.. _sphx_glr_api_gallery_fluids_potential_flow:

Potential Flow
--------------

Away from any boundary layer or wake, a great deal of real aerodynamics is
governed by a flow with no vorticity at all: a velocity potential obeying
Laplace's equation, which is linear, so entire flow fields can be built by
just adding together a handful of elementary solutions. The examples below
build up from a bare source/sink/doublet, through d'Alembert's paradox of
exactly zero drag on a circulation-free cylinder, to the textbook result
that made potential theory indispensable to early aerodynamics -- a
spinning cylinder, or by extension a cambered airfoil, generates lift in
direct proportion to its bound circulation, with no viscosity anywhere in
the calculation. Watch, in the streamline plots, how a source and sink
merge into a doublet as they're brought together, and how adding
circulation to a symmetric cylinder flow breaks its front-back symmetry and
shifts its stagnation points -- the geometric signature of lift.


.. 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="Superposing a uniform stream of speed U_\infty with a doublet of strength \kappa = 2\pi U_\infty R^2 places a circular streamline of radius R exactly at the origin -- a solid cylinder, in potential-flow terms. Adding a point vortex of circulation \Gamma at the same location keeps that circle intact (a vortex centered on the circle induces purely tangential velocity there) while breaking its front-back symmetry, generating lift with no viscosity anywhere in the calculation. This example builds that flow with flow_past_cylinder, plots its surface pressure coefficient">

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  .. image:: /api/gallery/fluids/potential_flow/images/thumb/sphx_glr_plot_cylinder_with_lift_thumb.png
    :alt:

  :doc:`/api/gallery/fluids/potential_flow/plot_cylinder_with_lift`

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      <div class="sphx-glr-thumbnail-title">Flow past a lifting cylinder</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Jean le Rond d&#x27;Alembert applied the equations of inviscid, irrotational flow to steady motion past a closed body and found -- to his own evident discomfort -- that the theory predicts exactly zero net drag on the body no matter its shape, a flatly unphysical result for anything actually moving through a real fluid. flow_past_cylinder with its default circulation=0.0 builds exactly this symmetric, inviscid flow: a uniform stream past a circular cylinder with no circulation added, and therefore no front-back (or top-bottom) asymmetry anywhere in the pressure field. Integrating the resulting surface pressure coefficient">

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  .. image:: /api/gallery/fluids/potential_flow/images/thumb/sphx_glr_plot_dalembert_paradox_thumb.png
    :alt:

  :doc:`/api/gallery/fluids/potential_flow/plot_dalembert_paradox`

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      <div class="sphx-glr-thumbnail-title">D'Alembert's paradox: zero net drag on a cylinder in inviscid flow</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="For a 2D flow with no vorticity (\nabla\times\mathbf{u}=0), the velocity is the gradient of a scalar potential, \mathbf{u}=\nabla\phi, and incompressibility (\nabla\cdot\mathbf{u}=0) then makes \phi satisfy Laplace&#x27;s equation,">

.. only:: html

  .. image:: /api/gallery/fluids/potential_flow/images/thumb/sphx_glr_plot_source_sink_doublet_thumb.png
    :alt:

  :doc:`/api/gallery/fluids/potential_flow/plot_source_sink_doublet`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Source, sink, and doublet: building blocks of potential flow</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/fluids/potential_flow/plot_cylinder_with_lift
   /api/gallery/fluids/potential_flow/plot_dalembert_paradox
   /api/gallery/fluids/potential_flow/plot_source_sink_doublet