.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/compressible/plot_06_de_laval_nozzle.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_06_de_laval_nozzle.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_06_de_laval_nozzle.py:


De Laval's converging-diverging nozzle
======================================

Gustaf de Laval's 1888 steam-turbine nozzle first converged and then
diverged, and accelerated steam beyond the speed of sound. The quasi-one-
dimensional isentropic relation :math:`A/A^*`
(:func:`~engineeringkit.aerodynamics.area_mach_ratio`) explains why: subsonic
flow speeds up in a converging duct, supersonic flow in a diverging one, and
the sonic point sits at the throat. Each area ratio has a subsonic and a
supersonic solution; this example finds both with a root bracket and traces
the Mach number through a nozzle.

.. GENERATED FROM PYTHON SOURCE LINES 16-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.optimize import brentq

    from engineeringkit.aerodynamics import area_mach_ratio








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Invert A/A* on each branch
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-37

.. code-block:: Python



    def mach_from_area(ratio, supersonic):
        lo, hi = (1.0, 50.0) if supersonic else (1e-6, 1.0)
        return brentq(lambda m: area_mach_ratio(m) - ratio, lo, hi)


    for ratio in [1.0, 1.6875, 4.0]:
        sub, sup = mach_from_area(ratio, False), mach_from_area(ratio, True)
        print(f"A/A* = {ratio:6.4f}: subsonic M = {sub:.4f}, supersonic M = {sup:.4f}")
    assert np.isclose(mach_from_area(1.6875, True), 2.0)





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

 .. code-block:: none

    A/A* = 1.0000: subsonic M = 1.0000, supersonic M = 1.0000
    A/A* = 1.6875: subsonic M = 0.3722, supersonic M = 2.0000
    A/A* = 4.0000: subsonic M = 0.1465, supersonic M = 2.9402




.. GENERATED FROM PYTHON SOURCE LINES 38-43

Mach number along a nozzle
--------------------------

A cosine-shaped converging section to a throat at x = 0.3 m, then a
conical diverging section to an exit area ratio of 4.

.. GENERATED FROM PYTHON SOURCE LINES 43-59

.. code-block:: Python


    x = np.linspace(0, 1, 301)
    throat = 0.3
    area = np.where(x < throat, 1 + 1.5 * (1 + np.cos(np.pi * x / throat)) / 2, 1 + 3 * (x - throat) / (1 - throat))
    mach = np.array([mach_from_area(ai, xi > throat) for xi, ai in zip(x, area, strict=True)])
    fig, axes = plt.subplots(2, 1, figsize=(7, 5), sharex=True)
    radius = np.sqrt(area / np.pi)
    axes[0].fill_between(x, -radius, radius, color="0.85")
    axes[0].plot(x, radius, "k", x, -radius, "k")
    axes[0].set(ylabel="radius [m]", title="Converging-diverging nozzle (A* = 1 m²)")
    axes[1].plot(x, mach)
    axes[1].axhline(1, color="gray", ls="--")
    axes[1].set(xlabel="x [m]", ylabel="Mach number")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/aerodynamics/compressible/images/sphx_glr_plot_06_de_laval_nozzle_001.png
   :alt: Converging-diverging nozzle (A* = 1 m²)
   :srcset: /api/gallery/aerodynamics/compressible/images/sphx_glr_plot_06_de_laval_nozzle_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.370 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_compressible_plot_06_de_laval_nozzle.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_06_de_laval_nozzle.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_06_de_laval_nozzle.ipynb <plot_06_de_laval_nozzle.ipynb>`

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

      :download:`Download Python source code: plot_06_de_laval_nozzle.py <plot_06_de_laval_nozzle.py>`

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

      :download:`Download zipped: plot_06_de_laval_nozzle.zip <plot_06_de_laval_nozzle.zip>`


.. only:: html

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

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