Pressure Distribution and Lift on a Piercy Aerofoil.

Chathura Lakmal Hewage
090418138
Contents Introduction 3 Background Theory 3 Aerofoil Design 3 Velocity And Pressure Distribution 4 Angle of Attack and lift 5 The Flow of Fluids 6 Lift 6 Resulting Lifting Force 7 Pressure Distribution 7 Center of Pressure 8 Calculations & Results 8 Discussion 13 Conclusion 13 References 13

Introduction
Aerodynamics is a branch of mechanics concerned with the motion of a fluid continuum the action of applied forces. The motion and general behaviour of a fluid is governed by the fundamental laws of classical mechanics and thermodynamics and plays an important role in such diverse fields as biology, meteorology, chemical engineering, and aerospace engineering. An introductory text on fluid mechanics, such as , surveys the basic concepts of fluid dynamics and the various mathematical models used to describe fluid flow under different restrictive assumptions. The objectives of this experiment are to investigate the way in which the static pressure varies on the surface of an aerofoil in low speed flow, and to deduce the lift force acting on the aerofoil. An aerofoil with a symmetrical section is used for the experiment, which is conducted in a wind tunnel at a wind speed low enough for the flow to be treated as incompressible. From the measured distribution of surface pressure around the aerofoil at a small angle of incidence, the lift will be calculated by numerical integration, and compared with the predictions of inviscid flow theory. The measured chord wise load distribution will also be compared with the results given for inviscid flow by thin aerofoil theory.

Background Theory
Aerofoil Design
A body immersed in a flowing fluid is exposed to both pressure and viscous forces. The sum of the forces that acts normal to the free-stream direction is the lift, and sum that acts parallel to the free-stream direction is the drag. The geometric and dynamic characteristics of aerofoils are shown in figure 1. This experiment is concerned with lift on a stationary aerofoil mounted in the test section of a wind tunnel and consider only two dimensional aerofoils where tip and root effects are neglected.
Because the velocity of the flow over the top of the aerofoil is greater than the free stream velocity. The pressure over the top is negative. This follows directly from the application of Bernoulli's equation. Similarly the velocity along the underside of the aerofoil is less than the free stream velocity and the pressure there is positive. Hence both the negative pressure over the top and the positive pressure along the bottom contribute to the lift.

There are a variety of ways to measure lift. In this experiment the lift force L on the aerofoil will be determine by integration of the measured pressure distribution over the aerofoil's surface. Typical pressure distribution on an aerofoil and its projection on the aerofoil normal are shown in figure 2

Velocity And Pressure Distribution
Velocity and pressure are dependent on each other - Bernoulli's equation says that increasing the velocity decreases the local pressure and vice versa. Thus the higher velocities on the upper airfoil side result in lower than ambient pressure whereas the pressure on the lower side is higher that the ambient pressure. It is possible to plot a pressure distribution instead of the velocity distribution (usually not the pressure, but the ratio of the local pressure to the stagnation pressure is plotted and called pressure coefficient Cp):
.
Summing up the pressure acting on the airfoil results in a total pressure force. Splitting up this total pressure force into a part normal to the flow and another one tangential to the flow direction, results in a lift force L and a drag force D. In some regions, the pressure force acting on the airfoil is of lower pressure than the surrounding pressure and in other regions, it is higher. The following image shows the pressure forces for the E64 at 2° angle of attack. It also shows, how the total force is split into lift and drag forces (the drag force due to pressure is usually very small, see below). The figure also shows, that most of the lift is caused by the low pressure area above the wing.

Pressure force distribution the surface of the Eppler E 64 aerofoil at 2 degrees angle of attack
(result from Drela's XFoil code).
If we had used a symmetrical aerofoil at no incidence, the distribution of velocity and thus the pressures along both surfaces would have been exactly the same, cancelling each other to a resulting total lift force of zero. The calculation of the forces can be performed by summing the pressure distribution of a plot of Cp versus chord. The total force corresponds to the area enclosed between the curves for the upper and the lower surface. It is easy to transform Bernoulli's equation, written once for a point in space far away from the aerofoil (where the velocity is ) and once for a point on the airfoil (velocity v) to arrive at the actual pressure p on the surface of the aerofoil
.
In general and aerofoil does not only create a lifting force, it also creates a moment, which tries to rotate the aerofoil to a different, angle of attack. For comparisons it is generally assumed that the lift force and this torque moment are acting at a fixed point, which is located 25% behind the leading edge on the x-Axis (often called «¼ chord point»). Typical cambered aerofoil create a moment, which tries to reduce the angle of attack (pitch down) - by definition its sign is negative. The moment acting at the ¼ chord point is nearly independent of angle of attack, as long as the flow stays attached to the aerofoil.
If no separation or compressibility effects are present, the pressure field around a two dimensional aerofoil creates a lifting force only, no drag. The drag of a two dimensional aerofoil is created by the friction of the air particles moving close to the surface. The aerofoil is surrounded by a boundary layer, which forms a thin sheet adjacent to the wall where the velocity is reduced from the free stream value down to zero on the wall. For the theoretical treatment of lift, the boundary layer effect is so small that it can be neglected as long as no separation occurs; this applies also to the moment of the aerofoil. But the boundary layer is the main source for the drag in two dimensional flow.
The lift, drag and moment of a wing depends not only on the aerofoil shape and its associated velocity distribution, but also on wing platform and on the wing area. Experiments show, that doubling the wing area or the fluid density also doubles lift and drag, but doubling the air speed yields four times as much lift. The forces and moments also depend on the density of the air and on the shape of the wing. It is possible to compare the aerodynamic properties of differently sized aerofoil or wings, if all forces and moments are normalized. These dimensionless properties (coefficients) are defined as follows: Lift Coefficient | | Drag Coefficient | | Moment Coefficient | |
Knowing these coefficients for a certain aerofoil section at a certain angle of attack, makes it possible to calculate the forces acting on wing sections of different sizes, mounted between walls at different flow velocities and air densities, but at the same angle of attack. To calculate the lift of a "real wing", the local flow conditions, which change from root to tip, have to be taken into account although. For "real wings" and aircraft the same coefficients are used, but the lower case identifiers l, d, and m are replaced by upper case characters L, D and M. If these coefficient are known for an aircraft, the total forces and moments for the complete aircraft can be calculated for different flow conditions.
Angle of Attack and lift
As shown in Figure below, the angle of attack is defined as the angle between the chord line and the relative wind. The angle of attack should not be confused with the angle of incidence which is the angle between the chord and the longitudinal axis of an airplane.

(a) Angle of Attack. (b) Angle of Incidence.

The Flow of Fluids
The ideal flow of fluid about objects is shown in next figure. Although ideal flow does not exist, it is a helpful in developing an understanding of lift and drag.

(a) Ideal flow around a circular cylinder. (b) Ideal flow around an airfoil

Lift
NASA Glenn Research Center published a Lift Theory that claims that the following theory is incorrect.
The author of this article is not a scientist and is merely republishing the lift theory as taught for decades by schools and government agencies.
Interested readers are encouraged to do their own research to form an educated opinion.
The next figure illustrates an uninterrupted airflow over an airfoil. The air above (blue) travels longer distance than the air below(red). The time it takes a particle to travel between A and B is constant, therefore the airflow above the airfoil is faster than below it. By applying the Bernoulli's principle the following is observed:
a. Higher speed - lower pressure above.
b. Lower speed - higher pressure below.

The velocity and pressure as air flows below and above an airfoil
The Glenn Research Center Lift Theory teaches that the following is the largest contributor to lift.
The air below (blue) the airfoil is deflected downward. The deflection has a horizontal and a vertical components(green). By applying Newton's Third Law of Motion we find that a force (red) that is equal in size to the vertical component of the deflection but in the opposite direction is acting on the wing. From Newton's Second Law of Motion F=ma we learn that a change of the velocity or the vertical component generates lift. "Lift is a force generated by turning a moving fluid" - Glenns Research Center Theory.

Lifting effect of the deflection
The differential pressure combined with the reaction force of the deflection (Newton's first law of motion), generates lift. The total lift of the airfoil is noted by a single vector that is perpendicular to the airflow.

Lift Notation
Resulting Lifting Force
As shown in this section, two forces result from the movement of an airfoil through air:
a. Lift
b. Drag
A graphic resolution of these two forces demonstrates a single resultant force that is a sum of the lift and drag. The resulting lifting force is perpendicular to the chord.

Lift and Drag Vector Resolution
Pressure Distribution
The understanding of airflow about aerodynamic objects is required for analyzing their aerodynamic characteristics. Bernoulli's principle does not cover the distribution of pressure above or below an airfoil. Because an airfoil's shape is curved by design, the airflow about it is subject to Circular Motion laws. The Centripetal Force* causes variations in the pressure over an airfoil as shown in next figure.

Momentum Influence Airflow over an Airfoil
The following illustrates a typical pressure distribution over an airfoil as its Angle of Attack varies. The (-) and (+) represent the pressure and the arrows represent the resulting forces.

Pressure Distribution versus Angle of Attack

Center of Pressure
The resultant force (green vector) of the pressures around an airfoil is shown in the next Figure. The point of the application of this force (lift) is noted as the Center of Pressure. For any Angle of Attack, the center of pressure is the point where the resultant force crosses the chord line.
It should be noted that the center of pressure is not a fixed point but is changing with the change in the Angle of Attack. An increase in angle of attack causes the center of pressure to move forward while a decrease in angle of attack moves the center of pressure backwards. The location of the center of pressure with respect to Center of Gravity is an important factor in airplane stability. The effect of the locations of the center of pressure and of the center of gravity will be discussed later.

The Center of Pressure and its Variations as Angle of Attack Changes.
Calculations & Results

1. Using equation one and two ambient air pressure and absolute temperature are calculated. Where ρHg=13.6×103 and tat=22C° Pat= ρHg×g×(Hat×10-3) ............(Eq1)
Pat=13.6×103×9.81×744.5×10-3
Pat=99.328kNm-2

Tat=tat+273.16K ………..(Eq2)
Tat=22+273.16K
Tat=295.16K

2.To calculate density and viscosity for the air in the wind tunnel, assuming that the values are the same as for the ambient atmosphere. these values are calculated using equation 3 and 4 where R = 287 , Tref=288.2, S=110.4 and μref=1.789×10-5 ρ=PRT ..............(Eq3) ρ=99.328×103287×295.16 ρ=1.173kgm-3 μ=μrefTref+ST+STTref32 ..............(Eq4) μ=1.789×10-5288.2+110.4295.16+110.4295.16288.232 μ=1.822×10-5kgm-1s-1

3.Speed of the wind tunnel and Reynolds number is also calculated using equation 5 and 6. V∞=2kPa-Pbρ=2k×ρH2O×g×HBetzρ ...........(Eq5)
V∞=21.03×1000×9.81×32×10-31.173
V∞=21.68ms-1 Rec=ρV∞cμ ...............(Eq6)
Rec=1.173×21.68×0.2541.822×10-5
4.This experiment was done for two angles of incidence all the results in the first table is for 3.3 Degree angle and the second table is for 8.8 Degree angle. The coefficient of pressure was calculated using equation 7 and mainly two graphs were plotted for each angle of incidence such as minus coefficient of pressure against x/c and ∆Cp against x/c. Cpn=ln-lBklA-lB .....................(Eq7) x/c | Manometer Length | Cp Upper | In | Cp Lower | Delta Cp | Delta Cp* x/c | Theoretical Value of Cp | | | | alpha=3.3deg | 0.0575958 | | | | | IA=35.7 | | | | | | | | IB=31.3 | | | | | | | | | | | | | | | 0.0000 | 34.1 | 0.617828773 | 0 | 0 | -0.617828773 | 0 | 0 | 0.0500 | 26.6 | -1.037069726 | 32.4 | 0.242718447 | 1.279788173 | 0.063989409 | 1.004217087 | 0.1000 | 27.4 | -0.86054722 | 31.7 | 0.088261253 | 0.948808473 | 0.094880847 | 0.6911496 | 0.1500 | 28.1 | -0.706090026 | 31.4 | 0.022065313 | 0.72815534 | 0.109223301 | 0.548421711 | 0.2000 | 28.4 | -0.639894086 | 31.2 | -0.022065313 | 0.617828773 | 0.123565755 | 0.4607664 | 0.2500 | 28.6 | -0.59576346 | 31.1 | -0.044130627 | 0.551632833 | 0.137908208 | 0.399035408 | 0.3000 | 28.8 | -0.551632833 | 31 | -0.06619594 | 0.485436893 | 0.145631068 | 0.351916151 | 0.3500 | 29.2 | -0.46337158 | 30.9 | -0.088261253 | 0.375110327 | 0.131288614 | 0.31395938 | 0.4000 | 29.5 | -0.39717564 | 31 | -0.06619594 | 0.3309797 | 0.13239188 | 0.282160643 | 0.4500 | 29.7 | -0.353045013 | 31.1 | -0.044130627 | 0.308914387 | 0.139011474 | 0.254698211 | 0.5000 | 29.8 | -0.3309797 | 31.1 | -0.044130627 | 0.286849073 | 0.143424537 | 0.2303832 | 0.6000 | 30.2 | -0.242718447 | 31.2 | -0.022065313 | 0.220653133 | 0.13239188 | 0.188107095 | 0.7000 | 30.5 | -0.176522507 | 31.3 | 0 | 0.176522507 | 0.123565755 | 0.150821208 | 0.8000 | 30.8 | -0.110326567 | 31.45 | 0.03309797 | 0.143424537 | 0.114739629 | 0.1151916 | 0.9000 | 31.2 | -0.022065313 | 31.6 | 0.06619594 | 0.088261253 | 0.079435128 | 0.0767944 | 1.0000 | | 0 | 31.7 | 0.088261253 | 0.088261253 | 0.088261253 | 0 |

For the 8.8 degree angle of incidence same procedure was carried out and below results were obtained. x/c | Manometer Length | Cp Upper | Manometer Length2 | Cp Lower | Delta Cp | Delta Cp* x/c | Theoretical Value of Cp | 0.0000 | 23.3 | -1.8338727 | | 0 | 1.83387271 | 0 | 0 | 0.0500 | 18.3 | -2.9126214 | 35.3 | 0.75512406 | 3.66774542 | 0.183387271 | 2.677913976 | 0.1000 | 25.5 | -1.3592233 | 34.3 | 0.53937433 | 1.89859763 | 0.189859763 | 1.8430668 | 0.1500 | 26.5 | -1.1434736 | 33.6 | 0.38834951 | 1.53182309 | 0.229773463 | 1.462458849 | 0.2000 | 27.2 | -0.9924488 | 33.2 | 0.30204962 | 1.29449838 | 0.258899676 | 1.2287112 | 0.2500 | 37.7 | 1.27292341 | 32.9 | 0.2373247 | -1.0355987 | -0.258899676 | 1.064095113 | 0.3000 | 28.2 | -0.776699 | 32.6 | 0.17259978 | 0.94929881 | 0.284789644 | 0.93844368 | 0.3500 | 38.7 | 1.48867314 | 32.5 | 0.15102481 | -1.3376483 | -0.468176915 | 0.837225558 | 0.4000 | 29.1 | -0.5825243 | 32.4 | 0.12944984 | 0.71197411 | 0.284789644 | 0.75242887 | 0.4500 | 29.5 | -0.4962244 | 32.4 | 0.12944984 | 0.62567422 | 0.281553398 | 0.679195671 | 0.5000 | 29.8 | -0.4314995 | 32.3 | 0.10787487 | 0.53937433 | 0.269687163 | 0.6143556 | 0.6000 | 30.3 | -0.3236246 | 32.2 | 0.08629989 | 0.40992449 | 0.245954693 | 0.501619247 | 0.7000 | 30.8 | -0.2157497 | 32.1 | 0.06472492 | 0.28047465 | 0.196332255 | 0.402190149 | 0.8000 | 31.3 | -0.1078749 | 32.1 | 0.06472492 | 0.17259978 | 0.138079827 | 0.3071778 | 0.9000 | 31.7 | -0.021575 | 32.1 | 0.06472492 | 0.08629989 | 0.077669903 | 0.2047852 | 1.0000 | | 0 | 32 | 0.04314995 | 0.04314995 | 0.043149946 | 0 |

6.To calculate coefficient of lift for each incidence trapezoidal rule were used this had to done separately due to the spacing between pressure tapping's is not uniform. The multiplied values of ∆Cp and x/c are already shown in the table therefore substituting the values in trapezoidal rule coefficient of lift value were obtained. CL=01∆CPdxc ………………Eq8 for the 3.3 degree incidence CL= CLover the front half+ CLover the rear half CL= 0.057480+0.056598
CL=0.114078
for the 8.8 degree incidence CL= CLover the front half+ CLover the rear half CL= 0.056041+0.081446
CL=0.137487
7. Finally a theoretical value of coefficient of pressure were calculated using the equation 9 and a graph was plotted.(refer graph 4) ∆CP=4α1-xx ......................(Eq9)

8. Finding the value of the lift curve slope dCLd∝, assuming that the graph CL vs. α is a straight line. dCLd∝=0.137487-0.1140780.15358-0.05759=0.24389 Discussion
Before doing any calculations a proper coordinates and numbering of surface pressure tapings on the aerofoil had to be made due to non-uniform spacing on the upper and lower surfaces of the aerofoil. So the first half being 0.05 over the front half of the wind section and 0.1 spacing over the rear half. Then using the manometer data coefficient of pressure was calculated using equation 7 for both upper and lower surface. By aid of excel rest of the required values also calculated such as difference of pressure in both upper and lower surfaces and minus coefficient pressure value by plotting a graphs of -CP againstxcand ∆CP against x/c for both angle of attack as seen in the graph for 3.3 degree angle the experiment graph and theoretical graph are similar so there is not much of a error . But for 8.8 degree angle of attack the graph show a flux from 0.3 to 0.5 maybe due turbulent flow but there is low chance of being the flow turbulent because the experiment was done for low speed such as 21.68m/s or shape of the can be because of a major experimental error or due to stalling but the rest of the graph is fine the experimental values are close to theoretical values. Then using the trapezoidal rule coefficient of lift were calculated this had to be done in two separate part because the pressure taping are not uniform throughout aerofoil, for the first part using step value of 0.05 coefficient of lift was calculated from 0 to 0.5 then using a step value of 0.1 again coefficient of lift were calculated from 0.5 to 1 by adding those to calculated values coefficient of lift was calculated(0.1374). In overall the results for the 3.3 degree angle were within a reasonable range and was close to the theoretical values.(refer graph 2) and for the 8.8 degree angle there is a error in calculated experimental values can be due to human error, turbulent flow or maybe due to stalling practically neither of these can't happen in low speed flow therefore this problem for 8.8 degree angle will be stay unsolved. Also the coefficient of lift did not come out as zero in both the leading and trailing edge but the calculated values were really small, practically there has to some small pressure when x = 0 but the theoretical value come as zero because when x=o the whole equation is zero. Also the lift curve slope was calculated to be 0.24389. In contrast by investigating the graph it can be concluded that throughout the aerofoil most largest pressure distribution and lift act on the first quarter of the aerofoil.
Conclusion
It has been shown that measurements of the pressure distribution over and aerofoil is an appropriate way for measuring the lift under dynamic conditions until the onset of stall reached. In contrast the coefficient of pressure values were expectable anyway by comparing with the theoretical values and by looking at the graphs it can be concluded there is a big error in this experiment maybe due to a calculation error or experimental error because for some unexplainable reason there are pressure flux in the middle of the aerofoil.
References
* Larsen J W 2005 Nonlinear Dynamics of Wind Turbine Wings PhD thesis pp 55 * Gupta S and Leishman J G 2006 Dynamic stall modelling of the S809 aerofoil and comparison with experiments Wind Energy 9 521-47, * Ramsay R R, Hoffman M J, Gregorek G M 1995 Effects of grit roughness and pitch oscillations on the S809 airfoil. Technical Report TP-442-7817 NREL Golden CO * Wernert P and Geissler W 1996 Experimental and numerical investigations of dynamic stall on a pitching airfoil AIAA Journal 3 4 5 982-989 * Althaus D Measurement of lift and drag in the laminar wind tunnel * http://www.iag.uni-stuttgart.de/laminarwindkanal/pdf-dateien/liftdrag2.pdf