.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/wings/plot_01_prandtl_lifting_line.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_aerodynamics_wings_plot_01_prandtl_lifting_line.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_aerodynamics_wings_plot_01_prandtl_lifting_line.py:


Prandtl's lifting-line theory and the finite-wing lift slope
============================================================

Ludwig Prandtl's *Tragflügeltheorie* (1918) modelled a wing as a bound vortex
of varying strength shedding a sheet of trailing vortices. The downwash they
induce lowers the effective angle of attack, so a finite wing's lift slope is
:math:`a = a_0/(1 + a_0/(\pi e AR))`
(:func:`~engineeringkit.aerodynamics.finite_wing_lift_slope`). This example
solves Glauert's Fourier form of the lifting-line equation for a rectangular
wing and compares the computed slope with the closed form.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import finite_wing_cl, finite_wing_lift_slope








.. GENERATED FROM PYTHON SOURCE LINES 21-29

Solve the monoplane equation for a rectangular wing
---------------------------------------------------

With :math:`\Gamma = 2bV\sum A_n\sin n\theta`, collocation at stations
:math:`\theta_k` gives
:math:`\sum_n A_n \sin n\theta_k\,(\mu n + \sin\theta_k) = \mu\alpha\sin\theta_k`
with :math:`\mu = a_0 c/(4b)`. Then :math:`C_L = \pi AR\,A_1` and
:math:`e = 1/(1 + \sum_{n>1} n A_n^2/A_1^2)`.

.. GENERATED FROM PYTHON SOURCE LINES 29-50

.. code-block:: Python



    def rectangular_wing(AR, alpha=0.1, n_terms=15, a0=2 * np.pi):
        n = np.arange(1, 2 * n_terms, 2)
        theta = np.linspace(0, np.pi / 2, n_terms + 1)[1:]
        mu = a0 / (4 * AR)
        M = np.sin(np.outer(theta, n)) * (mu * n + np.sin(theta)[:, None])
        A = np.linalg.solve(M, mu * alpha * np.sin(theta))
        cl = np.pi * AR * A[0]
        e = 1 / (1 + np.sum(n[1:] * (A[1:] / A[0]) ** 2))
        return cl / alpha, e


    aspect = np.linspace(2, 20, 37)
    slopes, efficiencies = np.array([rectangular_wing(ar) for ar in aspect]).T
    closed_form = finite_wing_lift_slope(aspect, efficiencies)
    print(f"AR = 8 rectangular wing: e = {efficiencies[12]:.3f}, slope {slopes[12]:.3f} vs closed form {closed_form[12]:.3f} per rad")
    # The closed form uses the drag-efficiency e in place of the lift-slope
    # factor tau; they agree to within a few percent at moderate AR.
    assert np.allclose(slopes, closed_form, rtol=0.03)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    AR = 8 rectangular wing: e = 0.937, slope 4.838 vs closed form 4.959 per rad




.. GENERATED FROM PYTHON SOURCE LINES 51-53

Slope approaches 2π only as the span becomes infinite
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 53-68

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(9, 3.6))
    axes[0].plot(aspect, slopes, "o", ms=3, label="lifting-line solution (rectangular)")
    axes[0].plot(aspect, finite_wing_lift_slope(aspect), label="closed form, e = 1")
    axes[0].axhline(2 * np.pi, color="gray", ls="--", label="2π (2-D section)")
    axes[0].set(xlabel="aspect ratio", ylabel="lift slope [1/rad]")
    axes[0].legend()
    alpha = np.radians(np.linspace(0, 10, 50))
    for ar in [4, 8, 30]:
        axes[1].plot(np.degrees(alpha), finite_wing_cl(alpha, ar), label=f"AR = {ar}")
    axes[1].set(xlabel="angle of attack [deg]", ylabel="$C_L$", title="Lower slope for stubby wings")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/aerodynamics/wings/images/sphx_glr_plot_01_prandtl_lifting_line_001.png
   :alt: Lower slope for stubby wings
   :srcset: /api/gallery/aerodynamics/wings/images/sphx_glr_plot_01_prandtl_lifting_line_001.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.216 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_wings_plot_01_prandtl_lifting_line.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/aerodynamics/wings/plot_01_prandtl_lifting_line.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_prandtl_lifting_line.ipynb <plot_01_prandtl_lifting_line.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_prandtl_lifting_line.py <plot_01_prandtl_lifting_line.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_prandtl_lifting_line.zip <plot_01_prandtl_lifting_line.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_