Aerofoil theory personal essay help English assignment help

Abstract In this experiment in a low speed flow the static pressure around an aerofoil will be observed and discussed. The lift on the aerofoil will also be calculated and compared with the theoretical value. The aerofoil being used in this particular experiment is symmetrical and is taking place in a wind tunnel with a speed of 18. 5m/s, therefore the flow is assumed to be incompressible. The different pressures along the surface of the aerofoil will be measured at an angle of attack of 4.

1 degrees and 6. 2 degrees. These values of pressure will then be analysed and graphs and calculations will be produced, the lift being calculated using the trapezium method in excel and these values and graphs will then be compared to the theoretical results for an inviscid flow from the thin aerofoil theory. The errors in the experiment will be quantified and any improvements to the experiment will be discussed. Table of Contents 1. Introduction 2. Experiment Description 3. Apparatus 4.

Calculations and Results 5. Discussion 6. Conclusion 7. References Introduction The thin aerofoil theory is very useful as it relates values of lift to small angles of attack for aerofoils with low camber and thickness without taking into account the viscosity of the flow. The thin aerofoil theory assumes that the flow is 2 dimensional, inviscid and incompressible. It can be used to predict pressures and forces on very thin cambered surfaces with the thickness approaching zero, along with finding the lift.

The assumptions for this theory is that the flow must be inviscid, two dimensional, incompressible, a small angle of attack, the maximum camber less than one, the maximum thickness to chord ratio must be less than one, and the Kutta condition is satisfied. The Kutta condition generally applies to bodies with sharp edges such as the trailing section of an aerofoil. The definition of this condition is that if an aerofoil has a sharp trailing edge moving through a fluid it will create circulation to hold the stagnation point at the trailing edge in place.

The fluid will approach the trailing edge from both surfaces and will flow away from the body with none of the flow remaining attached. Implementing this condition generally means there will be no pressure difference at the trailing edge between the upper and lower surfaces. In the thin aerofoil theory the changes in flow direction are small relative to the free stream and the velocity changes are small. The 2nd order terms in the velocity components are neglected and therefore the result is linear.

An aerofoil generally has a streamlined and thin cross section with a round leading edge to prevent separation of the flow and a sharp trailing edge to fix circulation. The chord of the aerofoil is the straight line joining the leading edge to the trailing edge with a length of c. The mean camber line is the line joining the points halfway between the upper and lower surfaces of the aerofoil. The camber is the biggest distance between the chord and the mean camber line and the greater this distance is the more camber the aerofoil has. The aerofoil thickness is the distance between the upper and lower surfaces.

In this report the theoretical results for an inviscid flow in relation to the thin aerofoil theory will be compared with the experimental results of an aerofoil in a low speed flow. The lift for both cases will be found and graphs will be plotted. This experiment is to test the theory of thin aerofoil with a small angle of attack. Experiment Description In this experiment the pressure distribution around an aerofoil is measured. The aerofoil used is an unswept wing with a constant chord and uniform cross section. This aerofoil is placed inside a 1. 00m by 0.

77m test section of an open return low speed wind tunnel, with a contraction ratio of 5. 6. This aerofoil is mounted on a turntable, so that the angle of attack to the horizontal can be adjusted. The wind tunnel is used in aerodynamics to research the effects of fluids moving around objects. The aerofoil has 30 pressure tappings along its surface, which are connected to tubes, which are then connected to a manometer inclined at 40 degrees. The manometer tubes are connected to tappings alternatively on the upper and lower surface with tapping 1 being at the leading edge and tapping 30 being at the trailing edge.

A Betz manometer is used to measure the tunnel pressure, which is given in mm0, which can then be used to calculate the wind tunnel speed. This pressure was given as 20. 25 mm0. The barometric pressure was recorded to be 761. 5 mmHg, which will be converted into Pascals. The ambient air temperature was taken to be 23. 5 The model is set to the angle of 4. 1 degrees and the 30 readings from the multitube manometer, which are connected to points on the aerofoil surface, are measured and recorded. Along with these readings the two tubes connected to the tunnel reference pressure tappings are recorded.

These steps are repeated but with an angle of attack of 6. 1 degrees. It was found from the experiment that the stall angle was around 8. 8 degrees. This was found by increasing the angle of attack until the pressures around the suction peak collapsed. Apparatus Queen Mary University of London, DEN233, Low Speed Aerodynamics, Lab Handout, (Accessed on 13th November 2013) http://www. grc. nasa. gov/WWW/k-12/airplane/tunoret. html [Accessed on 13th November 2013] Calculations and Results Using the equation the ambient air pressure is calculated to be 101565. 2. This was done by using 13600 as , 9.

807 as g and converting 761. 5 mmHg to N/. =101565. 2 N/ Having a temperature of 23. 5the absolute temperature was calculated using which gives a value of 296. 66 K. The value of is calculated using . By using R as 287. 3, and the values above for P and T, it can be calculated as being 1. 1917. By using the equation , Where = 288. 2 K, S =110. 4 K, and , can be calculated to give a value of The tunnel speed is calculated by Where This gives a value of 198. 59 and when put into the equation equals 18. 53 m/s, where k = 1. 03. The critical Reynolds number is calculated using .

It works out at 306464. 6 By using the equation The pressure coefficient will be found for each of the pressure tappings. is used to get the local chordwise loading where is the lower surface and is the upper surface at a given chord position. The values used will be from pressure tappings 2 and 3 at an angle of incidence of 4. 1, which are values of 11. 7 and 9. 8 inches, from the lower and upper surface respectively. These values are first converted from inches to millimetres by doing . This gives a value of 297. 18 mm. The same is done for the upper surface, , to give a value of 248. 92 mm.

The values of La and Lb are found from the free stream pressure tappings, and these are then converted in the same way to millimetres. The values for La and Lb are 8. 6 and 10. 1 and when converted are 218. 44 mm and 256. 54 mm. The pressure coefficient is then found using the equation above where k=1. 03, and this is found for both lower and upper surfaces. All the values for Cp are then inversed and a graph is plotted. Then to find these are subtracted to give a value of 1. 2298. 0. 1942-(-1. 0356)=1. 2298 The rest of these values are found in the same way and the graph of is plotted against x/c.

The trapezium rule is used in excel with the formula of , where A is the pressure coefficient and B is x/c. All these values are then added together to find the lift. The theoretical result is calculated by using the equation Where is the angle of incidence in radians and x is x/c. Once is found a graph of this against x/c is to be plotted. The lift is calculated using the same method as above, the trapezium rule. Using alpha as 4. 1 in radians is 0. 071558 is calculated to be 1. 248 with a value of 0. 05 for x/c. Figure 1 Figure 2 Lift Coefficient Experimental Theoretical Error 4. 1 0. 32686 0. 36360 10. 1% 6. 1 0.

45146 0. 53674 15. 9% Table 1 Discussion From the graph above in figure 1 of at different angles of attack some differences can be seen. When we compare both angles we can see that in both cases the experimental model does not correspond with the theoretical. The graphs show the difference in pressure along the aerofoil so the area under the curve will represent the lift. When comparing experimental and theoretical it can be observed that the theoretical line is very smooth with a higher suction peak whereas the experimental is very uneven. This theoretical smoothness is due to the viscosity being completely ignored.

However this is very unrealistic as viscosity will be present in the experimental flow. This viscosity reduces the velocity on the upper surface of the aerofoil, it removes the kinetic energy and therefore there is a greater loss in energy on the top surface compared with the bottom. This reduction in velocity causes a higher pressure according to Bernoulli’s law and therefore this higher pressure causes a lower lift, which corresponds with the results. So therefore if no viscosity is present the flow can accelerate without impedance of a boundary layer and therefore pressure will drop causing a higher lift.

In the graph above a frictionless inviscid flow is shown for theoretical meaning that viscosity has been neglected. This is typically an ideal fluid moving slowly with no boundary layer at the surface and no wake as the flow leaves the aerofoil. If there is no boundary layer then there will be no separation taking place and hence the flow will be entirely attached. This theory also means that there will be no drag on the aerofoil. This ideal fluid does not occur naturally and is very unrealistic as a small amount of viscosity is always present in a fluid.

Due to the fact that in the experiment the flow is not frictionless, viscosity will be present and it will remove energy from the flow and hence reduce the lift. If we think about an aerofoil with zero angle of attack there will be no lift. This is because the stagnation point will be at the very centre of the leading edge and the flow will accelerate at a constant rate over both sides of the aerofoil. This means the velocity will be equal on upper and lower surfaces and therefore the pressure is equal hence no lift is generated.

However this is only true for symmetric aerofoils. If a cambered aerofoil is tested the upper surface will be larger than the lower surface. This means that from the stagnation point at zero angle of attack the flow has further to travel along the upper surface, keeping the same time with the flow on the lower surface, which means a higher velocity occurs on the upper surface and lower pressure and this generates lift. Cambered aerofoils are generally used to maximise lift and minimise the stalling speed. Since our aerofoil was symmetrical it produced a lower lift.

As the angle of attack increases the lift will too increase in both the experimental and theoretical cases. A higher angle of attack means that air is deflected through a larger angle hence the vertical component of velocity increases resulting in a greater lift. The higher angle of attack produced a greater pressure difference, which is shown from the high peak just after the leading edge. In both cases the theoretical result produces a higher lift, which can be shown, as those lines are both higher than the experimental, and because the lift is the

area under the graph this will mean a larger area. This increase in lift will continue to happen until the critical angle of attack where the flow will become more turbulent and separation on the upper surface of the aerofoil becomes more pronounced which means there is less deflection downward so the increase in lift stops. This is called the stall angle of attack, in which it produces the maximum lift coefficient. Above this stall angle the flow will become more turbulent over the upper surface and the air separates, disrupting the flow.

The increase in angle of attack will increase lift due to the air rapidly accelerating over the top surface of the aerofoil, increasing the velocity and reducing the pressure. The flow reaches the stagnation point, which is now on the lower surface of the aerofoil, and the velocity is reduced to zero. As this point is directed on the lower surface the flow has more distance to travel over the top surface of the wing in a shorter amount of time as the air that is on the lower surface, therefore the flow has to increase its velocity therefore reducing pressure hence increasing lift.

In the –Cp graph in figure 2 it can be seen that looking at an angle of attack of 4. 1, on the upper surface of the aerofoil there is a low peak just after the leading edge where the pressure has dropped. After this peak the pressure does increase slightly and levels out as the flow passes over the aerofoil. On the lower surface there is a much higher pressure after the leading edge and this pressure decreases over time. For the angle of attack of 6. 1 it can be seen that the pressure on the upper surface is lower all along the aerofoil and on the lower surface the pressure is greatly increased.

In the case where the angle of attack is 6. 1, the pressure difference is shown to be greater than when the angle of attack is 4. 1 hence this means a greater lift, which corresponds with our lift values. As the pressures move along the aerofoil it can be seen that towards the trailing edge they become very similar which means that there is a little pressure difference there which corresponds with the Kutta condition, where there is no pressure difference between the upper and lower surfaces at the trailing edge.

The value of Cp should be highest at the stagnation point as this is where the velocity is reduced to zero hence should have the lowest pressure. Since the stagnation point is more directed on the lower surface of the aerofoil and not in the centre, the pressure drop happens just after zero chord length, which can be observed, on the graph. If the error between the experimental and theoretical results are considered, it can be seen by looking at the differences in lift in both cases. From the table it can be seen that for the angle of attack of 4.

1 degrees the theoretical result had an increase in lift of 10. 1% and moving to 6. 1 degrees this increased to 15. 9%. These results show that the theoretical produced a higher lift and therefore are the more accurate and preferable results. However it was to be expected that experimental results would produce some error whether it be human error or due to viscosity. It can be seen that as the angle of attack was increased the error in determining lift became greater. This could be a fault in the measurements of the pressure tappings.

It could also be because as there is a higher angle of attack the velocity moving over the upper surface of the aerofoil is larger and therefore the flow may become more turbulent and this causes problems such as separation of the boundary layer. The major source of inaccuracy in this experiment, which could account for the errors mentioned above, is a parallax error. When readings are taken of the pressure it should be made certain that the eye is level with the top of the liquid. To avoid this error a number of pressure readings could be taken to increase the reliability and an average could be taken.

More pressure tappings could have been added over the surface of the aerofoil and different angles of attack could be used to increase accuracy. When measurements are taken with the multitube manometer a digital sensor could have been used with an increased degree of accuracy. The measurements that were recorded were only measured to one degree of accuracy whereas a sensor could measure to two or three degrees making the result of the lift coefficient much more precise. To increase the overall reliability of the experiment and to decrease or eliminate any errors that occurred, it could

have been repeated a number of times for a greater accuracy. Conclusion From this report it can be concluded that as the angle of attack for an aerofoil increases the lift will also increase due to the greater pressure difference between the lower and upper surfaces of the aerofoil. This lift will continue to increase until the stall angle where the flow becomes turbulent and separation occurs. When comparing theoretical flows with experimental it was discussed that theoretical had the higher lift. This is because it was assumed by thin aerofoil theory that this flow was inviscid and frictionless.

This viscosity, which was present in the experimental, reduced the lift because the viscous forces caused a boundary layer, which impeded the flow, reducing the velocity and therefore increasing the pressure on the upper surface. This means that there is a decreased pressure difference between the two surfaces and therefore a decrease in lift. The results correspond with the theory however there were sources of error in the experiment as discussed previously, which could have been improved on. The thin aerofoil theory cannot predict drag as it is assumed to be an inviscid flow, therefore drag does not occur.