.. 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_03_bakhmeteff_specific_energy.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_03_bakhmeteff_specific_energy.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_03_bakhmeteff_specific_energy.py:


Bakhmeteff's specific energy (1912)
===================================

Boris Bakhmeteff, in a 1912 book on varied flow written at St Petersburg,
measured the energy of a stream from the channel bed instead of from a
fixed datum: the specific energy :math:`E = y + q^2/(2gy^2)`
(:func:`~engineeringkit.fluid_engineering.specific_energy`). Plotted
against depth, it has a least value at the critical depth
:math:`y_c = (q^2/g)^{1/3}`
(:func:`~engineeringkit.fluid_engineering.critical_depth`), where the
Froude number is 1, and any larger energy can be carried at a subcritical
and a supercritical alternate depth
(:func:`~engineeringkit.fluid_engineering.alternate_depths`). This example
draws the specific-energy diagram and uses it to find the flow over a
raised step in the bed.

.. GENERATED FROM PYTHON SOURCE LINES 20-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.fluid_engineering import alternate_depths, critical_depth, froude_number, specific_energy








.. GENERATED FROM PYTHON SOURCE LINES 26-28

The specific-energy diagram
---------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-40

.. code-block:: Python


    y = np.linspace(0.08, 3.0, 300)
    fig, ax = plt.subplots(figsize=(5, 5))
    for q in [1.0, 2.0, 4.0]:
        yc = critical_depth(q)
        ax.plot(specific_energy(y, q), y, label=f"q = {q:g} m²/s")
        ax.plot(1.5 * yc, yc, "ko")
    ax.plot([0, 3], [0, 3], "k:", lw=0.8, label="E = y")
    ax.set(xlim=(0, 3), ylim=(0, 3), xlabel="specific energy E [m]", ylabel="depth y [m]", title="Specific energy")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/fluid_engineering/open_channel/images/sphx_glr_plot_03_bakhmeteff_specific_energy_001.png
   :alt: Specific energy
   :srcset: /api/gallery/fluid_engineering/open_channel/images/sphx_glr_plot_03_bakhmeteff_specific_energy_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 41-47

Flow over a raised step
-----------------------

A stream 1.8 m deep carrying 2 m²/s meets a step 0.2 m high. With no
loss, the specific energy over the step is :math:`E_1 - \Delta z`, and
the subcritical stream stays subcritical: its surface dips.

.. GENERATED FROM PYTHON SOURCE LINES 47-55

.. code-block:: Python


    q, y1, dz = 2.0, 1.8, 0.2
    E1 = specific_energy(y1, q)
    deep, shallow = alternate_depths(E1 - dz, q)
    print(f"upstream: y = {y1} m, E = {E1:.3f} m, Fr = {froude_number(q / y1, y1):.2f}")
    print(f"over the step: y = {deep:.3f} m (the supercritical alternate would be {shallow:.3f} m)")
    print(f"water surface drops by {y1 - (deep + dz):.3f} m")





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

 .. code-block:: none

    upstream: y = 1.8 m, E = 1.863 m, Fr = 0.26
    over the step: y = 1.581 m (the supercritical alternate would be 0.402 m)
    water surface drops by 0.019 m




.. GENERATED FROM PYTHON SOURCE LINES 56-58

The largest step the stream can pass without backing up is the one that
brings the energy down to its minimum, :math:`3y_c/2`.

.. GENERATED FROM PYTHON SOURCE LINES 58-68

.. code-block:: Python


    yc = critical_depth(q)
    dz_max = E1 - 1.5 * yc
    print(f"critical depth {yc:.3f} m; largest step {dz_max:.3f} m")

    assert np.isclose(specific_energy(deep, q), E1 - dz) and deep > yc > shallow
    assert np.isclose(froude_number(q / yc, yc), 1.0)
    assert np.allclose(alternate_depths(E1 - dz_max, q), yc)

    plt.show()




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

 .. code-block:: none

    critical depth 0.742 m; largest step 0.751 m





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

   **Total running time of the script:** (0 minutes 0.103 seconds)


.. _sphx_glr_download_api_gallery_fluid_engineering_open_channel_plot_03_bakhmeteff_specific_energy.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_03_bakhmeteff_specific_energy.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_bakhmeteff_specific_energy.ipynb <plot_03_bakhmeteff_specific_energy.ipynb>`

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

      :download:`Download Python source code: plot_03_bakhmeteff_specific_energy.py <plot_03_bakhmeteff_specific_energy.py>`

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

      :download:`Download zipped: plot_03_bakhmeteff_specific_energy.zip <plot_03_bakhmeteff_specific_energy.zip>`


.. only:: html

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

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