressure difference

HOW?

There is a small hole in the surface at point A called static pressure orifice or static pressure tab as shown in the figure2. Since the surface is parallel to the flow, only random motion of the gas molecule will be felt by the surface itself.

2

HOW TO CALCULATE AIRSPEED OF INCOMPRESSIBLE FLOW?

We know static pressure + dynamic pressure= total pressure

.......1

The above equation only holds for incompressible flow

HOW TO CALCULATE AIRSPEED OF SUBSONIC FLOW?

The formulas which we have derived till now are only valid for flow with Mach number less than 0.3 (M<0.3) where we can assume flow to be incomprehensible.
Now, if we increase our Mach number greater than 0.3 and less then 1, compressibility will be taken into account
We will now derive the formula  to calculate the velocity of air vehicle for subsonic flow

we know that, from considering the enthalpy

from energy equation 

substitute equation 1 into 2

Speed of the sound is

Thus, equation 3 becomes  

The Mach number M=V/a

These are the fundamental and important equations in aerodynamics to calculate total pressure, temperature, density and Mach number.

The above equation says the ratio of total pressure and static pressure is direct measure of Mach number. thus, individual measurements of total pressure and static pressure in conjunction of the equation can be used to calibrate an instrument in the cockpit of an aeroplane called as mach metre, where the dial reads directly in terms of the flight Mach number of the aeroplane.

To obtain actually velocity we know that M=V/a

....5

equation 5 can be rearranged algebraically as

...6

The static temperature in the air surrounding the aeroplane is difficult to calculate.  With the help of equation 6, we will calculate actually velocity assuming that a is equal to standard seal-level.

The airspeed indicator is designed to sense the actual pressure difference.
equation 6 is calibrated as

....7

HOW TO CALCULATE AIRSPEED OF SUPERSONIC FLOW?

Flow which has mach number greater than 1 is called a supersonic flow.
supersonic flow has shock waves, hence, the flow passing through shock wave is nonisentropic. There are very large friction and thermal conduction effects hence, neither adiabatic nor frictionless condition holds.

The equation number is called a Rayleigh Pitot tube formula.