.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/airfoils/plot_03_hess_smith_panel_method.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_03_hess_smith_panel_method.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_03_hess_smith_panel_method.py:


The Hess-Smith panel method (1967)
==================================

At the Douglas Aircraft Company, John Hess and A. M. O. Smith replaced a
body by straight surface panels, each with its own uniform source strength,
and solved for the strengths that make the surface a streamline. Their 1967
review in *Progress in Aerospace Sciences* added a single vortex strength
shared by every panel. With the Kutta condition at the trailing edge, that
let the method carry lift. :func:`~engineeringkit.aerodynamics.panel_method`
implements it. This example checks it against the exact flow round a
circular cylinder, then shows what it adds to thin-airfoil theory: the
leading-edge suction peak and the extra lift slope of a thick section.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import naca4_airfoil, panel_method, thin_airfoil_cl








.. GENERATED FROM PYTHON SOURCE LINES 23-25

The circular cylinder is exact
------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-33

.. code-block:: Python


    t = np.linspace(0, -2 * np.pi, 49)
    circle = panel_method(np.column_stack((np.cos(t), np.sin(t))), 0.0)
    angle = np.arctan2(circle.y, circle.x)
    error = np.max(np.abs(circle.cp - (1 - 4 * np.sin(angle) ** 2)))
    print(f"cylinder, 48 panels: max |Cp - (1 - 4 sin^2)| = {error:.1e}")
    assert error < 1e-10





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

 .. code-block:: none

    cylinder, 48 panels: max |Cp - (1 - 4 sin^2)| = 3.6e-15




.. GENERATED FROM PYTHON SOURCE LINES 34-36

Pressure on the NACA 0012 and 2412
----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-46

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    for name in ("0012", "2412"):
        result = panel_method(naca4_airfoil(name, 101), np.radians(4.0))
        result.plot(ax=axes[0])
        axes[0].lines[-2].set_label(f"NACA {name} upper")
        axes[0].lines[-1].set_label(f"NACA {name} lower")
    axes[0].set_title("Surface pressure at 4°")
    axes[0].legend(fontsize=8)




.. image-sg:: /api/gallery/aerodynamics/airfoils/images/sphx_glr_plot_03_hess_smith_panel_method_001.png
   :alt: Surface pressure at 4°
   :srcset: /api/gallery/aerodynamics/airfoils/images/sphx_glr_plot_03_hess_smith_panel_method_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x7fa1ee1f7d40>



.. GENERATED FROM PYTHON SOURCE LINES 47-52

Lift slope grows with thickness
-------------------------------

Thin-airfoil theory gives :math:`2\pi` per radian whatever the thickness;
the panel method gives about :math:`2\pi(1 + 0.77\,t/c)`.

.. GENERATED FROM PYTHON SOURCE LINES 52-67

.. code-block:: Python


    alpha = np.radians(np.linspace(-4, 10, 8))
    for name in ("0006", "0012", "0018"):
        cl = np.array([panel_method(naca4_airfoil(name, 101), a).cl for a in alpha])
        axes[1].plot(np.degrees(alpha), cl, "o-", ms=3, label=f"NACA {name}")
        slope = np.polyfit(alpha, cl, 1)[0]
        thickness = int(name[2:]) / 100
        print(f"NACA {name}: slope {slope:.3f} per rad = 2 pi x {slope / (2 * np.pi):.3f} (1 + 0.77 t/c = {1 + 0.77 * thickness:.3f})")
        assert np.isclose(slope, 2 * np.pi * (1 + 0.77 * thickness), rtol=0.02)
    axes[1].plot(np.degrees(alpha), thin_airfoil_cl(alpha), "k--", label="thin airfoil, 2π")
    axes[1].set(xlabel="angle of attack [deg]", ylabel="$c_l$", title="Inviscid lift of symmetric sections")
    axes[1].legend()
    fig.tight_layout()

    plt.show()




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

 .. code-block:: none

    NACA 0006: slope 6.571 per rad = 2 pi x 1.046 (1 + 0.77 t/c = 1.046)
    NACA 0012: slope 6.828 per rad = 2 pi x 1.087 (1 + 0.77 t/c = 1.092)
    NACA 0018: slope 7.094 per rad = 2 pi x 1.129 (1 + 0.77 t/c = 1.139)





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

   **Total running time of the script:** (0 minutes 0.309 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_airfoils_plot_03_hess_smith_panel_method.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_03_hess_smith_panel_method.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_hess_smith_panel_method.ipynb <plot_03_hess_smith_panel_method.ipynb>`

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

      :download:`Download Python source code: plot_03_hess_smith_panel_method.py <plot_03_hess_smith_panel_method.py>`

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

      :download:`Download zipped: plot_03_hess_smith_panel_method.zip <plot_03_hess_smith_panel_method.zip>`


.. only:: html

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

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