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# Supersonic and Subsonic Aerodynamic (6MA008) Assignment

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## PART: 1 (Incompressible fluid)

Figure 1 shows a wind tunnel used to study to aerodynamic behavior of aircraft wing sections. The construction ratio of the wind nozzle is 12:1. The tunnel is turned on, and the pressure difference between the inlet (the setting chamber) and the test section is read as a height difference of 2 mm on a U-tube mercury manometer. The pressure and temperature of the flow in the inlet section are 1.013×10^{5} N/m^{2} and 293 K, respectively.

**a) Using the governing equations of the subsonic incompressible flow, express the Mach number of the air flow in the wind tunnel test section as a function of height difference of the manometer, density of fluid in the manometer, density and temperature and specific heat ratio of the air, and the contraction ratio of the nozzle.**

Here in this part of the assignment we need to perform a manual mathematical calculation of the Mach number of an incompressible fluid flow in a tunnel. For this particular case study, we need to calculate the various relationships of the Mach number and the various parameters of the incompressible fluid. Regarding the compressibility of the fluid we can say one of the most important assumptions that are; a fluid can never be compressible. The compressibility of the fluid is just a hypothesis only. So, when we calculated the Mach number of the fluid flow by considering the air flow we have to take all this thing into consideration.

The concept regarding the Mach number in any kind of fluid flow is to be clear first so that we can make the calculations properly. Mach number is the ratio of the object speed to the speed of sound on that particular medium.

For one dimensional fluid flow into a tunnel the continuity equation can be written as,

Since, the fluid is assumed to be an incompressible fluid so we can assume a constant density of the fluid is constant with pressure and temperature difference on the tunnel we have taken for consideration.

So the continuity equation becomes,

A_{1}V_{1} = A_{2}V_{2}

The differential Bernoulli’s equation is

Since, there is a relation between the Mach number and the pressure difference in different cross section of the tunnel we are using a manometer for finding out the pressure difference with the help of height gradient on the fluid present in the manometer tube attached with the tunnel.

*dp = -pgdh*

p_{2} – p_{1} = -p_{f} g (h_{2} – h_{1})

*p _{f}*= The density of the fluid in the manometer

* p _{1}*= The pressure at the height of h

_{1}

* p _{2}*= The pressure at the height of h

_{2}

We can apply the hydrostatic equation twice for the requirement of getting the relationship between the pressure and the density of the fluid in the tunnel.

*p _{3} – p_{1} = -p_{a} g (h_{3} – h_{1})*

*p*

_{1}– p_{2}= -p_{w}g (h_{1}– h_{2})By adding the above two equation we will get, *p _{3} – p_{1} = -p_{w} g (h_{1} – h_{2})*

Relationship between Mach number and contraction ratio,

Now the pressure difference and the velocity difference can be calculated for the tunnel and we will be able to find out the direct relationship between the Mach number and the area contraction.

For calculating the relation with the density of air we get to know that,

Equation (6) shows the relation between the Mach number and the density of the fluid.

Since the Mach number is the basic function of two velocities one is the object velocity and another one is the velocity of sound. So if we want to find out the relationship between the Mach number and the specific heat ratio we just need to check the two velocities are changing with the change of specific heat ratio or not. If the speed of object or the speed of sound is changing with the specific heat ration, then only we can say there is a relation between the Mach number and the specific heat ratio.

For the specific heat ratio,

Equation (7) shows the relation between the Mach number, specific heat ratio, velocity, gas constant and absolute temperature.

**b) Calculate Mach number and the velocity of the flow in the test section.**

For finding out the Mach number and the velocity at the test section we have the values as a given parameter,

Finding out the velocity at the test section for this time then we will find out the Mache number,

For incompressible fluid we know, *A _{1}V_{2} = A_{1}V_{1}*

It is the velocity of the air at the test section of the tunnel.

Now for finding out the Mach number at this position we have,

It shows the Mach number at the test position of the tunnel we have taken for consideration.

### PART: 2 (Aerodynamic forces)

An aircraft wing model with NACA 2415 aerofoil section is mounted in the test section of the previous wind tunnel, as shown in figure 1. the chord length of the wing is 71.5 mm and it completely spans the test section of 150 mm. the variations of lift and drag coefficients with angle of attack are shown in figure 2 and 3 respectively.

When the angle of attack is 8˚ calculate:

- a) Lift force
- b) Total drag force

Here in this part of the assignment we are considering a wing model of NACA 2415 aero foil section. It has a chord length of 71.5 mm and the span completely spread out to the test section of the tunnel is 150 mm. the variation of lift coefficient and the drag coefficient are shown in the figure attached just below this text.

We just need to find out the appropriate lift and drag coefficients from the two graphs and a calculation is to be done on the context of lift force exerted on the aero foil at an angle of 8 degrees. Along with it we also need to calculate the drag force too for this particular aero foil at an angle of 8 degrees.

From these two graphs given here we can visually say the lift coefficient and the drag coefficient for NACA 2415 aero foil. Here the lift coefficient is C_{l}= 1.050 for 8-degree angle of attack and the drag coefficient is C_{d}=0.019 for 8-degree angle of attack. We have already found out the velocity of air in the tunnel is about 12.56 m/s. now we just need to calculate the area of the foil. As the measurement is given in the problem statement the length can be taken as 71.5 mm as it is the chord length and the test tunnel is the overall distance the foil is attached is given as 150 mm so the area of the foil is about 10725 mm^{2} or 0.010725 m^{2}.

** a. For the first part we are calculating the lift force at an angle of attack 8 degree,**

** b. For the second part of this same problem statement we need to find out the drag force on the NACA 2415 aerofoil for the same parameters given in the problem statement.**

### TASK: 3 (Boundary layer)

A boundary layer is formed on the surface of the aerofoil, the transition of laminar boundary layer to turbulent layer occurs at Raynolds number equal to 6.5×10^{4}.

a) The total skin-friction drag on the wing.

b) The total pressure drag force.

The boundary layer is a layer which is formed on the outer surface of an airplane wing at its running condition. The layer of air just adjacent to the wing surface makes a thin layer of air throughout the boundary of the wing. The study of boundary layer is very much important because there are so many parameters are available which have some effects on changing of the boundary layer. At the adjacent surface of the wing the velocity of wind becomes zero for the boundary layer concept.

**a) The total skin-friction drag on the wing.**

In the first part of the problem statement we need to find out the skin-friction drag on the total wing. The detail calculation regarding the skin friction is given below.

Basically all the gases are not completely non viscous. When there is a flow of gas on any surface then just like every fluid its molecule attached with that particular surface and do not allow the adjacent molecules to go over them. They attract them also for this attachment and a boundary layer forms on the overall boundary of the aerofoil. This kind of momentum on the boundary surface makes a frictional force among it is known as skin friction.

We can say the overall mass flowing through the surface,

Here the assumption is no pressure on the surface of the aerofoil so we have,

Drag = Loss in Momentum

These are the basic calculation parts for finding out the frictional force on the laminar boundary layer.

**b) The total pressure drag force.**

For the pressure drag force, it is basically a drag force experienced by a body moving in air. The reason behind this kind of drag force is due to the assumption of the normal surface in front of the object. If the front facing object having a wide cross-section, then at the time of air flow on this surface the normal at this surface works as a wall in front of the air flow. The laminar flow then diverts and the layers creating as a boundary layer separated among themselves and this thing will be responsible for the pressure drag of that particular aerofoil.

The total drag in a very simple manner can be calculate by the help of this general formula if we can assume the front surface as a normal plane relatively to the air flow,

### TASK: 4 (Compressible flow)

The Mach number of the flow in the test section of the wind tunnel was increased isentropically to 0.4

**a) Calculate the lift force in the test section considering that the air properties in the settling chamber remain as mentioned in task 1.**

Here in this problem statement the Mach number is becomes 0.4 and we need to calculate the lift force on the aerofoil we have for consideration. For this process the general procedure to find out the lift fore is to calculate the velocity at the test section for a Mach number of 0.4 since we know Mach number is directly proportional to the velocity of the fluid. From this relation we will be able to find out the velocity and with the help of the velocity we can find out the lift force for the same aerofoil.

Let us finding out the velocity at 0.4 Mach number,

Since we have the velocity for a particular Mach number now we can find out the lift force of the aero foil at an attack angle of 8 degrees since in the problem statement it is given we can use the same assumption for this case study also.

By this method we can easily find out the velocity as well as lift force for compressible fluid for a particular Mach number.

**b) Discuss how the compressibility of air might affect the aerodynamic behavior of the previous wing.**

Here in this problem statement we need to explain about the relationship between the compressibility and the aerodynamic behavior of the wing.

If we consider a high speed of flow with the respect to an object, then the compressibility of the fluid plays an important role on the aerodynamic behavior of the wing of an airplane. The effects of compressibility of the air is more in the Mach number. The Mach number is the ratio of object velocity to the velocity of sound. Now if the compressibility of the fluid will increase that means the air pressure will also increase and along with it the density will be higher than normal condition. If the density will be higher, the wing will have to work less for floating on the air. If the density will increase, then the drag force will be higher also and the lift force needed will be less. Generally, the fluids are always incompressible but for air it is slightly compressible and then the parameters of the air or gas will change accordingly. In the Mach number derivation, we found there is a deep relationship between the air pressure and air density so along with it the all parameters of the aircraft will change with the compressibility of the fluid.

The two equations shown above are the evidence that the compressibility of fluid will affect to the aerodynamic behavior of the wing.

### TASK: 5 (Aerodynamic design)

**a) Discuss possible options that can increase the lift force generated by the wing in Task 4 and explain how these options might affect the overall aerodynamic performance of the wing. Consider that the flow conditions are same as those mentioned in Task 4.**

Here in this part of the assignment we need to discuss about the various aspects of the improvement of lifting force of the wing by considering the same condition of the tunnel air flow. For this consideration we have to say at-least three possible ways to show the possible ways can make a increment in the lift force of the wing.

The main parameters are directly related to the lift force of the wing for the current scenario is a wing tunnel and the wing is at 8-degree inclination. From the equation of the lift force we can say the lift force is directly proportional to the density of air flowing through the tunnel, it is also directly proportional to the area of the wing and along with it the lift force is also directly proportional to the coefficient of lift force. We can also say from the equation; the drag force is also directly proportional to the square of the velocity of the fluid. Since, all the parameters are discussed above directly proportional to the lift force of the wing then we can easily vary them and can get a higher lift force compared to the initial one.

For finding out the lifting force we have the equation in our hand,

From this equation we can say,

Since we cannot change the value of the coefficient of lift which is a constant term at the attacking angle of 8 degrees which is almost about 1.050 so there is not any option to change the value of coefficient of lift force. So by varying the density of the air we can increase the lift force for that particular wing. If we can increase the velocity by increasing the propeller blade we can also increase lift force. If the area of the test section can be increase, then also we can be able increase the lift force.

As of the velocity of the air flow is depends on the Mach number so we can say the lift force can also be increased by increasing the Mach number. The relation is stated below,

So, if we put the value of velocity of the air flow on the equation of lift force then we can get the relationship between the lift force to the Mach number, temperature at the test section and all. So the relationship becomes,

*Substituting the value of V in the left force equation we get,*

From this above equation we have in our hand can say the lift force is directly proportional to the square of the Mach number. And also we can say the lift force is directly proportional to the temperature at the test section.

For all the parameters we have discussed in this section to increase the lift force in the same arrangement we have, we can make an increment in some of the quantities those are having a proportional relationship with the lift force of the wing. The density can be increased by reducing the temperature of that test section. The velocity can be increased by speeding up the propeller blade. Since the arrangement is completely same here in this problem statement also we cannot increase the area of the test section. Another two methods are the increment of the Mach number at the test section; it can be easily increased by the increment of velocity. As well as by an increment of the temperature of the test section we can also increase the value of lift force of that particular wing.

**b) Provide a mathematical proof showing that at least two of the suggested options are capable of increasing the lift force of the wing.**

Here we have to give some mathematical proof so that we can say by increasing those parameters we can increase the lift force of the wing. Since, we have already stated all the related things on the previous part of this problem statement therefore it is very easy to give a mathematical proof of concept for this lift force,

We all know the lift force for a wing can be expressed as,

Here we can say the wing is at an attacking angle of 8 degrees for that reason we cannot change the value of coefficient of lift force, since we are performing with the same wing we cannot change the area so here what we can do? It is the main question coming from this part.

Assuming all the parameters on the left hand side of the above equation are constant excluding the velocity. If we are increasing the velocity it will not affect the density and the area also and we can get a higher lift force for the same wing.

Since, we can say the lift force ‘L’ is also depends on the Mache number; the reason behind it is the velocity is a function of Mach number. So, here indirectly if we increase the value of Mach number then also we will be able to increase the lift force for the wing we have. The relation stated below,

(It shows the relation between the lift force and the Mach number of the wing)

### TASK: 6 (Supersonic Flow)

**a) Design the divergent section of the previous wind tunnel, presented in Task 1, which can provide a supersonic flow at its exit with a minimum Mach number of 1.6 and calculate the velocity of the flow at the exit.**

For designing purpose of a supersonic wind tunnel we need to consider a convergent divergent duct in a useful manner. In convergent duct the process is to increasing the cross-sectional area of the diffuser and decreasing the velocity form supersonic to subsonic velocity of fluid. At the same time with the help of the wind tunnel we can converge the mid part and can increase the cross-sectional area of the duct portion them we can also convert a subsonic velocity into a supersonic velocity fluid flow. All the related assumptions and possible calculations are shown below. From the conservation of mass we can say,

At the time of defining the wind speed in the wind tunnel it is necessary to define the Mach number carefully and taken into consideration. The Mach number is a unit less quantity can be defines as.

To relate the all the parameters like temperature, pressure and density we need to show an expression it is shown below,

Sine, the main fundamental need is to consider the Mach number here so we need to relate the temperature, pressure and density with the Mach number and we have,

The critical temperature, density and pressure ratios can be expressed as,

All the calculations in terms of temperature, density and pressure are to be considering convergent divergent wind tunnel for supersonic flow of the fluid.

For calculating the exit velocity of the duct we have to clear the concept of the Mach number and for the exit velocity we have,

**b) Determine the reading of a Pitot tube which is inserted at the exit of the previous supersonic nozzle.**

The Pitot tube is a device that connects at the end points of the diffuser of the wind tunnel at the time of testing any aerodynamic material like a wing. It is basically used to measure the various parameters at the time of performing those experiments. At the external duct the measuring instrument measures the overall parameters of the fluid flow like air flow speed and in case of water jet it is also used to measure the water flow speed at the exit duct of the tunnel.

Here we have discussed the procedure of measuring the velocity of the air flow with the help of Pitot tube, the derivation stated below in this part.

The static pressure is given by ‘P’

The dynamic pressure is given by, *1/2 pV ^{2}*

And the stagnation pressure is given by the equation shown below,

Where,

Now for measuring the velocity we can follow the following rules from the Pitot tube we have connected at the end of the wind tunnel.

For such kind of calculation, we need to select a reference plane on the same height of the Pitot tube we have. At this condition we can say the stagnation pressure becomes,

For calculating the velocity at the exit point it is very easy to calculate the value of pressure difference of the third Pitot tube we have shown below.

### TASK: 7 (Applied aerodynamic)

**With the aid of appropriate research and figures, discuss at least three uses of the aerofoil shaped surfaces for applications related to aerospace engineering. The discussion shoul revolve around the aerodynamic behavior and function of the selected surfaces and the examples should cover both subsonic and supersonic flow.**

In this part of the assignment we have discussed the various research topologies on the context of the air craft wings as a consideration of aero foil. With the help of their diagrams and the related parameters for them we have the intention to find out the exact parameters for those particular Aerofoils. On the context of this research we have covered both the supersonic and subsonic properties of them. We have also given some examples in this part of the assignment so that we got the exact parameters and can cover both the supersonic and subsonic characteristics of the examples.

**1. **As the most important and basic one usage of the aerofoil to get the lift force on it. The lift force is the force developed by the streamlined air flow through the wing of the aircraft. Mainly the wing can be considered as the aerofoil. When the stream like air flow is there on the aerofoil then with the help of its surface and angle of attack controlling we can produce a downward thrust of the air flow and that provides the lift up force to the aircraft.

**2. **Another most important topic on the usage of the aerofoil is the drag. The drag is the parameter on the aerodynamics is the opposite thrust on the object relative to the fluid flow. The aerofoil and its shape size and all play an important role on it. Since, we are using the aerofoil in terms of producing the lift force along with it we can also use the aerofoil as an item for producing a drag for reducing the object velocity when necessary.

At the time of designing the aerofoil for a particular purpose will be used in aerospace engineering one should also consider the drag and for this purpose we need to take the front cross-section of the foil into consideration so that it can produce a sufficient amount of drag also.

**3. **In some cases, if we consider the general purpose of the aerofoil the speeding up and the more lifting up and general purpose both this kind of aerofoil are used in various requirements. For this situation at the design time we need to follow the design parameter for their exact usage. If we want a higher amount of lift, then we need a special kind of aerofoil. If there is a need of subsonic velocity in air so at that time also we need some special kind of aerofoil, if we need supersonic speed of a aircraft on air then we have to reduce the amount of drag so that it can be applied to have a supersonic velocity. So we can say for different purposes we need some special shapes of the aerofoil wing. The different shapes of them are given below to show both the examples of subsonic and supersonic velocities for the wings of aircraft.

In such kind of condition, we have to consider the Mach number also for designing of the aerofoil. The reason behind it is, the Mach number is the property of an aerofoil aerodynamic test where we can get the exact result. After considering those results we can say if they are in which region. If an aerofoil is experiencing a Mach number less than 1 then the entire system will be known a subsonic. If the value of Mach number is exactly 1 then it will be known as the sonic region. If the value of the Mach number will be more than 1 then it will be known as supersonic region. So in the usage of the different kind of airfoils we need to be careful about their material used in the aerofoil. The exact amount of drag and lift experiencing by the aerofoil are to be found out on this context.

For different kind of aerofoil we have to observe its shape due to the boundary layer concept. Regarding this concept, we can say at the time of designing or using any aerofoil for a particular purpose we should take the boundary laminations into consideration. Since, there is a deep relationship between the skin friction and the drag coefficient for boundary lamination on the surface of the wing so; we need to apply them to use a proper wing for aerodynamic application in a proper purpose.

Since, the discussion in this part is based on the usage of the different kind of aerofoil in different application as an example. So, we can say the angle of attach is also an important topic for this kind of understanding. The angle of attach is the angle of the wing for betterment of the lift force. Since we have seen earlier in this assignment that the coefficient of lift force having a directly proportional relation with the lift force so we need to check before the application of any aerofoil about its maximum angle of attach because a more lift force shows a more thrust on the wing also.

These are the basic discussion on the different kind of aerofoil are used in aircraft engineering for betterment of their physical condition in different aspects.

### Reference List

Li, Z. N. (2010). Aircraft Structure Science. 2^{nd} Edition, Beihang University Press, Beijing.

Henne, P. A. (1990). Applied Computational Aerodynamics, American Institute ofn Aeronautics and Astronautics.

Puckett, A. E. (1946). Supersonic Wave Drag of Thin Airfoils. Journal of the Aeronautical Sciences, 13, 475-484. http://doi.org/10.2514/8.11425

Hicks, R. M. & Henne, P. A. (1978). Wing Design by Numerical Optimization. Journal of Aircraft, 15, 407-412. http://doi.org/10.2514/3.58379

Qian, Y. J. (2004). Aerodynamics.

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