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Joukowski_airfoil_mapping.py
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Joukowski_airfoil_mapping.py
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import numpy as np
import math
import matplotlib.pyplot as plt
from scipy import interpolate
from scipy import integrate
def joukowski(beta, a, c, alpha):
# define constants
i = np.sqrt(-1+0j)
U_inf = 1
rho = 1
# meshing/plotting parameters
D = 0.7
dx = 0.01
fs = 18
dtheta = 0.01
# calculate center of circle
zeta0 = a - c * np.exp(-i*beta)
# form matrix of values in computational domain
something = int(round(2.0*D/dx+1))
nx, ny = (something, something)
xi1 = np.linspace(-D,D,nx)
eta1 = np.linspace(-D,D,nx)
xi, _ = np.meshgrid(xi1, eta1)
eta = np.transpose(xi)
zeta = xi + i * eta
for j in range(0,len(xi)):
for k in range(0,len(eta)):
if abs(zeta[j][k]-zeta0) < c:
zeta[j][k] = float('nan')
# create subplots
f, ((ax1, ax2, ax3),(ax4, ax5, ax6)) = plt.subplots(2, 3)
# compute flow solution in computational domain
gamma = 4*math.pi*U_inf*c*math.sin(alpha+beta)
print 'Circulation:', gamma
# F is complex, stores Psi and Phi, F = Phi + i*Psi
F = (U_inf*np.exp(-i*alpha)*((zeta-zeta0) + (c**2*np.exp(2*i*alpha))/(zeta-zeta0)) + (i*gamma)/(2*np.pi)*np.log((zeta-zeta0)*np.exp(-i*alpha)/c))
# plot the cylinder surface in computational domain
theta = np.linspace(0,2*math.pi,100)
circle = zeta0 + c*np.exp(i*theta)
ax1.plot(np.real(circle), np.imag(circle))
ax1.set_xlim(-D,D), ax1.set_ylim(-D,D)
ax1.set_xlabel('$\\xi$')
ax1.set_ylabel('$\\eta$')
ax1.set_title('Comp. Domain')
# plot streamlines in computational domain
# (lines of constant Psi)
levels = [-0.4, -0.2, 0.0, 0.2, 0.4, 0.6]
cp2 = ax2.contour(np.real(zeta),np.imag(zeta),np.imag(F), levels, colors='k')
ax2.contourf(np.real(zeta),np.imag(zeta),np.imag(F))
ax2.clabel(cp2, inline=True, fontsize=10, colors='k')
ax2.plot(np.real(circle), np.imag(circle))
ax2.set_xlim(-D,D), ax2.set_ylim(-D,D)
ax2.set_xlabel('$\\xi$', fontsize=16)
ax2.set_ylabel('$\\eta$', fontsize=16)
ax2.set_title('Comp. Domain')
# Joukowski: perform the calculation here that
# finds the mapped points in the z-plane
zT = zeta + a**2 / zeta
# plot the airfoil surface in physical domain
airfoil = circle+a**2/circle
ax3.plot(np.real(airfoil), np.imag(airfoil))
ax3.set_xlim(-D,D), ax3.set_ylim(-D,D)
ax3.set_xlabel('x', fontsize=16)
ax3.set_ylabel('y', fontsize=16)
ax3.set_title('Physical Domain')
# plot streamlines in physical domain
# (lines of constant psi)
cp4 = ax4.contour(np.real(zT),np.imag(zT),np.imag(F), levels, colors='k')
ax4.contourf(np.real(zT),np.imag(zT),np.imag(F))
ax4.clabel(cp4, inline=True, fontsize=10, colors='k')
ax4.plot(np.real(airfoil), np.imag(airfoil))
ax4.set_xlim(-D,D), ax4.set_ylim(-D,D)
ax4.set_xlabel('x', fontsize=16)
ax4.set_ylabel('y', fontsize=16)
ax4.set_title('Physical Domain')
# find chord length
xi_min = np.amin(np.real(circle))
eta_min = np.imag(circle)[np.argmin(np.real(circle))]
xi_max = np.amax(np.real(circle))
eta_max = np.imag(circle)[np.argmax(np.real(circle))]
x_1 = xi_min + a**2 / xi_min
y_1 = eta_min + a**2 / eta_min
x_2 = xi_max + a**2 / xi_max
y_2 = eta_max + a**2 / eta_max
chord_length = np.real(np.sqrt((x_2-x_1)**2 + (y_2-y_1)**2))
print 'Chord Length:' , chord_length
print 'Angle of Attack', alpha, 'rads'
# plotting u and v
# differentiate F
dFdzeta = U_inf*np.exp(-i*alpha) - U_inf*a**2*np.exp(i*alpha)/(zeta - zeta0)**2 + i*gamma/(2*math.pi*(zeta - zeta0))
# u and v
velsU = np.real(dFdzeta)
velsV = -np.imag(dFdzeta)
# mag u and v
surface_vels = np.sqrt(velsV**2+velsU**2)
# find values on the shape
values = []
for j in range(0,len(surface_vels)):
if np.isnan(surface_vels[j]).any():
row = surface_vels[j]
for k in range(0,len(row)-1):
if np.isfinite(row[k]) and (np.isnan(row[k+1]) or np.isnan(row[k-1])):
values.append((xi1[k],eta1[j],surface_vels[j][k]))
values.sort(key=lambda x: x[0])
xi3 = []
eta3 = []
vels_values = []
for elmt in values:
xi3.append(elmt[0])
eta3.append(elmt[1])
vels_values.append(elmt[2])
xi3 = np.asarray(xi3)
eta3 = np.asarray(eta3)
vels_values = np.asarray(vels_values)
# map values to physical domain & plot
x3 = xi3 * (xi3**2+eta3**2+a**2)/(xi3**2+eta3**2)
ax5.scatter(x3, vels_values)
ax5.set_xlabel('x', fontsize=16)
ax5.set_ylabel('Velocity', fontsize=16)
ax5.set_title('Surface Velocities')
# plot Cp as a function of x/c, one curve for top of airfoil,
# another curve for bottom of airfoil
Cp = 1 - (vels_values / U_inf)**2
ax6.scatter(x3/c, Cp)
ax6.set_xlabel('x/c', fontsize=16)
ax6.set_ylabel('Coeff. of Pressure', fontsize=16)
ax6.set_title('x/c vs. Cp')
#plt.show()
# Integrate Cp on the airfoil to find the Lift
# & Compare with Theory
Cl = get_Cl(gamma, a, U_inf, alpha)
Lift = Cl * 0.5 * rho * U_inf**2
AnalyticalLift = rho*U_inf*gamma
print 'Lift:', Lift
print 'Analytical Lift:', AnalyticalLift
return AnalyticalLift/(0.5 * rho * U_inf**2)
# compute the coefficient of lift
def get_Cl(Gamma, R, U, alpha):
top = integralTopX(-R,R,Gamma,R,U,alpha)
bottom = integralBottomX(-R,R,Gamma,R,U, alpha)
return 1.0/(2.0*float(R)) * (bottom - top)
# integrate the top half of the cylinder in the x-dir
def integralTopX(x1, x2, Gamma, R,U, alpha):
def integr(x):
y = math.sqrt(R**2-x**2)
theta = math.atan2(y,x) - alpha
u_theta = -2*U*math.sin(theta) - Gamma/(2*math.pi*R)
return 1.0 - (u_theta/U)**2
return integrate.quad(integr, x1, x2)[0]
# integrate the bottom half of the cylinder in the x-dir
def integralBottomX(x1, x2, Gamma, R,U, alpha):
def integr(x):
y = -math.sqrt(R**2-x**2)
theta = math.atan2(y,x) - alpha
u_theta = -2*U*math.sin(theta) - Gamma/(2*math.pi*R)
return 1.0 - (u_theta/U)**2
return integrate.quad(integr, x1, x2)[0]
joukowski(0.35, 0.1, 0.115, 0)