Assignment 1
I this assignment, you are going to use the thin airfoil theory and the XFOIL software to compute
airfoil characteristics of the NACA four-digit airfoil sections. The coordinates of the NACA four-digit
airfoil first derived in 1932 with the intention of having some simple parameters to design airfoils
that resembled some of the most popular airfoils used at that time (see Abbott and Doenhoff [1]).
The thickness distribution, which was selected to correspond closely to the popular Clark Y and
Göttingen airfoils, is given by the following equation:
yt  5ct 0.2969   0.1260  0.3516 2  0.2843 3  0.1015 4  ,
(1)
Where c is the chord length, t is the maximum thickness expressed as a fraction of the chord, and
  x / c is the non-dimensional coordinate along the airfoil, going from zero at the leading edge to
1 at the trailing edge. The shape of the camber line is expressed analytically as two parabolic arcs,
which are tangent at the position of the maximum camber line ordinate. The equations defining the
camber line are
yc 
mc
2 p   2  ,   p;
2 
p
yc 
mc
1  p 
2
1  2 p   2 p   2  ,   p ,
(2)
Where m is the maximum value of yc , expressed as a fraction the chord, and p is the value of x/c
corresponding to this maximum.
The numbering system for the four-digit airfoils is based on the geometry. The first integer equals
100m, the second equals 10p, and the final two taken together equal 100t. Thus the NACA 4412
airfoil has 4% camber, at x=0.4c from the leading edge and is 12% thick.
1. Use thin airfoil theory to determine an expression for the lift coefficient, Cl  Cl ( ) , where
 is the angle of attack, and the moment coefficient around the c/4-point, CM c/4 . If you do it
both numerically and analytically, you can validate the numerical solution against the exact
analytical value.
2. Based on the curvature of the camber line of the NACA 44xx airfoil, compute the resulting
pressure coefficient distribution along the airfoil, Cp 
pu  pl
, for angles of attack,
½ U 2
  50 ,  2.50 , 00 , 2.50 , 50 , and plot the results as curves on a common figure. Here 
denotes the density of air, U is the freestream velocity, and pu  pl is the difference
between the pressure on the upper and lower sides of the camber line.
3. For an angle of attack   50 , use XFOIL to compute the inviscid pressure distributions,
p0  p
, for NACA 4406, 4412 and 4415 airfoils. Here p0 denotes the static
½ U 2
undisturbed pressure. Plot the pressure distributions C p  C p ( x / c) on a common figure.
Cp 
4. From the pressure distributions computed by XFOIL, determine Cp 
pu  pl
for the three
½ U 2
different airfoils. Plot the distributions together with the one obtained from thin airfoil
theory and discuss the accuracy of the approximation of the thin airfoil theory for airfoils of
finite thickness.
5. Use XFOIL to compute lift and drag coefficient distributions for the 4418 airfoil at Reynolds
numbers 105, 106 and 5x106 for angles of attack going from -100 to 300. Plot and compare the
polars on the same figure. Try different transition criteria and assess the impact of these on
the behavior of the aerodynamic coefficients. Compare the results to measured data and
discuss possible differences. Search the open literature for relevant experimental sources.
Literature
[1] Abbott, I.H. and Doenhoff, A.E. ‘Theory of wing sections’. New York. Dover. 1959.