Plotting a NACA 4-series airfoil

I’m trying to plot an airfoil from the formula as described on this wikipedia page.

This Jupyter notebook can be viewed on this github page.

%matplotlib inline
import math
import matplotlib.pyplot as pyplot

def frange( start, stop, step ):
    yield start
    while start <= stop:
        start += step
        yield start

#https://en.wikipedia.org/wiki/NACA_airfoil#Equation_for_a_cambered_4-digit_NACA_airfoil
def camber_line( x, m, p, c ):
    if 0 <= x <= c * p:
        yc = m * (x / math.pow(p,2)) * (2 * p - (x / c))
    #elif p * c <= x <= c:
    else:
        yc = m * ((c - x) / math.pow(1-p,2)) * (1 + (x / c) - 2 * p )
    return yc

def dyc_over_dx( x, m, p, c ):
    if 0 <= x <= c * p:
        dyc_dx = ((2 * m) / math.pow(p,2)) * (p - x / c)
    #elif p * c <= x <= c:
    else:
        dyc_dx = ((2 * m ) / math.pow(1-p,2)) * (p - x / c )
    return dyc_dx

def thickness( x, t, c ):
    term1 =  0.2969 * (math.sqrt(x/c))
    term2 = -0.1260 * (x/c)
    term3 = -0.3516 * math.pow(x/c,2)
    term4 =  0.2843 * math.pow(x/c,3)
    term5 = -0.1015 * math.pow(x/c,4)
    return 5 * t * c * (term1 + term2 + term3 + term4 + term5)

def naca4( m, p, t, c=1 ):
    for x in frange( 0, 1.0, 0.01 ):
        dyc_dx = dyc_over_dx( x, m, p, c )
        th = math.atan( dyc_dx )
        yt = thickness( x, t, c )
        yc = camber_line( x, m, p, c )
        xu = x  - yt * math.sin(th)
        xl = x  + yt * math.sin(th)
        yu = yc + yt * math.cos(th)
        yl = yc - yt * math.cos(th)
        yield (xu, yu), (xl, yl)

#naca2412 
m = 0.02
p = 0.4
t = 12

naca4points = naca4( m, p, t )
for (xu,yu),(xl,yl) in naca4points:
    pyplot.plot( xu, yu, 'r,')
    pyplot.plot( xl, yl, 'r,')
pyplot.ylabel('y')
pyplot.xlabel('x')
pyplot.axis('equal')
figure = pyplot.gcf()
figure.set_size_inches(16,16,forward=True)

The result looks like .

I expected it to look more like .

Questions: Why is the line not completely smooth? There seems to be a discontinuity where the beginning and end meet. Why does it not look like the diagram on wikipedia? How do I remove the extra loop at the trailing edge? How do I fix the chord so that it runs from 0.0 to 1.0?

Answer

First, t should be 0.12 not 12. Second, to make a smoother plot, increase the sample points.

It is also a good idea to use vectorize method in numpy:

%matplotlib inline
import math
import matplotlib.pyplot as plt
import numpy as np

#https://en.wikipedia.org/wiki/NACA_airfoil#Equation_for_a_cambered_4-digit_NACA_airfoil
def camber_line( x, m, p, c ):
    return np.where((x>=0)&(x<=(c*p)),
                    m * (x / np.power(p,2)) * (2.0 * p - (x / c)),
                    m * ((c - x) / np.power(1-p,2)) * (1.0 + (x / c) - 2.0 * p ))

def dyc_over_dx( x, m, p, c ):
    return np.where((x>=0)&(x<=(c*p)),
                    ((2.0 * m) / np.power(p,2)) * (p - x / c),
                    ((2.0 * m ) / np.power(1-p,2)) * (p - x / c ))

def thickness( x, t, c ):
    term1 =  0.2969 * (np.sqrt(x/c))
    term2 = -0.1260 * (x/c)
    term3 = -0.3516 * np.power(x/c,2)
    term4 =  0.2843 * np.power(x/c,3)
    term5 = -0.1015 * np.power(x/c,4)
    return 5 * t * c * (term1 + term2 + term3 + term4 + term5)

def naca4(x, m, p, t, c=1):
    dyc_dx = dyc_over_dx(x, m, p, c)
    th = np.arctan(dyc_dx)
    yt = thickness(x, t, c)
    yc = camber_line(x, m, p, c)  
    return ((x - yt*np.sin(th), yc + yt*np.cos(th)), 
            (x + yt*np.sin(th), yc - yt*np.cos(th)))


#naca2412 
m = 0.02
p = 0.4
t = 0.12
c = 1.0

x = np.linspace(0,1,200)
for item in naca4(x, m, p, t, c):
    plt.plot(item[0], item[1], 'b')

plt.plot(x, camber_line(x, m, p, c), 'r')
plt.axis('equal')
plt.xlim((-0.05, 1.05))
# figure.set_size_inches(16,16,forward=True)