.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/airfoils/plot_04_thin_airfoil_pitching_moment.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_airfoils_plot_04_thin_airfoil_pitching_moment.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_airfoils_plot_04_thin_airfoil_pitching_moment.py:


Thin-airfoil pitching moment and the aerodynamic centre (1922-1926)
===================================================================

Munk's and Glauert's thin-airfoil theory predicts more than lift. The
pitching moment about the quarter chord,
:math:`c_{m,c/4} = \tfrac{\pi}{4}(A_2 - A_1)`
(:func:`~engineeringkit.aerodynamics.naca4_moment_coefficient`), depends
only on the camber line and not on the angle of attack. The quarter chord is
therefore the *aerodynamic centre*. A measured section's centre sits at
:math:`x_{ac}/c = \tfrac14 - m_0/a_0`
(:func:`~engineeringkit.aerodynamics.aerodynamic_center`), where :math:`m_0`
is its residual moment slope. This example reproduces Anderson's
:math:`c_{m,c/4} = -0.053` for the NACA 2412 and finds the aerodynamic
centre from panel-method moments.

.. GENERATED FROM PYTHON SOURCE LINES 19-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import aerodynamic_center, naca4_airfoil, naca4_moment_coefficient, panel_method

    cm = naca4_moment_coefficient(0.02, 0.4)
    print(f"NACA 2412 thin-airfoil c_m,c/4 = {cm:.4f}")
    assert np.isclose(cm, -0.053, atol=5e-4)





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

 .. code-block:: none

    NACA 2412 thin-airfoil c_m,c/4 = -0.0531




.. GENERATED FROM PYTHON SOURCE LINES 29-35

Moment about different reference points
---------------------------------------

Moving the reference point from the quarter chord to :math:`x` adds
:math:`c_l(x/c - \tfrac14)`. Only about the aerodynamic centre does the
moment stay flat as the angle of attack changes.

.. GENERATED FROM PYTHON SOURCE LINES 35-58

.. code-block:: Python


    alpha = np.radians(np.linspace(-4, 8, 7))
    results = [panel_method(naca4_airfoil("2412", 121), a) for a in alpha]
    cl = np.array([r.cl_pressure for r in results])
    cm_quarter = np.array([r.cm_quarter_chord for r in results])
    a0 = np.polyfit(alpha, cl, 1)[0]
    m0 = np.polyfit(alpha, cm_quarter, 1)[0]
    x_ac = float(aerodynamic_center(a0, m0))
    print(f"panel method: a0 = {a0:.3f} per rad, m0 = {m0:.4f} per rad, x_ac/c = {x_ac:.4f}")
    assert np.isclose(x_ac, 0.25, atol=0.02)

    fig, ax = plt.subplots(figsize=(7, 4))
    for x_ref in (0.0, 0.25, x_ac, 0.5):
        ax.plot(np.degrees(alpha), cm_quarter + cl * (x_ref - 0.25), "o-", ms=3, label=f"about x/c = {x_ref:.3f}")
    ax.axhline(cm, color="k", ls="--", lw=0.8, label="thin airfoil, c/4")
    ax.set(xlabel="angle of attack [deg]", ylabel="$c_m$", title="NACA 2412: the moment is flat about the aerodynamic centre")
    ax.legend(fontsize=8)
    fig.tight_layout()

    flat = cm_quarter + cl * (x_ac - 0.25)
    assert np.ptp(flat) < 0.1 * np.ptp(cm_quarter - cl * 0.25)

    plt.show()



.. image-sg:: /api/gallery/aerodynamics/airfoils/images/sphx_glr_plot_04_thin_airfoil_pitching_moment_001.png
   :alt: NACA 2412: the moment is flat about the aerodynamic centre
   :srcset: /api/gallery/aerodynamics/airfoils/images/sphx_glr_plot_04_thin_airfoil_pitching_moment_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    panel method: a0 = 6.802 per rad, m0 = -0.0653 per rad, x_ac/c = 0.2596





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

   **Total running time of the script:** (0 minutes 0.179 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_airfoils_plot_04_thin_airfoil_pitching_moment.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/airfoils/plot_04_thin_airfoil_pitching_moment.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_thin_airfoil_pitching_moment.ipynb <plot_04_thin_airfoil_pitching_moment.ipynb>`

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

      :download:`Download Python source code: plot_04_thin_airfoil_pitching_moment.py <plot_04_thin_airfoil_pitching_moment.py>`

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

      :download:`Download zipped: plot_04_thin_airfoil_pitching_moment.zip <plot_04_thin_airfoil_pitching_moment.zip>`


.. only:: html

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

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