Skip to content

Potential-Flow Builder

Learning goal

Feel superposition in your hands: because Laplace's equation is linear, elementary flows add. Uniform stream + doublet is a cylinder; add a vortex and stagnation points migrate — the origin of lift.

Things to try

  1. Cylinder preset — find the two stagnation points. Why do streamlines never enter the disk?
  2. Lifting cylinder — increase \(\Gamma\) and watch the stagnation points slide around the body and (at large \(\Gamma\)) merge and detach. The asymmetry of streamline spacing above/below is the lift (Bernoulli: tighter lines = faster = lower pressure).
  3. Source + stream — the dividing streamline is the Rankine half-body; everything inside it comes from the source.
  4. Source–sink pair — a closed Rankine oval: the potential-flow ancestor of every streamlined body.

The mathematics on display

\[\phi = Ux + \frac{Q}{2\pi}\ln r + \frac{K}{r}\cos\theta - \frac{\Gamma}{2\pi}\theta\]

Streamlines are traced by integrating \(\mathbf{v} = \nabla\phi\) with a midpoint (RK2) scheme — the same numerical-integration machinery studied in PHY653. Color encodes speed.

Potential flow · Conformal mapping · Laplace's equation