.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/compressible/plot_08_prandtl_meyer_expansion.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_08_prandtl_meyer_expansion.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_08_prandtl_meyer_expansion.py:


The Prandtl-Meyer expansion fan (1907-1908)
===========================================

Ludwig Prandtl (1907) and his student Theodor Meyer (1908) showed that a
supersonic stream turning *away* from itself around a corner expands
isentropically through a centred fan of Mach waves. The turning angle that
takes a sonic stream to Mach :math:`M` is the Prandtl-Meyer function
:math:`\nu(M)` (:func:`~engineeringkit.aerodynamics.prandtl_meyer_angle`),
so a turn by :math:`\theta` sets :math:`\nu(M_2) = \nu(M_1) + \theta`
(:func:`~engineeringkit.aerodynamics.prandtl_meyer_expansion`). Combined
with Meyer's oblique shocks this gives *shock-expansion theory*, which this
example applies to a flat plate at incidence and checks against Ackeret's
linear supersonic lift :math:`c_l = 4\alpha/\sqrt{M^2-1}`.

.. GENERATED FROM PYTHON SOURCE LINES 18-30

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import (
        oblique_shock,
        prandtl_meyer_angle,
        prandtl_meyer_expansion,
        stagnation_pressure,
    )

    gamma = 1.4








.. GENERATED FROM PYTHON SOURCE LINES 31-33

Expansion round a 20° corner at Mach 1.5
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-39

.. code-block:: Python


    M2 = float(prandtl_meyer_expansion(1.5, np.radians(20.0)))
    p_ratio = float(stagnation_pressure(1.0, 1.5) / stagnation_pressure(1.0, M2))
    print(f"nu(1.5) = {np.degrees(prandtl_meyer_angle(1.5)):.2f} deg -> M2 = {M2:.3f}, p2/p1 = {p_ratio:.4f}")
    assert np.isclose(np.degrees(prandtl_meyer_angle(M2) - prandtl_meyer_angle(1.5)), 20.0)





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

 .. code-block:: none

    nu(1.5) = 11.91 deg -> M2 = 2.207, p2/p1 = 0.3398




.. GENERATED FROM PYTHON SOURCE LINES 40-46

Shock-expansion theory for a flat plate
---------------------------------------

The upper surface turns away from the stream through a fan and the lower
surface turns into it through an oblique shock. The pressure difference
acts normal to the plate.

.. GENERATED FROM PYTHON SOURCE LINES 46-73

.. code-block:: Python



    def flat_plate_cl(M, alpha):
        p_inf = 1.0
        p0 = stagnation_pressure(p_inf, M, gamma)
        upper = stagnation_pressure(1.0, prandtl_meyer_expansion(M, alpha, gamma), gamma)
        p_upper = p0 / upper
        p_lower = p_inf * oblique_shock(M, alpha, gamma).pressure_ratio
        q = gamma / 2 * p_inf * M**2
        return (p_lower - p_upper) / q * np.cos(alpha)


    alpha = np.radians(np.linspace(0.5, 15, 30))
    fig, ax = plt.subplots(figsize=(7, 4))
    for M in (2.0, 3.0, 4.0):
        exact = np.array([flat_plate_cl(M, a) for a in alpha])
        (line,) = ax.plot(np.degrees(alpha), exact, label=f"shock-expansion, M = {M:g}")
        ax.plot(np.degrees(alpha), 4 * alpha / np.sqrt(M**2 - 1), "--", color=line.get_color())
    ax.set(xlabel="angle of attack [deg]", ylabel="$c_l$", title="Flat plate: shock-expansion (solid) vs linear theory (dashed)")
    ax.legend()
    fig.tight_layout()

    small = np.radians(1.0)
    print(f"M = 2, alpha = 1 deg: c_l = {flat_plate_cl(2.0, small):.5f}, linear theory {4 * small / np.sqrt(3):.5f}")
    assert np.isclose(flat_plate_cl(2.0, small), 4 * small / np.sqrt(3), rtol=0.01)

    plt.show()



.. image-sg:: /api/gallery/aerodynamics/compressible/images/sphx_glr_plot_08_prandtl_meyer_expansion_001.png
   :alt: Flat plate: shock-expansion (solid) vs linear theory (dashed)
   :srcset: /api/gallery/aerodynamics/compressible/images/sphx_glr_plot_08_prandtl_meyer_expansion_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    M = 2, alpha = 1 deg: c_l = 0.04031, linear theory 0.04031





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

   **Total running time of the script:** (0 minutes 0.200 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_compressible_plot_08_prandtl_meyer_expansion.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_08_prandtl_meyer_expansion.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_08_prandtl_meyer_expansion.ipynb <plot_08_prandtl_meyer_expansion.ipynb>`

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

      :download:`Download Python source code: plot_08_prandtl_meyer_expansion.py <plot_08_prandtl_meyer_expansion.py>`

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

      :download:`Download zipped: plot_08_prandtl_meyer_expansion.zip <plot_08_prandtl_meyer_expansion.zip>`


.. only:: html

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

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