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.

WIND TUNNEL

The wind tunnel is a ground-based experimental facility, which gives us an enormous amount to experimental data of natural flow simulation. we will see two types of wind tunnel here 
1. Supersonic wind tunnel 
2. Subsonic wind tunnel 
 We have divergent nozzle in the supersonic wind tunnel and convergent nozzle in the subsonic wind tunnel.  How do we decide this?  

Let’s derive an equation

Differentiation we get

Recalling the moment equation and Euler’s equation

Substituting 6 into 7

Since the flow is isentropic

Equation becomes

Rearranging

The equation number 9 is a very important equation of aerodynamics

• Cases 1. Subsonic flow M<1, for the velocity to increase (dV positive) the area must decrease (dA negative ), from equation 9, when the flow is subsonic area must decrease to increase the velocity, hence we need a converging nozzle for subsonic flow  as shown in figure a
• Case 2. Supersonic flow M>1, for the velocity to increase (dV positive) the area must also increase, from equation 9, when the flow is supersonic area must increase to increase the velocity hence, we need a divergent nozzle for supersonic flow, as shown in figure b
• Case 3. Transient flow M=1   

dA/A = 0, stream tube has min area at M=1. This minimum area is called the throat.shown in figure c

SUBSONIC WIND TUNNEL 

1

Figure 1 is a subsonic  wind tunnel

Where

• V₁= velocity in the nozzle
• p₁=pressure at nozzle
• A₁=area of nozzle
• V₂=velocity in the test section
• p₂=pressure at the test section
• A₂=area of the test section
• V₃= velocity in the diffuser
• p₃=pressure at diffuser
• A₃= area of the diffuser   

                    

  Mach number will be less than 1 as we are dealing for subsonic wind tunnel, we will assume flow to be incompressible.

Air is passed in the nozzle with the help blower at low velocity. The nozzle converges to a small area A₂ at the test section, velocity increase, from equation 1 (the continuity equation). Then the air is passed into a diffuser or diverging duct, where the area (A₃) is increased and velocity (V₃) decreases, from equation 2 (the continuity equation) 

The pressure at various location in the wind tunnel is related to the velocity through Bernoulli’s the equation for incompressible flow

As V increases p decreases hence, p₂<p₁ i.e. is the test section the pressure is smaller than the reservoir pressure upstream of the nozzle. In much subsonic wind tunnel, all or part of the test section is open, or vent, to the surrounding air. In such cases, the outside air pressure is communicated directly to the flow of the test section, and p₂=1atm. Downstream of the WindTunnel is a diffuser, where the pressure (p₃) increases and velocity (V₃) decreases.

If A3=A1 then, from equation 1, V₃=V₁; and from equation 3, p₃=p₁ 

Note: In the actual wind tunnel, the aerodynamic drag created by the flow over the model in the test section causes a loss of momentum not included in the derivation of the Bernoulli’s equation; hence, in reality, p₃ is slightly less than p₁, because of such losses

The test section of the velocity is derived as, from equation 3

Substitute equation 1 into 4

ρ1 in the above equation is called as dynamic pressure 

Solve for V2

In equation 5 area the ratio is fixed A₂/A₁ quantity, as it is given at the time of design only. The controlling nob is basically, pressure difference (p₁-p₂) which allows the wind tunnel to operate to control the value of the test section velocity V₂

WHERE WILL WE GET PRESSURE DIFFERENCE FROM?

Manometer we will get the pressure difference, how?

We will use u tube manometer, the left side of the tube is connected to p1 and the right side is connected to pressure p2. The difference in height of the fluid on both side gives us the pressure difference as shown in figure 2

In the modern wind tunnel, the manometer has been replaced to pressure transducers and electrical digital display

How to different obtain values of pressure difference?

Well, we can obtain different values of pressure difference by increasing and decreasing velocity. For example, if I want to double the pressure difference I have to increase the velocity by approximately 42%.

If we increase the contraction ratio while keeping the pressure constant, velocity difference across the nozzle is increased, although the actual velocity at the t and exit of the nozzle. 

SUPERSONIC WIND TUNNEL 

 Supersonic flow M>1, for the velocity to increase (dV positive) the area must also increase, from the equation 9, when the flow is supersonic area must increase to increase the velocity hence, we need a divergent nozzle for supersonic flow.

Just passing the air from the divergent nozzle is enough to generate supersonic flow in wind tunnel

No, it’s not enough

HOW TO GENERATE SUPERSONIC FLOW IN WIND TUNNEL??

Starting with the stagnant gas in a reservoir, the preceding discussion says that a duct of sufficiently converging- diverting shape must be used, as shown in figure  

The flow starts out with very low-velocity V @ 0 in the reservoir, expands to high subsonic speeds in the convergent section reach Mach 1 at the throat, then it is passed through a divergent nozzle to get supersonic flow.

For rocket engines, the flow quality is not important, but the weight of the nozzle is a major concern, for the Wight to be minimized, the engine’s length is minimized, which gives rise to a rapidly diverging, bell-like shape for a supersonic section.

The real flow through the nozzle is closely approximated by isentropic flow because little or no heat added or taken away through the nozzle walls and the vast the core of the flow is virtually frictionless. Equation 10 to 12 apply to nozzle flow .these equation demonstrates the power of Mach number in aerodynamic calculation.

Mach number continuously increases through the nozzle, going from near zero in the reservoir to M=1 at the throat and to supersonic valises of downstream of the throat.

INDUCED DRAG

Is it possible to generate lift without Drag? Well, the answer is no, we can’t produce lift without drag. Here we will see how lift is related to drag.

Wingtip vortices tend to drag the surrounding air around that induces a small velocity component in the downwash direction at the wing this downward component is called as downwash. Denoted by symbol w.

As u can see in figure 1.a Relative wind and downwash add vectorially to produce a ‘’local’’ relative wind that is canted downward from the original direction of relative wind There is an increase in drag. This increase is called induced drag.

1.a

Calculation of induced drag

Consider a finite wing as shown in figure 1.b

1.b

Here,

Ø  R₁=Resultant aerodynamic force in an imaginary situation with no vortices

Ø  D₁= The component of R₁ parallel to v_∞

Ø  R= Resultant aerodynamic force in an actual situation with vortices

Ø  D= The component of R parallel to v_∞

Ø  D_i= The induced drag, difference between D and D1

·         D₁ is an imaginary case is due to skin friction and pressure drag due to flow separation.

·         D is an actual drag which includes the effect of the changed pressure distribution due to the wingtip vortices as well as skin friction and pressure drag due to flow separation.

As u can see in figure R is tiled backwards relatively to R₁, then D>D₁

To calculate the magnitude of D_i

We will consider finite wing as shown in figure 1.c

1.c

Here,

Geometric the angle of attack (a): The angle of attack defined between the mean chord of the wing and the direction of v∞ is called a geometric angle of attack

Induced angle of attack (a_i): The angle between the local flow direction and the free-stream direction is called the induced angle of attack

The airfoil section seeing an effective angle of attack

The local flow direction in the vicinity of the wing is inclined downward with respect to the free stream. The lift vector remains perpendicular to the relative wind, tilted back through angle ai , as shown in the figure, still considering drag to be parallel to be a free stream, tilted-lift vector contributes a certain amount of drag. This drag is called induced drag.

From figure 1.c

Value of a_i are generally small

Hence,

Note that in equation 3 ai should be in radians

ai for a given section of a finite wing depends on the distribution of downwash along of the span of the wing. To see this in more detail consider figure 1.d which is showing the front view of the finite wing.

1.d

The lift per span varies as the function of the distance along the wing

1.       The chord may vary in length along the wing

2.       The wing may be twisted in such way that each airfoil section of the wing is at a different geometric angle of attack

3.       The shape of the airfoil section may changes along the span.

In an elliptical lift distribution which in turn, produces a uniform downwash distribution

In the case of, incompressible flow theory predicts that

Here,

C_L is the lift coefficient of finite wing

AR is aspect ratio b²/S

Substitute the value of AR and ai into equation 3

Defining, the induced drag coefficient as

This result holds only for elliptical lift distribution, the wing no twist and same airfoil shape across the span.

For all wing in general, a span efficiency factor e can be defined such that for elliptical platform e=1 and for other e<1

From equation 5 is an important relation as it demonstrates

1.       To minimise induced drag we need to increase your AR

2.       Induced drag various as the square of the lift coefficient

 TOTAL DRAG
   

 Profile drag is the sum of skin friction drag and pressure drag (c_d).C_D,i for infinite wing (infinite aspect ratio) is zero.