.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/wings/plot_04_glauert_fourier_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_04_glauert_fourier_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_04_glauert_fourier_lifting_line.py:


Glauert's Fourier solution of the lifting line for any planform (1926)
======================================================================

Prandtl's lifting-line equation is an integro-differential equation for the
spanwise circulation. In *The Elements of Aerofoil and Airscrew Theory*
(1926) Hermann Glauert solved it for any planform. He wrote the circulation
as a Fourier sine series in :math:`\theta`, with
:math:`y = -\tfrac b2\cos\theta`, and enforced the equation at as many
stations as there are terms (:func:`~engineeringkit.aerodynamics.lifting_line`).
The first coefficient fixes the lift and the others add induced drag, so the
span efficiency is :math:`e = 1/(1 + \delta)` with
:math:`\delta = \sum_{n>1} nA_n^2/A_1^2`. This example recovers the elliptic
optimum exactly, then sweeps the taper ratio of straight wings to reproduce
Glauert's induced-drag factor :math:`\delta`.

.. GENERATED FROM PYTHON SOURCE LINES 19-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import finite_wing_cl, lifting_line

    b, alpha = 10.0, np.radians(5.0)








.. GENERATED FROM PYTHON SOURCE LINES 27-29

The elliptic wing has uniform downwash and e = 1
------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-35

.. code-block:: Python


    elliptic = lifting_line(b, lambda y: 1.2 * np.sqrt(1 - (2 * y / b) ** 2), alpha)
    print(f"elliptic: AR = {elliptic.aspect_ratio:.2f}, CL = {elliptic.CL:.4f}, e = {elliptic.span_efficiency:.6f}")
    assert np.isclose(elliptic.span_efficiency, 1.0)
    assert np.isclose(elliptic.CL, finite_wing_cl(alpha, elliptic.aspect_ratio))





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

 .. code-block:: none

    elliptic: AR = 10.61, CL = 0.4613, e = 1.000000




.. GENERATED FROM PYTHON SOURCE LINES 36-43

Taper ratio and Glauert's delta
-------------------------------

For a fixed aspect ratio of 8, a straight tapered wing with tip-to-root chord
ratio :math:`\lambda` near 0.4 loads almost elliptically. A rectangular wing
(:math:`\lambda = 1`) wastes about 6% and a pointed one (:math:`\lambda = 0`)
far more.

.. GENERATED FROM PYTHON SOURCE LINES 43-69

.. code-block:: Python


    AR = 8.0
    taper = np.linspace(0.0, 1.0, 41)
    delta = []
    for lam in taper:
        root = 2 * b / (AR * (1 + lam))
        wing = lifting_line(b, lambda y, r=root, t=lam: r * (1 - (1 - t) * np.abs(2 * y / b)), alpha, n_terms=60)
        delta.append(1 / wing.span_efficiency - 1)
    delta = np.array(delta)
    best = taper[np.argmin(delta)]
    print(f"AR = 8: delta = {delta[-1]:.4f} rectangular, minimum {delta.min():.4f} at taper ratio {best:.3f}")
    assert 0.25 < best < 0.45 and delta.min() < 0.02

    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    axes[0].plot(taper, delta)
    axes[0].set(xlabel=r"taper ratio $\lambda = c_t/c_r$", ylabel=r"induced-drag factor $\delta$", title="AR = 8 straight wings")
    for lam in (1.0, 0.35, 0.0):
        root = 2 * b / (AR * (1 + lam))
        wing = lifting_line(b, lambda y, r=root, t=lam: r * (1 - (1 - t) * np.abs(2 * y / b)), alpha, n_terms=60)
        axes[1].plot(2 * wing.y / b, wing.cl / wing.CL, label=rf"$\lambda$ = {lam:g}")
    axes[1].axhline(1, color="gray", ls="--", lw=0.8)
    axes[1].set(xlabel="$2y/b$", ylabel="$c_l / C_L$", title="Section lift: where each wing stalls first")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/aerodynamics/wings/images/sphx_glr_plot_04_glauert_fourier_lifting_line_001.png
   :alt: AR = 8 straight wings, Section lift: where each wing stalls first
   :srcset: /api/gallery/aerodynamics/wings/images/sphx_glr_plot_04_glauert_fourier_lifting_line_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    AR = 8: delta = 0.0676 rectangular, minimum 0.0126 at taper ratio 0.375





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

   **Total running time of the script:** (0 minutes 0.212 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_wings_plot_04_glauert_fourier_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_04_glauert_fourier_lifting_line.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_glauert_fourier_lifting_line.ipynb <plot_04_glauert_fourier_lifting_line.ipynb>`

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

      :download:`Download Python source code: plot_04_glauert_fourier_lifting_line.py <plot_04_glauert_fourier_lifting_line.py>`

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

      :download:`Download zipped: plot_04_glauert_fourier_lifting_line.zip <plot_04_glauert_fourier_lifting_line.zip>`


.. only:: html

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

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