.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/hydrology/plot_02_chezy_open_channel_formula.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_hydrology_plot_02_chezy_open_channel_formula.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_hydrology_plot_02_chezy_open_channel_formula.py:


Chézy's open-channel formula (1775)
===================================

Antoine de Chézy was asked to size a canal bringing the water of the Yvette to
Paris. He reasoned that in uniform flow the weight of the water down the slope
balances a bed friction that grows as the square of the velocity, so the mean
velocity is ``C sqrt(R S)``, with ``R`` the cross-sectional area over the wetted
perimeter. The example evaluates a rectangular canal: deepening it raises the
hydraulic radius and the velocity, which tends to the wide-channel limit where
``R`` equals the depth.

.. GENERATED FROM PYTHON SOURCE LINES 13-32



.. image-sg:: /api/gallery/hydrology/images/sphx_glr_plot_02_chezy_open_channel_formula_001.png
   :alt: Chézy: v = C sqrt(R S), C = 50, S = 4e-4
   :srcset: /api/gallery/hydrology/images/sphx_glr_plot_02_chezy_open_channel_formula_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.hydrology import chezy_velocity, hydraulic_radius

    depth = np.linspace(0.05, 3, 120)
    fig, ax = plt.subplots()
    for width in (1.0, 4.0, 20.0):
        radius = hydraulic_radius(width, depth)
        ax.plot(depth, chezy_velocity(50, radius, 4e-4), label=f"width {width:g} m")
    wide = chezy_velocity(50, depth, 4e-4)
    ax.plot(depth, wide, "k--", label="wide channel, R = depth")
    assert np.all(chezy_velocity(50, hydraulic_radius(20.0, depth), 4e-4) < wide)
    assert np.isclose(chezy_velocity(50, 1.0, 4e-4), 2 * chezy_velocity(50, 0.25, 4e-4))
    ax.set(xlabel="flow depth [m]", ylabel="mean velocity [m/s]", title="Chézy: v = C sqrt(R S), C = 50, S = 4e-4")
    ax.legend()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.098 seconds)


.. _sphx_glr_download_api_gallery_hydrology_plot_02_chezy_open_channel_formula.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/hydrology/plot_02_chezy_open_channel_formula.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_chezy_open_channel_formula.ipynb <plot_02_chezy_open_channel_formula.ipynb>`

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

      :download:`Download Python source code: plot_02_chezy_open_channel_formula.py <plot_02_chezy_open_channel_formula.py>`

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

      :download:`Download zipped: plot_02_chezy_open_channel_formula.zip <plot_02_chezy_open_channel_formula.zip>`


.. only:: html

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

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