.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fluid_engineering/measurement/plot_04_thomson_v_notch_weir.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_measurement_plot_04_thomson_v_notch_weir.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_measurement_plot_04_thomson_v_notch_weir.py:


James Thomson's triangular notch (1858)
=======================================

James Thomson, engineer and elder brother of Lord Kelvin, reported to the
British Association in 1858 that a V-shaped notch measures small flows
better than a rectangular weir. The flow section keeps the same shape as
the head changes, so the discharge coefficient hardly varies, and the flow
grows as :math:`H^{5/2}`
(:func:`~engineeringkit.fluid_engineering.v_notch_weir_flow`) instead of
:math:`H^{3/2}` (:func:`~engineeringkit.fluid_engineering.rectangular_weir_flow`),
so a small flow still gives a readable head. This example compares the
rating curves of the two weirs and finds how precisely each must be read.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.fluid_engineering import rectangular_weir_flow, rehbock_discharge_coefficient, v_notch_weir_flow








.. GENERATED FROM PYTHON SOURCE LINES 23-28

Rating curves
-------------

A 90° notch and a 0.5 m rectangular weir standing 0.4 m above the bed of
a laboratory flume, with Rehbock's coefficient for the rectangle.

.. GENERATED FROM PYTHON SOURCE LINES 28-41

.. code-block:: Python


    H = np.linspace(0.005, 0.3, 200)
    P, b = 0.4, 0.5
    Q_vee = v_notch_weir_flow(H)
    Q_rect = rectangular_weir_flow(H, b, rehbock_discharge_coefficient(H, P))

    fig, ax = plt.subplots(figsize=(6, 4))
    ax.loglog(Q_vee * 1000, H, label="90° V-notch, $Q \\propto H^{5/2}$")
    ax.loglog(Q_rect * 1000, H, label="0.5 m rectangular weir, $Q \\propto H^{3/2}$")
    ax.set(xlabel="flow [L/s]", ylabel="head over the crest [m]", title="Sharp-crested weir rating curves")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/fluid_engineering/measurement/images/sphx_glr_plot_04_thomson_v_notch_weir_001.png
   :alt: Sharp-crested weir rating curves
   :srcset: /api/gallery/fluid_engineering/measurement/images/sphx_glr_plot_04_thomson_v_notch_weir_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 42-47

Head for a small flow
---------------------

For 2 L/s the notch stands several times higher than the rectangle, so a
1 mm reading error matters far less.

.. GENERATED FROM PYTHON SOURCE LINES 47-54

.. code-block:: Python


    Q = 2e-3
    h_vee = np.interp(Q, Q_vee, H)
    h_rect = np.interp(Q, Q_rect, H)
    for name, h, n in [("V-notch", h_vee, 2.5), ("rectangular", h_rect, 1.5)]:
        print(f"{name}: head {h * 1000:.1f} mm; a 1 mm error is a {100 * n * 1e-3 / h:.1f}% flow error")





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

 .. code-block:: none

    V-notch: head 73.4 mm; a 1 mm error is a 3.4% flow error
    rectangular: head 16.9 mm; a 1 mm error is a 8.9% flow error




.. GENERATED FROM PYTHON SOURCE LINES 55-56

The slopes of the log-log curves are the exponents of the head.

.. GENERATED FROM PYTHON SOURCE LINES 56-64

.. code-block:: Python


    slope_vee = np.polyfit(np.log(H), np.log(Q_vee), 1)[0]
    print(f"fitted exponent of the notch: {slope_vee:.3f}")

    assert np.isclose(slope_vee, 2.5)
    assert h_vee > 2.5 * h_rect

    plt.show()




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

 .. code-block:: none

    fitted exponent of the notch: 2.500





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

   **Total running time of the script:** (0 minutes 0.188 seconds)


.. _sphx_glr_download_api_gallery_fluid_engineering_measurement_plot_04_thomson_v_notch_weir.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/measurement/plot_04_thomson_v_notch_weir.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_thomson_v_notch_weir.ipynb <plot_04_thomson_v_notch_weir.ipynb>`

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

      :download:`Download Python source code: plot_04_thomson_v_notch_weir.py <plot_04_thomson_v_notch_weir.py>`

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

      :download:`Download zipped: plot_04_thomson_v_notch_weir.zip <plot_04_thomson_v_notch_weir.zip>`


.. only:: html

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

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