Aerodynamics Lab Exercise
Ed Hudson
Feb. 27, 2003
Objective:
This experiment determined the lift characteristics of an airfoil as a function of angle relative to the air stream, and compared the measured data to standard reference values.
The airfoil tested was a standard NACA 4412 configuration. The values obtained were compared to standard reference values.
Theory:
As wind flows over and under an airfoil it creates high and low pressure areas. The shape and size of the airfoil dictates the amount of differential pressure available at a given speed. The amount of “lift force” an airfoil is capable of providing can be roughly calculated using Equation 1. For a more accurate assessment of lift characteristics, other variables must be considered. Equation 3 is a more comprehensive calculation, as it includes a lift coefficient to compensate for variables such as airfoil shape.
Approximate lift force for a typical fixed shape thin airfoil is calculated using the equation:
_{} Equation 1
Where:
L = Lift Force
ρ = air density
U = Velocity
A = planform area of the airfoil
_{} is the dynamic pressure of the fluid.
Note that the density of air varies with altitude, affecting lift.
For a thin airfoils it is customary to use the planform area:
A = BC.
Where:
B is the length of the airfoil
C is the “chord length” (The chord length is the length from the leading edge to the trailing edge.)
Equation 1 represents a very simplified approach to calculating lift. A more comprehensive equation adds a coefficient to characterize the many variables such as body shape, inclination, air viscosity, and compressibility. This variable is called the lift coefficient, (Cl). Experimentation in wind tunnels has determined Cl’s for many different shapes and air conditions. ^{(2)}
For simple flow conditions and geometries such as a flat plate, the value of Cl can be determined mathematically using Equation 2.
_{}. Equation 2
Where:
П=3.14159
a= angle in radians
For more complex geometries such as the NACA 4412 airfoil, this parameter is typically determined experimentally using models in a wind tunnel. The basic equation with the lift coefficient included is:
_{} or _{} Equation 3
Where:
L = Lift Force
ρ = air density
U = Velocity
A = planform area of the airfoil
Cl = Lift Coefficient
Drag is calculated using the equation:
_{} Equation 4
Where:
A = Planform area
ρ = air density
U = Velocity
Cd = Drag Coefficient
Cd is a function of various parameters such as Reynolds number, Re, Mach number, Ma, Froude number, and relative roughness of the surface. ^{(1)}
Typically lift increases as the angle of attack increases up to the point of viscosityinduced boundary layer separation. This point is known as a “stall” condition, and lift is lost.
Typically the lift coefficient (Cl) increases and the drag coefficient decreases with an increase in aspect ratio.
Results:
The NACA 4412 airfoil specimen was tested in the ATU University Mechanical Lab Wind Tunnel at varying angles relative to the flow stream. Table 1 contains the recorded data. Table 2 contains calculated Lift Coefficients for the various angles tested.
Table 1, Collected Data from Wind Tunnel Experiment
Differential Pressure in inches of water (ΔP) 
Angle a 
Lift Transducer Indication 
Corrected Lift Force (set 12.86 lbs to zero) 
Variable Speed Drive Frequency 
Calculated Wind Speed (Velocity) 
Temperature of Air 
0 (no flow) 
0 
12.86 lbs. 
0 
0 
0 
64.3 deg. F 
0.45 
0 
10.88 lbs 
1.98 lbs 
21.7 
36.24 ft/sec 
64.3 deg. F 
0.5 
4 deg. 
10.11 lbs. 
2.75 lbs 
21.7 
36.24 ft/sec 
64.3 deg. F 
0.45 
6 deg. 
9.66 lbs 
3.2 lbs. 
21.7 
36.24 ft/sec 
63.3 deg. F 
0.5 
9 deg. 
9.63 lbs. 
3.23 lbs. 
21.7 
36.24 ft/sec 
63.3 deg. F 
0.55 
11 deg. 
9.62 lbs. 
3.24 lbs 
21.7 
36.24 ft/sec 
64.3 deg. F 
0.45 
12.5 deg 
10.2 lbs 
2.66 lbs 
21.7 
36.24 ft/sec 
64.3 deg. F 
0.45 
14 deg. 
10.7 lbs 
2.16 lbs 
21.7 
36.24 ft/sec 
64.3 deg. F 
0.45 
21 deg. 
10.5 lbs 
2.36 lbs 
21.7 
36.24 ft/sec 
64.3 deg. F 
Table 2, Calculated values obtained from test results
Angle a in radians 
Planform Area “A” A = BC, in^{2} 
Calculated Air Density ρ (interpolated from tabled data using test temperatures)^{(1)} 
Simple Lift Coefficient calculated from Equation 2 _{} 
Recorded Lift Force 
Actual Calculated Lift Coefficient using Equation 3 _{} 
0 
108 
.007528 lbm/ft^{3} 
0 
1.98 lbs 
0.534 
0.0698 
108 
.007528 lbm/ft^{3} 
0.877 
2.75 lbs 
0.742 
0.105 
108 
.007528 lbm/ft^{3} 
0.66 
3.2 lbs. 
0.863 
0.157 
108 
.007528 lbm/ft^{3} 
0.986 
3.23 lbs. 
0.871 
0.192 
108 
.007528 lbm/ft^{3} 
1.2 
3.24 lbs 
0.874 
0.218 
108 
.007528 lbm/ft^{3} 
1.37 
2.66 lbs 
0.717 
0.244 
108 
.007528 lbm/ft^{3} 
1.533 
2.16 lbs 
0.583 
0.367 
108 
.007528 lbm/ft^{3} 
2.31 
2.36 lbs 
0.637 
Note that units for all measurements had to be properly converted and matched for the Lift Coefficient calculation.
Figure 2, Measured Lift vs. Lift Coefficient



Results Discussion:
The results were very similar to the standard reference values for a NACA 4412 airfoil specimen but varied more than instrument uncertainties can account for. Review of measurements taken, reveals that flow was calculated from a pitot tube located in front of the sample. Since the area of the wind tunnel is relatively small, the flow velocity would have been affected by the airfoil itself, and measurements taken at a location just behind the airfoil would likely have revealed slightly different flow velocity.
It can clearly be seen in Figures 3 and 4 when stall occurs, although some slight lift is available as high as the maximum tested angle of 21 deg. It is obvious from Table 2, that the simple lift coefficient calculation described in Equation 2 is very inaccurate for a true airfoil shape.
Figure 5, Test Specimen setup in Wind Tunnel
Figure 6, Drawing of Experiment setup in Wind Tunnel
Uncertainty Calculation:
Dy = Jf/Jx dx + Jf/Jx_{2 }dx_{2 }+_{ }Jf/Jx_{3 }dx_{3 }+ Jf/Jx_{4 }dx_{4 }+ Jf/Jx_{5 }dx_{5}
_{ }
Dy/y = Jf/Jx Dx/y + Jf/Jx_{2 }Dx_{2}/y +_{ }Jf/Jx_{3 }Dx_{3}/y_{ }+ Jf/Jx_{4 }Dx_{4}/y_{ }+ Jf/Jx_{5 }Dx_{5}/y
C_{L} = 2W/rU^2A
C_{D} = 2W/rU^2A
y = C_{L} Coefficient of Lift
x = W Weight
x_{2 }=_{ }r Density of Air
x_{3 }= U Speed
x_{4 }= A Area
Jf/Jx = 2/rU^{2}A
Jf/Jx_{2} = 2W/r^{2}U^{2}A
Jf/Jx_{3} = 4W/rU^{3}A
Jf/Jx_{4} = 2W/rU^{2}A^{2}
Jf/Jx Dx/y = 2/rU^{2}A * DWrU^{2}A/^{ }2W ^{ }=^{ }DW/W
Jf/Jx_{2 }Dx_{2}/y = 2W/r^{2}U^{2}A * DrrU^{2}A/^{ }2W^{ }= Dr/r
Jf/Jx_{3 }Dx_{3}/y = 4W/rU^{3}A * DUrU^{2}A/^{ }2W^{ }= 2DU/U
Jf/Jx_{4 }Dx_{4}/y = 2W/rU^{2}A^{2 }* DrU^{2}A/^{ }2W^{ }= DA/A
Dp/p= DW/W+ Dr/r + 2DU/U + DA/A
References:
(1) Fundamentals of Fluid Mechanics, Third Edition, Munson, Young, Okiishi, Von Hoffmann Press, Inc. 1998
(2) NASA Web Site: http://wright.nasa.gov/airplane/lifteq.html
(3) http://www.allstar.fiu.edu/aero/lift_drag.htm
(4) http://expaircraft.com/library/heintz/airfoils.html Chris Heintz, Zenith Aircraft