.. _sphx_glr_api_gallery_fluids_navier_stokes:

Navier-Stokes and Turbulence
----------------------------

The general-purpose engine behind this package's instability simulations:
a doubly periodic, pseudo-spectral solver for the full 2D incompressible
vorticity-transport equation, exact up to machine precision and aliasing in
space and 4th-order accurate in time. The first example shows its most basic
signature -- a vortex patch's peak vorticity bleeding away under viscous
diffusion, at a rate no inviscid simulation could reproduce. The second
pushes the same solver into a genuinely turbulent regime: many vortices of
mixed sign interacting nonlinearly, cascading kinetic energy from the large
scales they were seeded at down to small scales where viscosity finally
dissipates it. Watch, in the energy spectrum plot, how a real inertial range
lines up with Kolmogorov's predicted -5/3 power law over more than a decade
of wavenumber -- one of the most-tested predictions in classical physics,
reproduced here from nothing but the bare incompressible equations.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Navier (1822) and Stokes (1845) added a viscous friction term to Euler&#x27;s inviscid equations of motion, giving the equation that still governs essentially all of fluid dynamics today,">

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  .. image:: /api/gallery/fluids/navier_stokes/images/thumb/sphx_glr_plot_decaying_vortex_patch_thumb.png
    :alt:

  :doc:`/api/gallery/fluids/navier_stokes/plot_decaying_vortex_patch`

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      <div class="sphx-glr-thumbnail-title">Viscous decay under the Navier-Stokes equations</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="NavierStokes2D integrates the 2D incompressible vorticity-transport equation \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&#x27;s kinetic energy only decays, redistributed across scales by the nonlinear advection term before viscosity ultimately dissipates it. Kolmogorov&#x27;s 1941 (K41) theory predicts that, in the &quot;inertial range&quot; of scales between where the flow was initially seeded and where viscosity dissipates it, the kinetic energy spectrum follows a universal power law,">

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  .. image:: /api/gallery/fluids/navier_stokes/images/thumb/sphx_glr_plot_turbulent_cascade_thumb.png
    :alt:

  :doc:`/api/gallery/fluids/navier_stokes/plot_turbulent_cascade`

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      <div class="sphx-glr-thumbnail-title">The Kolmogorov -5/3 cascade in decaying 2D turbulence</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/fluids/navier_stokes/plot_decaying_vortex_patch
   /api/gallery/fluids/navier_stokes/plot_turbulent_cascade