.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/boundary_layer/plot_03_turbulent_skin_friction.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_aerodynamics_boundary_layer_plot_03_turbulent_skin_friction.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_aerodynamics_boundary_layer_plot_03_turbulent_skin_friction.py:


Turbulent skin friction and the one-fifth-power law
===================================================

Prandtl (1927) carried Blasius's empirical pipe-friction law over to the
turbulent flat plate, with a :math:`u \propto y^{1/7}` velocity profile. The
result, :math:`C_f = 0.074\,Re_L^{-1/5}`
(:func:`~engineeringkit.aerodynamics.skin_friction_turbulent`), decays far
more slowly with Reynolds number than the laminar
:math:`1.328\,Re_L^{-1/2}`; a turbulent layer can carry several times the
friction drag. This example compares the two and the payoff of delaying
transition.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import skin_friction_laminar, skin_friction_turbulent








.. GENERATED FROM PYTHON SOURCE LINES 22-24

Laminar versus turbulent friction coefficient
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-35

.. code-block:: Python


    Re = np.logspace(5, 8, 200)
    cf_lam = skin_friction_laminar(Re)
    cf_turb = skin_friction_turbulent(Re)
    for r in [5e5, 5e6, 5e7]:
        ratio = float(skin_friction_turbulent(r) / skin_friction_laminar(r))
        print(f"Re = {r:.0e}: turbulent / laminar C_f = {ratio:.1f}")
    slope = np.polyfit(np.log(Re), np.log(cf_turb), 1)[0]
    print(f"log-log slope of the turbulent law: {slope:.3f}")
    assert np.isclose(slope, -0.2)





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

 .. code-block:: none

    Re = 5e+05: turbulent / laminar C_f = 2.9
    Re = 5e+06: turbulent / laminar C_f = 5.7
    Re = 5e+07: turbulent / laminar C_f = 11.4
    log-log slope of the turbulent law: -0.200




.. GENERATED FROM PYTHON SOURCE LINES 36-42

A laminar run before transition
-------------------------------

A plate that stays laminar up to :math:`Re_{tr}` then turns turbulent has
:math:`C_f = 0.074/Re_L^{1/5} - A/Re_L` with
:math:`A = Re_{tr}(0.074\,Re_{tr}^{-1/5} - 1.328\,Re_{tr}^{-1/2})`.

.. GENERATED FROM PYTHON SOURCE LINES 42-55

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(6, 4))
    ax.loglog(Re, cf_lam, label="laminar (Blasius)")
    ax.loglog(Re, cf_turb, label="turbulent (Prandtl, 1/5 power)")
    for re_tr in [5e5, 3e6]:
        A = re_tr * (skin_friction_turbulent(re_tr) - skin_friction_laminar(re_tr))
        mixed = np.where(Re < re_tr, cf_lam, cf_turb - A / Re)
        ax.loglog(Re, mixed, "--", label=f"transition at Re = {re_tr:.0e}")
    ax.set(xlabel="plate Reynolds number $Re_L$", ylabel="average $C_f$", title="Flat-plate skin friction")
    ax.legend(fontsize=8)
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/aerodynamics/boundary_layer/images/sphx_glr_plot_03_turbulent_skin_friction_001.png
   :alt: Flat-plate skin friction
   :srcset: /api/gallery/aerodynamics/boundary_layer/images/sphx_glr_plot_03_turbulent_skin_friction_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.291 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_boundary_layer_plot_03_turbulent_skin_friction.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/aerodynamics/boundary_layer/plot_03_turbulent_skin_friction.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_turbulent_skin_friction.ipynb <plot_03_turbulent_skin_friction.ipynb>`

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

      :download:`Download Python source code: plot_03_turbulent_skin_friction.py <plot_03_turbulent_skin_friction.py>`

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

      :download:`Download zipped: plot_03_turbulent_skin_friction.zip <plot_03_turbulent_skin_friction.zip>`


.. only:: html

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

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