.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/compressible/plot_04_prandtl_glauert.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_compressible_plot_04_prandtl_glauert.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_compressible_plot_04_prandtl_glauert.py:


The Prandtl-Glauert compressibility correction
==============================================

Hermann Glauert (1928), building on remarks by Prandtl, showed that the
linearised subsonic potential equation is Laplace's equation in stretched
coordinates. Every pressure coefficient of a thin body is therefore amplified
by :math:`1/\sqrt{1-M_\infty^2}`
(:func:`~engineeringkit.aerodynamics.prandtl_glauert_cp`), a factor that
blows up at :math:`M = 1` and fed the myth of an impenetrable "sound
barrier". The Kármán-Tsien rule (1939) is plotted alongside: the two agree at
low Mach and diverge as the flow becomes transonic.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import prandtl_glauert_cp








.. GENERATED FROM PYTHON SOURCE LINES 22-24

Amplification of a suction peak
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-40

.. code-block:: Python


    M = np.linspace(0, 0.9, 181)
    fig, ax = plt.subplots(figsize=(6, 4))
    for cp0 in [-0.3, -0.6]:
        pg = prandtl_glauert_cp(cp0, M)
        beta = np.sqrt(1 - M**2)
        kt = cp0 / (beta + M**2 / (1 + beta) * cp0 / 2)
        ax.plot(M, pg, label=f"Prandtl-Glauert, $C_{{p,0}}$ = {cp0}")
        ax.plot(M, kt, "--", label=f"Kármán-Tsien, $C_{{p,0}}$ = {cp0}")
        print(f"C_p0 = {cp0}: at M = 0.7, PG {float(prandtl_glauert_cp(cp0, 0.7)):.3f}, KT {np.interp(0.7, M, kt):.3f}")
        assert np.isclose(pg[0], cp0)
    ax.set(xlabel="free-stream Mach number", ylabel="pressure coefficient $C_p$", title="Subsonic compressibility corrections")
    ax.invert_yaxis()
    ax.legend(fontsize=8)
    fig.tight_layout()




.. image-sg:: /api/gallery/aerodynamics/compressible/images/sphx_glr_plot_04_prandtl_glauert_001.png
   :alt: Subsonic compressibility corrections
   :srcset: /api/gallery/aerodynamics/compressible/images/sphx_glr_plot_04_prandtl_glauert_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    C_p0 = -0.3: at M = 0.7, PG -0.420, KT -0.447
    C_p0 = -0.6: at M = 0.7, PG -0.840, KT -0.955




.. GENERATED FROM PYTHON SOURCE LINES 41-46

Lift slope rises the same way
-----------------------------

The same factor multiplies the lift-curve slope:
:math:`2\pi/\sqrt{1-M^2}` is 25% higher at :math:`M = 0.6`.

.. GENERATED FROM PYTHON SOURCE LINES 46-50

.. code-block:: Python


    print(f"lift-slope factor at M = 0.6: {float(prandtl_glauert_cp(1.0, 0.6)):.3f}")

    plt.show()




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

 .. code-block:: none

    lift-slope factor at M = 0.6: 1.250





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

   **Total running time of the script:** (0 minutes 0.159 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_compressible_plot_04_prandtl_glauert.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/compressible/plot_04_prandtl_glauert.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_prandtl_glauert.ipynb <plot_04_prandtl_glauert.ipynb>`

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

      :download:`Download Python source code: plot_04_prandtl_glauert.py <plot_04_prandtl_glauert.py>`

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

      :download:`Download zipped: plot_04_prandtl_glauert.zip <plot_04_prandtl_glauert.zip>`


.. only:: html

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

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