Dihedral angle effect

Effect of dihedral on the lift coefficient slope of rectangular wings.

References

1

Katz, J. et al., Low-Speed Aerodynamics, 2nd ed, Cambridge University Press, 2001: figure 12.21

Out:

Elapsed time: 1.1781010627746582 s

import time

import matplotlib.pyplot as plt
import numpy as np

import ezaero.vlm.steady as vlm

start = time.time()

# dihedral angles grid
deltas = np.array([-45, -30, -15, 0, 15, 30]) * np.pi / 180

# define mesh parameters and flight conditions
mesh = vlm.MeshParameters(m=8, n=30)
# slope for each dihedral calculated using two flight conditions
flcond_0 = vlm.FlightConditions(ui=100.0, aoa=0.0, rho=1.0)
flcond_1 = vlm.FlightConditions(ui=100.0, aoa=np.pi / 180, rho=1.0)
cla_list = []  # container for the lift coefficient slope
for delta in deltas:
    # The figure in the book uses an aspect ratio of 4. It does not
    # correspond to the planform, but the "real" wingspan, hence we project
    # the wingspan with the dihedral angle
    bp = 4 * np.cos(delta)
    # define rectangular wing (same cr and ct), with no sweep (theta).
    wing = vlm.WingParameters(
        root_chord=1.0,
        tip_chord=1.0,
        planform_wingspan=bp,
        sweep_angle=0,
        dihedral_angle=delta,
    )
    res_0 = vlm.Simulation(wing=wing, mesh=mesh, flight_conditions=flcond_0).run()
    res_1 = vlm.Simulation(wing=wing, mesh=mesh, flight_conditions=flcond_1).run()
    d_cl = res_1.cl_wing - res_0.cl_wing
    d_alpha = flcond_1.aoa - flcond_0.aoa
    slope = d_cl / d_alpha * np.cos(delta)  # project load
    cla_list.append(slope)

end = time.time()
elapsed = end - start

print("Elapsed time: {} s".format(elapsed))

fig = plt.figure()
plt.plot(deltas * 180 / np.pi, cla_list, "o-")
plt.xlabel(r"$\delta$[deg]")
plt.ylabel(r"CL$_\alpha$")
plt.ylim(0, 4)
plt.grid()
plt.xlim(deltas.min() * 180 / np.pi, deltas.max() * 180 / np.pi)
plt.show()