.. 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_02_prandtl_blasius_boundary_layer.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_02_prandtl_blasius_boundary_layer.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_02_prandtl_blasius_boundary_layer.py:


Prandtl's boundary layer and the Blasius solution
=================================================

At the 1904 Heidelberg congress Ludwig Prandtl proposed that viscosity
matters only in a thin *boundary layer* next to a surface. His student Heinrich
Blasius (1908) reduced the flat-plate layer to the ordinary differential
equation :math:`f''' + \tfrac12 f f'' = 0`. Solving it by shooting
recovers the constants behind
:func:`~engineeringkit.aerodynamics.boundary_layer_laminar`
(:math:`\delta_{99} \approx 5x/\sqrt{Re_x}`) and
:func:`~engineeringkit.aerodynamics.skin_friction_laminar`
(:math:`C_f = 1.328/\sqrt{Re_L}`).

.. GENERATED FROM PYTHON SOURCE LINES 17-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.integrate import solve_ivp
    from scipy.optimize import brentq

    from engineeringkit.aerodynamics import boundary_layer_laminar, reynolds_number, skin_friction_laminar








.. GENERATED FROM PYTHON SOURCE LINES 25-27

Shoot for the wall shear f''(0)
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-46

.. code-block:: Python



    def blasius(t, y):
        return [y[1], y[2], -0.5 * y[0] * y[2]]


    def edge_velocity(curvature):
        return solve_ivp(blasius, (0, 12), [0, 0, curvature], rtol=1e-10, atol=1e-12).y[1, -1] - 1


    fpp0 = brentq(edge_velocity, 0.1, 1.0)
    eta = np.linspace(0, 8, 801)
    sol = solve_ivp(blasius, (0, 8), [0, 0, fpp0], t_eval=eta, rtol=1e-10, atol=1e-12)
    eta99 = np.interp(0.99, sol.y[1], eta)
    print(f"f''(0) = {fpp0:.5f} -> local C_f sqrt(Re_x) = {2 * fpp0:.4f}, average = {4 * fpp0:.4f}")
    print(f"delta_99 sqrt(Re_x) / x = {eta99:.3f}")
    assert np.isclose(4 * fpp0, skin_friction_laminar(1.0), rtol=1e-3)
    assert abs(eta99 - 4.91) < 0.01





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

 .. code-block:: none

    f''(0) = 0.33206 -> local C_f sqrt(Re_x) = 0.6641, average = 1.3282
    delta_99 sqrt(Re_x) / x = 4.910




.. GENERATED FROM PYTHON SOURCE LINES 47-49

Growth of the layer on a 1 m plate
----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-64

.. code-block:: Python


    x = np.linspace(1e-3, 1.0, 200)
    Re_x = reynolds_number(1.225, 5.0, x, 1.789e-5)
    fig, axes = plt.subplots(1, 2, figsize=(9, 3.6))
    axes[0].plot(sol.y[1], eta)
    axes[0].axhline(eta99, color="gray", ls="--", label=f"99% at eta = {eta99:.2f}")
    axes[0].set(xlabel="u / U", ylabel="similarity variable eta", title="Blasius velocity profile")
    axes[0].legend()
    axes[1].plot(x, boundary_layer_laminar(x, Re_x) * 1e3, label="5 x / sqrt(Re_x)")
    axes[1].plot(x, eta99 * x / np.sqrt(Re_x) * 1e3, "--", label="Blasius 4.91 x / sqrt(Re_x)")
    axes[1].set(xlabel="distance from leading edge [m]", ylabel="thickness [mm]", title="Air at 5 m/s")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/aerodynamics/boundary_layer/images/sphx_glr_plot_02_prandtl_blasius_boundary_layer_001.png
   :alt: Blasius velocity profile, Air at 5 m/s
   :srcset: /api/gallery/aerodynamics/boundary_layer/images/sphx_glr_plot_02_prandtl_blasius_boundary_layer_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.318 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_boundary_layer_plot_02_prandtl_blasius_boundary_layer.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_02_prandtl_blasius_boundary_layer.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_prandtl_blasius_boundary_layer.ipynb <plot_02_prandtl_blasius_boundary_layer.ipynb>`

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

      :download:`Download Python source code: plot_02_prandtl_blasius_boundary_layer.py <plot_02_prandtl_blasius_boundary_layer.py>`

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

      :download:`Download zipped: plot_02_prandtl_blasius_boundary_layer.zip <plot_02_prandtl_blasius_boundary_layer.zip>`


.. only:: html

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

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