.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fluid_engineering/pipe_flow/plot_05_colebrook_equation.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_pipe_flow_plot_05_colebrook_equation.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_pipe_flow_plot_05_colebrook_equation.py:


Colebrook's turbulent friction factor (1939)
============================================

Commercial pipes do not behave like Nikuradse's sand: their roughness is
uneven, and friction leaves the smooth law gradually rather than with
Nikuradse's dip. Cyril Colebrook, after experiments with Cedric White on
pipes coated with mixed grains, added the smooth and rough laws inside one
logarithm in 1939,

.. math::

    \frac{1}{\sqrt f} = -2\log_{10}\left(\frac{\varepsilon/D}{3.7} + \frac{2.51}{Re\sqrt f}\right),

(:func:`~engineeringkit.fluid_engineering.colebrook_friction_factor`). The
equation is implicit in :math:`f`. This example solves it by the
fixed-point iteration engineers used by hand and compares it with the two
limits it joins.

.. GENERATED FROM PYTHON SOURCE LINES 22-31

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.fluid_engineering import (
        colebrook_friction_factor,
        nikuradse_friction_factor,
        smooth_pipe_friction_factor,
    )








.. GENERATED FROM PYTHON SOURCE LINES 32-38

Fixed-point iteration
---------------------

With :math:`x = 1/\sqrt f` the equation reads
:math:`x = -2\log_{10}(\varepsilon/3.7D + 2.51x/Re)`. The right side
changes slowly with :math:`x`, so substitution converges in a few steps.

.. GENERATED FROM PYTHON SOURCE LINES 38-53

.. code-block:: Python


    Re, rel = 2e5, 1e-3
    x = [3.0]
    for _ in range(8):
        x.append(-2 * np.log10(rel / 3.7 + 2.51 * x[-1] / Re))
    f_iter = 1 / np.array(x) ** 2
    f_ref = colebrook_friction_factor(Re, rel)
    for k, fk in enumerate(f_iter[:5]):
        print(f"step {k}: f = {fk:.6f}")
    print(f"bracketed root: f = {f_ref:.6f}")

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(9, 3.8))
    ax1.semilogy(np.abs(f_iter - f_ref) + 1e-17, marker="o")
    ax1.set(xlabel="iteration", ylabel="|f - f*|", title="Substitution converges fast")




.. image-sg:: /api/gallery/fluid_engineering/pipe_flow/images/sphx_glr_plot_05_colebrook_equation_001.png
   :alt: Substitution converges fast
   :srcset: /api/gallery/fluid_engineering/pipe_flow/images/sphx_glr_plot_05_colebrook_equation_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    step 0: f = 0.111111
    step 1: f = 0.020274
    step 2: f = 0.021057
    step 3: f = 0.021033
    step 4: f = 0.021034
    bracketed root: f = 0.021034

    [Text(0.5, 12.633333333333331, 'iteration'), Text(58.222222222222214, 0.5, '|f - f*|'), Text(0.5, 1.0, 'Substitution converges fast')]



.. GENERATED FROM PYTHON SOURCE LINES 54-60

A smooth bridge between two laws
--------------------------------

Colebrook's curve sits above both the smooth and the fully rough laws and
approaches each at its end of the range. The largest excess is in the
transition, where both mechanisms count.

.. GENERATED FROM PYTHON SOURCE LINES 60-80

.. code-block:: Python


    Re_range = np.geomspace(4e3, 1e8, 200)
    cb = colebrook_friction_factor(Re_range, rel)
    smooth = smooth_pipe_friction_factor(Re_range)
    rough = nikuradse_friction_factor(rel)
    ax2.loglog(Re_range, cb, label="Colebrook, ε/D = 0.001")
    ax2.loglog(Re_range, smooth, "k--", label="smooth")
    ax2.axhline(rough, color="k", ls=":", label="fully rough")
    ax2.set(xlabel="Reynolds number", ylabel="friction factor", title="Colebrook joins the limits")
    ax2.legend(fontsize="small")
    fig.tight_layout()

    excess = cb / np.maximum(smooth, rough) - 1
    print(f"largest excess over the larger limit: {excess.max():.0%} at Re = {Re_range[excess.argmax()]:.1e}")

    assert abs(f_iter[-1] - f_ref) < 1e-10 * f_ref
    assert np.all(cb >= np.maximum(smooth, rough) * (1 - 2e-3))
    assert abs(cb[0] / smooth[0] - 1) < 0.12 and abs(cb[-1] / rough - 1) < 0.01

    plt.show()




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

 .. code-block:: none

    largest excess over the larger limit: 18% at Re = 6.6e+04





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

   **Total running time of the script:** (0 minutes 0.080 seconds)


.. _sphx_glr_download_api_gallery_fluid_engineering_pipe_flow_plot_05_colebrook_equation.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/pipe_flow/plot_05_colebrook_equation.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_colebrook_equation.ipynb <plot_05_colebrook_equation.ipynb>`

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

      :download:`Download Python source code: plot_05_colebrook_equation.py <plot_05_colebrook_equation.py>`

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

      :download:`Download zipped: plot_05_colebrook_equation.zip <plot_05_colebrook_equation.zip>`


.. only:: html

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

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