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.