.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/wings/plot_02_induced_drag_elliptic_loading.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_02_induced_drag_elliptic_loading.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_02_induced_drag_elliptic_loading.py:


Induced drag and the optimality of elliptic loading
===================================================

Prandtl and Max Munk (1918-1919) showed that the trailing vortices carry away
kinetic energy at a rate that appears as an *induced* drag,
:math:`C_{D,i} = C_L^2/(\pi e AR)`
(:func:`~engineeringkit.aerodynamics.induced_drag_coefficient`), and that the
elliptic spanwise load (:math:`e = 1`) minimises it for a given span and lift.
This example builds spanwise loads, computes their induced drag from the
Fourier coefficients, and confirms the ellipse wins.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import induced_drag_coefficient








.. GENERATED FROM PYTHON SOURCE LINES 21-27

Span efficiency from the load shape
-----------------------------------

For :math:`\Gamma \propto \sum A_n \sin n\theta` with lift fixed by
:math:`A_1`, :math:`e = 1/(1 + \sum_{n>1} n A_n^2/A_1^2)`: any harmonic
content beyond the ellipse adds drag.

.. GENERATED FROM PYTHON SOURCE LINES 27-41

.. code-block:: Python


    theta = np.linspace(0, np.pi, 201)
    y = -np.cos(theta)
    loads = {"elliptic": [1.0, 0.0, 0.0], "fuller tips (A3 = -0.1)": [1.0, -0.1, 0.0], "peaky centre (A3 = +0.15)": [1.0, 0.15, 0.02]}
    AR, CL = 8.0, 0.5
    fig, axes = plt.subplots(1, 2, figsize=(9, 3.6))
    for label, (a1, a3, a5) in loads.items():
        gamma = a1 * np.sin(theta) + a3 * np.sin(3 * theta) + a5 * np.sin(5 * theta)
        e = 1 / (1 + (3 * a3**2 + 5 * a5**2) / a1**2)
        axes[0].plot(y, gamma, label=f"{label}: e = {e:.3f}")
        print(f"{label:28s} e = {e:.4f}, C_Di = {float(induced_drag_coefficient(CL, AR, e)):.5f}")
    axes[0].set(xlabel="spanwise station 2y/b", ylabel="circulation (normalised)", title="Same lift, different loads")
    axes[0].legend(fontsize=8)




.. image-sg:: /api/gallery/aerodynamics/wings/images/sphx_glr_plot_02_induced_drag_elliptic_loading_001.png
   :alt: Same lift, different loads
   :srcset: /api/gallery/aerodynamics/wings/images/sphx_glr_plot_02_induced_drag_elliptic_loading_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    elliptic                     e = 1.0000, C_Di = 0.00995
    fuller tips (A3 = -0.1)      e = 0.9709, C_Di = 0.01025
    peaky centre (A3 = +0.15)    e = 0.9350, C_Di = 0.01064

    <matplotlib.legend.Legend object at 0x7f0aec5e7110>



.. GENERATED FROM PYTHON SOURCE LINES 42-44

Induced drag grows with the square of lift
------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 44-53

.. code-block:: Python


    cl = np.linspace(0, 1.4, 50)
    for ar in [5, 8, 12, 30]:
        axes[1].plot(cl, induced_drag_coefficient(cl, ar), label=f"AR = {ar}")
    axes[1].set(xlabel="$C_L$", ylabel="$C_{D,i}$", title="Elliptic loading, e = 1")
    axes[1].legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.183 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_wings_plot_02_induced_drag_elliptic_loading.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_02_induced_drag_elliptic_loading.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_induced_drag_elliptic_loading.ipynb <plot_02_induced_drag_elliptic_loading.ipynb>`

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

      :download:`Download Python source code: plot_02_induced_drag_elliptic_loading.py <plot_02_induced_drag_elliptic_loading.py>`

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

      :download:`Download zipped: plot_02_induced_drag_elliptic_loading.zip <plot_02_induced_drag_elliptic_loading.zip>`


.. only:: html

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

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