.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fluid_engineering/open_channel/plot_02_belanger_hydraulic_jump.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_fluid_engineering_open_channel_plot_02_belanger_hydraulic_jump.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_fluid_engineering_open_channel_plot_02_belanger_hydraulic_jump.py:


Bélanger's hydraulic jump (1828)
================================

Water shooting fast and shallow from under a sluice gate cannot slow down
smoothly: it rises abruptly, in a turbulent roller, to a deeper, slower
stream. Jean-Baptiste Bélanger studied this hydraulic jump in his 1828
essay on steady flow in open channels. The rule used today, that the
pressure force plus the momentum flux is the same on both sides, gives the
sequent depth
:math:`y_2/y_1 = (\sqrt{1 + 8\mathrm{Fr}_1^2} - 1)/2`
(:func:`~engineeringkit.fluid_engineering.sequent_depth`). The energy
balance cannot be used across the jump, because the roller dissipates
energy (:func:`~engineeringkit.fluid_engineering.hydraulic_jump_head_loss`).
This example finds the jump below a sluice gate and how much of the
stream's energy it destroys.

.. GENERATED FROM PYTHON SOURCE LINES 20-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.fluid_engineering import G, froude_number, hydraulic_jump_head_loss, sequent_depth, specific_energy








.. GENERATED FROM PYTHON SOURCE LINES 26-30

The jump below a sluice gate
----------------------------

A gate releases 4 m²/s per metre of width at a depth of 0.4 m.

.. GENERATED FROM PYTHON SOURCE LINES 30-39

.. code-block:: Python


    q, y1 = 4.0, 0.4
    v1 = q / y1
    y2 = sequent_depth(y1, v1)
    loss = hydraulic_jump_head_loss(y1, y2)
    E1 = specific_energy(y1, q)
    print(f"Fr1 = {froude_number(v1, y1):.2f}: depth rises from {y1} m to {y2:.2f} m (Fr2 = {froude_number(q / y2, y2):.2f})")
    print(f"head lost in the jump {loss:.2f} m, {100 * loss / E1:.0f}% of the upstream specific energy")





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

 .. code-block:: none

    Fr1 = 5.05: depth rises from 0.4 m to 2.66 m (Fr2 = 0.29)
    head lost in the jump 2.72 m, 49% of the upstream specific energy




.. GENERATED FROM PYTHON SOURCE LINES 40-45

Momentum, not energy
--------------------

The momentum function :math:`M = y^2/2 + q^2/(gy)` is equal at the two
depths, while the specific energy drops.

.. GENERATED FROM PYTHON SOURCE LINES 45-49

.. code-block:: Python


    M1, M2 = y1**2 / 2 + q**2 / (G * y1), y2**2 / 2 + q**2 / (G * y2)
    print(f"M1 = {M1:.3f} m², M2 = {M2:.3f} m²; E1 = {E1:.2f} m, E2 = {specific_energy(y2, q):.2f} m")





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

 .. code-block:: none

    M1 = 4.159 m², M2 = 4.159 m²; E1 = 5.50 m, E2 = 2.78 m




.. GENERATED FROM PYTHON SOURCE LINES 50-56

Stronger jumps
--------------

Stilling basins below spillways are designed around the upstream Froude
number: a weak undular jump near 1 loses little energy, a steady jump at
4.5 to 9 loses most of it.

.. GENERATED FROM PYTHON SOURCE LINES 56-74

.. code-block:: Python


    Fr1 = np.linspace(1, 10, 100)
    ratio = sequent_depth(1.0, Fr1 * np.sqrt(G))
    loss_fraction = hydraulic_jump_head_loss(1.0, ratio) / specific_energy(1.0, Fr1 * np.sqrt(G))
    fig, ax = plt.subplots(figsize=(6, 4))
    ax.plot(Fr1, ratio, label="depth ratio $y_2/y_1$")
    ax.plot(Fr1, 10 * loss_fraction, label="energy lost × 10")
    ax.axvspan(4.5, 9, color="0.9", zorder=0, label="steady jump")
    ax.set(xlabel="upstream Froude number $Fr_1$", title="Bélanger's hydraulic jump")
    ax.legend()
    fig.tight_layout()

    assert np.isclose(M1, M2)
    assert np.isclose(loss, E1 - specific_energy(y2, q))
    assert froude_number(q / y2, y2) < 1 < froude_number(v1, y1)
    assert np.isclose(ratio[0], 1.0) and loss_fraction[0] == 0

    plt.show()



.. image-sg:: /api/gallery/fluid_engineering/open_channel/images/sphx_glr_plot_02_belanger_hydraulic_jump_001.png
   :alt: Bélanger's hydraulic jump
   :srcset: /api/gallery/fluid_engineering/open_channel/images/sphx_glr_plot_02_belanger_hydraulic_jump_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.067 seconds)


.. _sphx_glr_download_api_gallery_fluid_engineering_open_channel_plot_02_belanger_hydraulic_jump.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/fluid_engineering/open_channel/plot_02_belanger_hydraulic_jump.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_belanger_hydraulic_jump.ipynb <plot_02_belanger_hydraulic_jump.ipynb>`

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

      :download:`Download Python source code: plot_02_belanger_hydraulic_jump.py <plot_02_belanger_hydraulic_jump.py>`

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

      :download:`Download zipped: plot_02_belanger_hydraulic_jump.zip <plot_02_belanger_hydraulic_jump.zip>`


.. only:: html

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

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