.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/wings/plot_03_oswald_drag_polar.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_03_oswald_drag_polar.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_03_oswald_drag_polar.py:


Oswald's efficiency factor and the parabolic drag polar
=======================================================

W. Bailey Oswald's NACA Report 408 (1933) folded every source of
lift-dependent drag of a whole aeroplane into one *efficiency factor*, so its
drag polar becomes :math:`C_D = C_{D0} + C_L^2/(\pi e AR)`
(:func:`~engineeringkit.aerodynamics.drag_polar`). The maximum lift-to-drag
ratio lies where a line from the origin is tangent to the polar, at
:math:`C_L = \sqrt{\pi e AR C_{D0}}`
(:func:`~engineeringkit.aerodynamics.best_glide_cl`). The empirical
:func:`~engineeringkit.aerodynamics.oswald_efficiency_estimate` supplies
:math:`e` for straight wings.

.. GENERATED FROM PYTHON SOURCE LINES 17-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import best_glide_cl, drag_polar, glide_ratio, oswald_efficiency_estimate
    from engineeringkit.aerodynamics.visualizers import plot_drag_polar








.. GENERATED FROM PYTHON SOURCE LINES 24-26

Polars for three aspect ratios
------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-43

.. code-block:: Python


    cd0 = 0.022
    cl = np.linspace(-0.4, 1.6, 401)
    fig, ax = plt.subplots(figsize=(6, 4.5))
    for AR in [5.0, 8.0, 10.0]:
        e = float(oswald_efficiency_estimate(AR))
        plot_drag_polar(cl, drag_polar(cl, cd0, AR, e), ax=ax, label=f"AR = {AR:.0f}, e = {e:.2f}")
        cl_star = float(best_glide_cl(cd0, AR, e))
        ld_max = float(glide_ratio(cl_star, drag_polar(cl_star, cd0, AR, e)))
        print(f"AR = {AR:4.1f}: e = {e:.3f}, best-glide C_L = {cl_star:.3f}, (L/D)max = {ld_max:.2f}")
        # At the tangent point induced drag equals parasite drag.
        assert np.isclose(drag_polar(cl_star, cd0, AR, e), 2 * cd0)
        assert np.isclose(ld_max, 0.5 * np.sqrt(np.pi * e * AR / cd0))
    ax.set_title("Parabolic drag polars and best-glide tangents")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/aerodynamics/wings/images/sphx_glr_plot_03_oswald_drag_polar_001.png
   :alt: Parabolic drag polars and best-glide tangents
   :srcset: /api/gallery/aerodynamics/wings/images/sphx_glr_plot_03_oswald_drag_polar_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    AR =  5.0: e = 0.901, best-glide C_L = 0.558, (L/D)max = 12.68
    AR =  8.0: e = 0.811, best-glide C_L = 0.669, (L/D)max = 15.22
    AR = 10.0: e = 0.757, best-glide C_L = 0.723, (L/D)max = 16.44





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

   **Total running time of the script:** (0 minutes 0.157 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_wings_plot_03_oswald_drag_polar.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_03_oswald_drag_polar.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_oswald_drag_polar.ipynb <plot_03_oswald_drag_polar.ipynb>`

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

      :download:`Download Python source code: plot_03_oswald_drag_polar.py <plot_03_oswald_drag_polar.py>`

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

      :download:`Download zipped: plot_03_oswald_drag_polar.zip <plot_03_oswald_drag_polar.zip>`


.. only:: html

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

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