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 viscosity-induced 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, in2 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/ft3 0 1.98 lbs 0.534 0.0698 108 .007528 lbm/ft3 0.877 2.75 lbs 0.742 0.105 108 .007528 lbm/ft3 0.66 3.2 lbs. 0.863 0.157 108 .007528 lbm/ft3 0.986 3.23 lbs. 0.871 0.192 108 .007528 lbm/ft3 1.2 3.24 lbs 0.874 0.218 108 .007528 lbm/ft3 1.37 2.66 lbs 0.717 0.244 108 .007528 lbm/ft3 1.533 2.16 lbs 0.583 0.367 108 .007528 lbm/ft3 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

 Figure 3, Angle of Attack vs. Lift Coefficient

 Figure 4, Lift Force vs. Angle

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/Jx2 dx2 + Jf/Jx3 dx3 + Jf/Jx4 dx4 + Jf/Jx5 dx5

Dy/y =  Jf/JDx/y + Jf/JxDx2/y + Jf/JxDx3/y + Jf/JxDx4/y + Jf/JxDx5/y

CL = 2W/rU^2A

CD = 2W/rU^2A

y = CL  Coefficient of Lift

x = W  Weight

x2 = r   Density of Air

x3 = U  Speed

x4 = A  Area

Jf/Jx  = 2/rU2A

Jf/Jx2 = 2W/r2U2A

Jf/Jx3 = 4W/rU3A

Jf/Jx4 = 2W/rU2A2

Jf/JDx/y = 2/rU2A * DWrU2A/ 2W  = DW/W

Jf/JxDx2/y = 2W/r2U2A * DrrU2A/ 2W  = Dr/r

Jf/JxDx3/y = 4W/rU3A  * DUrU2A/ 2W  = 2DU/U

Jf/JxDx4/y = 2W/rU2A2 * DrU2A/ 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