.. 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_01_pitot_tube.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_01_pitot_tube.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_01_pitot_tube.py:


Pitot's tube (1732)
===================

In 1732 Henri Pitot lowered a glass tube bent at a right angle into the
Seine, its mouth facing upstream, and watched the water inside rise above
the river surface. The rise is the velocity head :math:`v^2/(2g)`, and it
showed, against the belief of the day, that the current is faster near the
surface than at depth. Turned around, the stagnation rise gives the speed,
:math:`v = \sqrt{2\Delta p/\rho}`
(:func:`~engineeringkit.fluid_engineering.pitot_velocity`). This example
reads river and air speeds from a Pitot tube and finds the Mach number at
which the incompressible formula starts to mislead an airspeed indicator.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.fluid_engineering import G, dynamic_pressure, pitot_velocity, velocity_head








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Water rising in the tube
------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-32

.. code-block:: Python


    for v in [0.5, 1.0, 2.0]:
        print(f"river at {v:.1f} m/s: water rises {100 * velocity_head(v):.1f} cm in the tube")

    rise = 0.051
    print(f"a {100 * rise:.1f} cm rise means {pitot_velocity(998.0 * G * rise, 998.0):.2f} m/s")





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

 .. code-block:: none

    river at 0.5 m/s: water rises 1.3 cm in the tube
    river at 1.0 m/s: water rises 5.1 cm in the tube
    river at 2.0 m/s: water rises 20.4 cm in the tube
    a 5.1 cm rise means 1.00 m/s




.. GENERATED FROM PYTHON SOURCE LINES 33-40

Airspeed and compressibility
----------------------------

In a gas the stagnation rise is
:math:`p[(1 + \frac{\gamma - 1}{2}M^2)^{\gamma/(\gamma - 1)} - 1]`, larger
than :math:`\tfrac12\rho v^2`. Feeding it into the incompressible formula
overestimates the speed.

.. GENERATED FROM PYTHON SOURCE LINES 40-56

.. code-block:: Python


    gamma, p, T, R = 1.4, 101325.0, 288.15, 287.05
    rho, a = p / (R * T), np.sqrt(gamma * R * T)
    M = np.linspace(0.01, 0.8, 80)
    dp = p * ((1 + (gamma - 1) / 2 * M**2) ** (gamma / (gamma - 1)) - 1)
    error = pitot_velocity(dp, rho) / (M * a) - 1

    fig, ax = plt.subplots(figsize=(6, 4))
    ax.plot(M, 100 * error)
    ax.axhline(1, color="k", ls=":")
    ax.set(xlabel="Mach number", ylabel="overestimate of speed [%]", title="Incompressible Pitot formula in air")
    fig.tight_layout()

    M_1pct = np.interp(0.01, error, M)
    print(f"the error reaches 1% at Mach {M_1pct:.2f}")




.. image-sg:: /api/gallery/fluid_engineering/measurement/images/sphx_glr_plot_01_pitot_tube_001.png
   :alt: Incompressible Pitot formula in air
   :srcset: /api/gallery/fluid_engineering/measurement/images/sphx_glr_plot_01_pitot_tube_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    the error reaches 1% at Mach 0.28




.. GENERATED FROM PYTHON SOURCE LINES 57-59

The tube inverts the dynamic pressure exactly, and stays within 1% in air
below about Mach 0.28.

.. GENERATED FROM PYTHON SOURCE LINES 59-65

.. code-block:: Python


    v = np.array([0.3, 3.0, 30.0])
    assert np.allclose(pitot_velocity(dynamic_pressure(998.0, v), 998.0), v)
    assert 0.25 < M_1pct < 0.3

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.026 seconds)


.. _sphx_glr_download_api_gallery_fluid_engineering_measurement_plot_01_pitot_tube.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_01_pitot_tube.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_pitot_tube.ipynb <plot_01_pitot_tube.ipynb>`

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

      :download:`Download Python source code: plot_01_pitot_tube.py <plot_01_pitot_tube.py>`

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

      :download:`Download zipped: plot_01_pitot_tube.zip <plot_01_pitot_tube.zip>`


.. only:: html

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

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