5 Example - Thin Airfoil Theory

Example: Thin Airfoil Theory The camber line of the NACA 2612 airfoil is represented by two polynomial expressions. The first polynomial provides the camber curve forward of the location of maximum camber, while the second provides the curve aft of this location. The point of maximum camber is denoted as xcmax (here xcmax = 0.6c), while the chord length of the airfoil is c. The variable z, is the distance of the camber line above the chord line. (note: there is additional workspace on the next page and a table of integrals is provided on the last page) 2 z = 0.0555⎡1.2⎛⎜ x ⎞⎟ − ⎛⎜ x ⎞⎟ ⎤ ⎢ ⎝c⎠ ⎝c⎠ ⎥ c ⎣ ⎦

for 0 < x/c < xcmax/c

2 z = 0.125⎡− 0.2 + 1.2⎛⎜ x ⎞⎟ − ⎛⎜ x ⎞⎟ ⎤ ⎢ c ⎝ c ⎠ ⎝ c ⎠ ⎥⎦ ⎣

for xcmax/c < x/c < 1.0

Using thin airfoil theory, find: a) the angle of attack for zero lift ( 4 marks ) b) the lift coefficient at an angle of attack of 5 degrees ( 1 mark ) c) the moment coefficient about the leading edge at α = 5 degrees ( 4 marks ) d) the location of the center of pressure (xcp/c) for the lift coefficient given in b) ( 1 mark ) Solution: 1

Find the location of maximum camber:

cos

1.772 Camber line slopes: ⁄ ⁄

0.0666 ⁄ ⁄

Using the transformation:

0.15

1 1 2 ⁄ ⁄

0.111

0.25

cos

0.0111

0.0555 cos

0.6

⁄ ⁄

0.025

0.125 cos

Part a), αL=0 1

cos

1

0.0555 cos

cos

.

1

0.0111

0.025

0.125 cos

1 cos

1

.

1

0.019434

0.16158

0.04525

2.59

Part b), CL at α = 5o 2

2 0.08726

0.04525

0.8326

Part c), CMLe at α = 5o 2

dz cos dx

.

2

0.0111

0.0555 cos

cos

0.025

0.125 cos

cos

0.125 cos

cos 2

.

2 2

0.054615

0.07334

0.081459

.

0.0111

0.0555 cos

cos 2

0.025 .

2

0.017401

4 Part d), xcp

1 1 4

4

0.03919

0.01387 0.283

0.340