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Posted: January 18th, 2025
When an aircraft moves through the atmosphere there are many physical and chemical changes that take place around its surface. But if the speed is low enough, chemical changes are insignificant. To study the physical changes over the aircraft we take a cross section of its wing and study them in a wind tunnel.
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In this experiment, an unswept, untapered aerofoil with symmetrical section is mounted in a transonic wind tunnel, so as to measure the surface pressure distribution. This is done with different free-stream mach numbers ranging from subsonic to supercritical. These measurements are used to assess the validity of the Prandtl-Glauret Law, which relates the pressure coefficient at a point on the surface of an aerofoil in sub-critical, compressible flow to that at the same point in incompressible flow.
Initially, the pressure coefficients are calculated using the measured surface pressures. Which are then compared with the theoretical pressure distributions predicted by the Prandtl-Glauret Law. Using the free-steam Mach numbers, the critical pressure coefficient (Cp*) is calculated and the critical Mach number (M∞crit) is also deduced.
Furthermore, these results will be used as the basis for discussing the changes in flow properties as the Mach number increases into the transonic regime.
When the airfoil is located in a free stream air, the velocity of air over the upper surface increases whereas the pressure decreases and vice-versa. The effect of changes in pressure over a wing is critical in the study of aerodynamics, as its properties highly affect the flight. Pressure is an effect which occurs when a force is applied on a surface. It is the amount of force acting on a unit area.
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Every point in a fluid flow field has its own unique pressure, which is called pressure coefficient,Cp. It is a very useful parameter for studying the flow of incompressible fluids such as water, and also the low-speed flow of compressible fluids such as air.
This is given by the formula;
where:
P : Pressure at the point where pressure coefficient is to be calculated
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P∞: Freestream pressure
: Fluid density in freestream
: Freestream velocity of the fluid.
For compressible fluids such as air or high speed flows, the difference between stagnation and static pressure is no longer an accurate measure of dynamic pressure. As a result, pressure coefficients can be greater than one in compressible flows.[1]
Hence for compressible flows, the previous formula can be more conveniently re-written in the form
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(M∞ is the free-stream Mach number)
Mach number is a quantity that defines how quickly a vehicle travels with respect to the speed of sound. The Mach number (M) is simply the ratio of the vehicle’s velocity (V) divided by the speed of sound at that altitude (a).[2]
Where;
is the Mach number
is the relative velocity of the source to the medium and
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is the speed of sound in the medium
Since it is defined as the ratio of two speeds, it’s a dimensionless number. It is highly dependent on temperature and atmospheric composition. The Mach number is commonly used both with objects travelling at high speed in a fluid, and with high-speed fluid flows inside wind tunnels.
Generally the flow is divided into five different conditions:
With the increasing mach numbers shock waves are produced and the temperature, pressure, and density also increase.
The free stream mach number of the undisturbed flow, M∞, is related to static/stagnation pressures, P∞/Po∞, by the equation:
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A “critical Mach number” is the speed of an aircraft (below Mach 1) when the air flowing over some area of the airfoil has reached the speed of sound. For instance, if the air flowing over a wing reaches Mach 1 when the wing is only moving at Mach 0.8, then the wing’s critical Mach number is 0.8.[3]
When an aircraft is moving, the airflow around the aircraft is not exactly the same as the airspeed of the aircraft due to the airflow speeding up and slowing down to travel around the aircraft structure. At the Critical Mach number, local airflow in some areas near the airframe reaches the speed of sound, even though the aircraft itself has airspeed lower than Mach 1.0. This creates a weakshock wave.
The critical Mach number differs from wing to wing. It depends on the geometry of the wing. Since a thicker wing accelerates the airflow to a faster speed than a thinner one, therefore a thicker wing will have a lower Critical Mach number.
In theory, it posses a limit for the aircrafts before crossing the sound barrier. Any aircraft that has the ability to surpass the critical Mach number are called transonic aircrafts.
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An aircraft is known to be in transonic region when the Mach number is between 0.8-1.2. It is the moment when the aircraft is concurrently below, at, and above thespeed of sound. The transonic period is dependent on aircraft speed and the pressure and temperature of the local environment.
Transonic is a term used by aircraft designers to describe those high subsonic speeds – usually above Mach 0.7 – where an aircraft is travelling below the speed of sound but shockwaves are still present.
This unstable formation of shock waves leads to wave drag. It is the main form of drag in transonic flights. At transonic and supersonic speeds, there is a substantial increase in the total drag of the airplane due to fundamental changes in the pressure distribution. These physical changes induce flow separation over the aircraft surfaces.
The drag coefficient of the airplane is greater in transonic range than in the supersonic because of the erratic shock formation and general flow instabilities. However, once a supersonic flow has been established, the flow stabilizes and the drag coefficient is reduced.[4]
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One of the important changes that take place in the transonic region is the sudden increase in drag. The Mach number at which the drag of the airplane increases markedly is called thedrag-divergence Mach number. Its value is typically greater than 0.6; therefore it is atransoniceffect and its value is also close to, and always greater than, the critical Machnumber. Usually the drag coefficient increases rapidly at Mach 1 and begins to decrease at around Mach 1.2 when the flow shifts to supersonic regime.
WIND TUNNEL
The tunnel used in this experiment has a transonic test section.
The width and height of the working section are 89mm and 178mm respectively.
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It is equipped with liners which are ventilated so as to reduce interference and blockage at transonic speeds.
The tunnel Mach number is controlled by varying the injector pressure. And the maximum speed that can be achieved is about 0.88.
The stagnation pressure is taken to be equal to settling chamber pressure, and
The reference stagnation pressure is taken from a pressure tapping in the floor well upstream of the model.
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THE MODEL
The model to be tested is a NACA 0012 symmetric section and is unswept untapered model.
It has a maximum thickness/chord ratio of 12% and has a chord length of 90mm.
There are 9 pressure tapings altogether, built into the model.
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These tapings are connected to a multi-tube mercury manometer which has a locking mechanism that freezes the mercury level, helpful for taking the readings.
Tapings 1-8 are on the upper-surface of the model, while one is at the lower surface called 3a.
This pressure tapping 3a is at the same chord wise position as the tapping 3 on the upper side, which enables the model to be set at zero incidence.
Initially the barometric pressure Pat in inches of mercury was recorded.
The manometer bank slope angle was set at 450.
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The injector pressure Pj was initially set to 115 psi and the manometer readings were recorded, along with atmospheric pressure(Iat), static pressure (I∞), stagnation pressure (Io∞), airfoil pressure tapings.
The injector pressure Pj was changed six times and different readings were recorded at every value.
The readings taken in the manometer were all in inches, which were later on converted to absolute pressure.
1) To convert the manometer reading to absolute pressure, following equation was used
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Calculating the value for P1, for example
P1 = 32.7 +/- l12.8 – 33.2l sin(45)
Therefore, P1 = 18.28 pa
All the other pressure readings were converted to absolute pressures in this manner.
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2) To calculate the free-stream Mach number, following equation was used
Calculating M∞ at the injector pressure 115psi,
M∞ = sqrt(2/0.4){(P∞/Po∞)^-(0.4/1.4) – 1
Therefore,
M∞ = 0.82
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Similarly, the other Mach numbers were calculated
3) Pressure coefficient (Cp) was calculated using formula,
Using the calculated value of M∞, P, P∞ and substituting in the equation of Cp gives,
Therefore,
Cp = 0.296
Other values of Cp were calculated using the same formula.
4) The Prandtl-Glauret law pressure coefficient (Cpc) were calculated using,
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Where Cpi is the theoretical pressure coefficient in incompressible flow.
Substituting the values
Cpc = 0.396/sqrt(1-o.82^2)
Cpc = 0.6919
Other values of Cpc were calculated similarly.
5) The equation used to calculate critical pressure coefficient (Cp*) was,
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Calculating for initial Mach number i.e. M∞ = 0.82
Therefore,
Cp* = 0.379
Remaining five values of Cp* were calculated using similar format.
Converted manometer readings to absolute pressures;
From the above graphs, it can be seen that the Mcritical value is approximately 0.77. Hence for the NACA 0012 wing, shockwaves will only occour if the Mach number is above this value. Mach numbers below (or equal to) this value will not experience any formation of shockwave. As in this experiment, there wont be any formation of local shockwaves for Mach numbers 0.38, 0.55, 0.64 and 0.76.
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As stated previously, transonic is a term used by aircraft designers to describe those high subsonic speeds, where an aircraft is travelling below the speed of sound but shockwaves are still present. It usually is above Mach 0.7.
The airflow around a moving airplane is considerd to be incompressible at subsonic speeds. The compressibility effects have only minor effects on the flow pattern and drag, up to a free-stream Mach number of about 0.7 to 0.8. The flow is subsonic everywhere. The local Mach number at the airfoil surface becomes higher than the free-stream Mach number because the flow must speed up as it proceeds about the aerofoil.
There eventually occurs a free-stream Mach number called the critical Mach number at which a supersonic point appears near the maximum thickness. Indicating that the flow at that point has reached Mach 1.
Larger regions of supersonic flow appear on the airfoil surface, as the free-stream Mach number is increased beyond the critical Mach number and approaches Mach 1. The flow must pass through a shock in order to return to subsonic flow from supersonic flow. This loss of velocity is accompanied by an increase in temperature, that is, a production of heat. This heat represents an expenditure of propulsive energy that may be presented as wave drag.
At transonic speeds the main form of drag is wave drag which is due to the formation of shock waves. They create a lot of pressure difference over the aerofoil. And also induce flow seperation. These shocks appear anywhere on the airplane (wing, fuselage, engine etc.)
Large increases in thrust is required to produce any further increases in airplane speed.
Due to the formation of these erratic shockwaves, the coefficient of drag is higher in transonic flow than in the initial supersonic flow.
As determind previously, the critical mach number for this NACA 0012 aerofoil is 0.77. Thus, in the graphs for M = 0.79 and 0.82, we should expect a shockwave over the body of the aerofoil.
As the formation of shock wave induces a large pressure difference, The pressure before the shock wave is very high, which decreaes suddenly after the shock. So to determine the position of shock wave, the pressure readings should be checked at those individual mach numbers.
Hence, from the manometer readings of the pressure tappings it was seen that, the pressure suddenly decreases between the x/c% 0.45 and 0.55, that is, between the pressure tapping 5 and 6. This was for Mach number 0.79.
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Whereas for Mach number 0.82, the pressure suddenly decreased between the x/c% 0.55 and 0.65, that is, between the pressure tappings 6 and 7.
This can be tabulated in the form
M
Approximate position of shock (x/c)
0.79
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0.45-0.55
0.82
0.55-.65
It can be seen from the above table that the shockwave progressively moves more downstream as the free steam mach number increases.
From the graphs of Cpexp Vs Cpthoery(Cpc), it was seen that the experimental value corelates with the theoritical values for smaller mach numbers. But as the Mach numbers increases the experimental value and theoretical values start to deviate. For the Mach numbers 0.38, 0.55 and 0.64 the graphs almost overlap each other. But for higher mach numbers such as 0.76, 0.79 and 0.82 they differ with great amount.
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This is mainly because of the prediction of Prandtl-Glauret theory about the aerofoil being thin. The theory breaks down completely in super-critical flow, when regions of locally supersonic flow occur and shock waves start to form over the surface of the aerofoil. Whereas the theory does not take shockwaves into account. Hence when the critical mach number is reached, the theory is of no importance.[4]
The acuuracy would have been increased if better equipments were used for pressure tappings and if the number of presssure tappings were increased. Because the initial pressure tapping was at x/c, 6.5% of the chord lenght and the final tapping was at x/c,75%. Whereas, important changes take place at the leading and trailing edges when the speed is super-critical. Even though there were some discrepancies, the experiment gave some fairly accurate values in determining the critical mach number and the position of the shockwaves.
To conclude, the aims of the experiment were achieved. The experimental results bear close relationship with the theoretical predictions of the Prandtl-Glauret law upto the critical mach numner (Mcrit). After which they highly differentiate due to the formation of shock waves. It was seen that due to the formation of shockwaves there was a considerable drop in the graphs of Cp, and also the location of the shockwaves moved further downstream as M increased. The experiment gave a better understanding of the behaviour of flow over the aerofoil in the transonic regime. There were some errors in the experiment which included human errors and some approximation of results. More accuracy in the results would have been achieved if the errors mentioned in the discussion were minimized. Finally, even though there were some error, the experiment was a success.
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