.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fluids/navier_stokes/plot_turbulent_cascade.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_fluids_navier_stokes_plot_turbulent_cascade.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_fluids_navier_stokes_plot_turbulent_cascade.py:


The Kolmogorov -5/3 cascade in decaying 2D turbulence
=========================================================

:class:`~physicskit.fluids.systems.navier_stokes.NavierStokes2D` integrates
the 2D incompressible vorticity-transport equation
:math:`\partial_t \omega + (\mathbf{u}\cdot\nabla)\omega = \nu \nabla^2
\omega` pseudo-spectrally on a doubly periodic domain, with no external
forcing: once set going, the flow's kinetic energy only decays, redistributed
across scales by the nonlinear advection term before viscosity ultimately
dissipates it. Kolmogorov's 1941 (K41) theory predicts that, in the
"inertial range" of scales between where the flow was initially seeded and
where viscosity dissipates it, the kinetic energy spectrum follows a
universal power law,

.. math::

    E(k) \propto k^{-5/3},

where :math:`E(k)` is defined so that :math:`\int E(k)\,dk` is the total
kinetic energy per unit mass. This example seeds the solver's initial
vorticity field with many randomly placed Gaussian vortex blobs of mixed
sign and random position, lets their nonlinear interactions cascade energy
across scales, and checks the resulting spectrum -- via
:func:`~physicskit.fluids.utils.spectral_analysis.energy_spectrum` -- against
that -5/3 law.

.. GENERATED FROM PYTHON SOURCE LINES 28-37

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.fluids.systems.navier_stokes import NavierStokes2D
    from physicskit.fluids.utils.spectral_analysis import energy_spectrum
    from physicskit.fluids.visualizers.flow_fields import plot_vorticity_field
    from physicskit.fluids.visualizers.spectra import plot_energy_spectrum








.. GENERATED FROM PYTHON SOURCE LINES 38-40

Seed many random vortex blobs
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 40-54

.. code-block:: Python


    n, length = 128, 2 * np.pi
    solver = NavierStokes2D(n=n, length=length, nu=2e-4)

    rng = np.random.default_rng(0)
    n_blobs, core = 40, 0.12
    omega0 = np.zeros_like(solver.X)
    for _ in range(n_blobs):
        xc, yc = rng.uniform(0, length, size=2)
        sign = rng.choice([-1.0, 1.0])
        dx = np.minimum(np.abs(solver.X - xc), length - np.abs(solver.X - xc))
        dy = np.minimum(np.abs(solver.Y - yc), length - np.abs(solver.Y - yc))
        omega0 += sign * np.exp(-(dx**2 + dy**2) / (2 * core**2))








.. GENERATED FROM PYTHON SOURCE LINES 55-57

Let the nonlinear interactions cascade energy across scales
-----------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 57-64

.. code-block:: Python


    result = solver.simulate(omega0, dt=0.002, steps=800)

    fig, ax = plot_vorticity_field(solver.X, solver.Y, result["omega"], result["u"], result["v"])
    ax.set_title("Decaying 2D turbulence")
    fig.tight_layout()




.. image-sg:: /api/gallery/fluids/navier_stokes/images/sphx_glr_plot_turbulent_cascade_001.png
   :alt: Decaying 2D turbulence
   :srcset: /api/gallery/fluids/navier_stokes/images/sphx_glr_plot_turbulent_cascade_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 65-69

Check the energy spectrum against Kolmogorov's -5/3 law
-------------------------------------------------------------
A genuine inertial range shows up as a stretch of the measured spectrum
running parallel to the reference line on these log-log axes.

.. GENERATED FROM PYTHON SOURCE LINES 69-75

.. code-block:: Python


    k, E = energy_spectrum(result["u"], result["v"], length)
    fig, ax = plot_energy_spectrum(k, E)
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/fluids/navier_stokes/images/sphx_glr_plot_turbulent_cascade_002.png
   :alt: Kinetic energy spectrum
   :srcset: /api/gallery/fluids/navier_stokes/images/sphx_glr_plot_turbulent_cascade_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 5.931 seconds)


.. _sphx_glr_download_api_gallery_fluids_navier_stokes_plot_turbulent_cascade.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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

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

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

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

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

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


.. only:: html

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

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