.. 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_01_manning_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_fluid_engineering_open_channel_plot_01_manning_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_fluid_engineering_open_channel_plot_01_manning_formula.py:


Manning's open-channel formula (1889)
=====================================

Robert Manning, chief engineer of the Irish Board of Works, compared the
open-channel formulas of his day against the best measurements in 1889 and
proposed that the mean velocity of uniform flow is
:math:`V = R^{2/3}S^{1/2}/n` in SI units
(:func:`~engineeringkit.fluid_engineering.manning_velocity`), with
:math:`R` the hydraulic radius
(:func:`~engineeringkit.fluid_engineering.hydraulic_radius`), :math:`S`
the bed slope and :math:`n` a roughness coefficient. It is Chézy's formula
:math:`V = C\sqrt{RS}` with :math:`C = R^{1/6}/n`. This example finds the
normal depth of a concrete channel and checks whether the flow is
subcritical.

.. GENERATED FROM PYTHON SOURCE LINES 19-25

.. code-block:: Python

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

    from engineeringkit.fluid_engineering import froude_number, hydraulic_radius, manning_flow, manning_velocity








.. GENERATED FROM PYTHON SOURCE LINES 26-30

Rating curve of a rectangular channel
-------------------------------------

A concrete channel (:math:`n = 0.013`) 3 m wide on a slope of 1 in 1000.

.. GENERATED FROM PYTHON SOURCE LINES 30-40

.. code-block:: Python


    n, b, S = 0.013, 3.0, 1e-3
    y = np.linspace(0.05, 2.5, 100)
    Q = manning_flow(n, b * y, b + 2 * y, S)

    fig, ax = plt.subplots(figsize=(6, 4))
    ax.plot(Q, y)
    ax.set(xlabel="flow [m³/s]", ylabel="normal depth [m]", title="Rating curve, n = 0.013, S = 0.001")
    fig.tight_layout()




.. image-sg:: /api/gallery/fluid_engineering/open_channel/images/sphx_glr_plot_01_manning_formula_001.png
   :alt: Rating curve, n = 0.013, S = 0.001
   :srcset: /api/gallery/fluid_engineering/open_channel/images/sphx_glr_plot_01_manning_formula_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 41-45

Normal depth for a design flow
------------------------------

Manning's formula is solved for the depth that carries 10 m³/s.

.. GENERATED FROM PYTHON SOURCE LINES 45-55

.. code-block:: Python


    Q_design = 10.0
    y_n = optimize.brentq(lambda d: manning_flow(n, b * d, b + 2 * d, S) - Q_design, 0.01, 10.0)
    R = hydraulic_radius(b * y_n, b + 2 * y_n)
    V = manning_velocity(n, R, S)
    Fr = froude_number(V, y_n)
    ax.plot(Q_design, y_n, "ko")
    print(f"normal depth {y_n:.3f} m, R = {R:.3f} m, V = {V:.2f} m/s, Fr = {Fr:.2f}")
    print("subcritical: disturbances travel upstream" if Fr < 1 else "supercritical")





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

 .. code-block:: none

    normal depth 1.619 m, R = 0.779 m, V = 2.06 m/s, Fr = 0.52
    subcritical: disturbances travel upstream




.. GENERATED FROM PYTHON SOURCE LINES 56-61

Chézy's coefficient
-------------------

Manning's :math:`C = R^{1/6}/n` changes slowly with depth, which is why
Chézy's constant-C formula worked within a single river.

.. GENERATED FROM PYTHON SOURCE LINES 61-70

.. code-block:: Python


    C = R ** (1 / 6) / n
    print(f"Chézy C = {C:.1f} m^(1/2)/s")

    assert np.isclose(manning_flow(n, b * y_n, b + 2 * y_n, S), Q_design)
    assert np.isclose(V, C * np.sqrt(R * S))
    assert Fr < 1 and np.all(np.diff(Q) > 0)

    plt.show()




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

 .. code-block:: none

    Chézy C = 73.8 m^(1/2)/s





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

   **Total running time of the script:** (0 minutes 0.027 seconds)


.. _sphx_glr_download_api_gallery_fluid_engineering_open_channel_plot_01_manning_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/fluid_engineering/open_channel/plot_01_manning_formula.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_manning_formula.ipynb <plot_01_manning_formula.ipynb>`

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

      :download:`Download Python source code: plot_01_manning_formula.py <plot_01_manning_formula.py>`

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

      :download:`Download zipped: plot_01_manning_formula.zip <plot_01_manning_formula.zip>`


.. only:: html

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

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