How the demo computes it

The playground is not playing back an animation — it solves the flow around whatever shape you give it, in real time, with a panel method. Here is exactly how, so the numbers feel earned rather than magic.

Potential flow: one equation for the outer flow

Outside the thin boundary layer the flow is very nearly inviscid and irrotational, so the velocity can be written as the gradient of a scalar velocity potential $\phi$, and mass conservation reduces to Laplace’s equation:

Laplace’s equation is linear, which is the key that unlocks everything: we can build a complicated flow by adding up simple building-block solutions.

Building blocks: sources and vortices

Two elementary solutions of (1) do all the work. A source pushes fluid out radially — it gives a body thickness. A vortex swirls the flow around — it provides the circulation, and therefore the lift. Smear these continuously over the wing’s surface and we have enough freedom to represent any shape generating any amount of lift.

Discretise: chop the surface into panels

We split the outline into $N$ short straight panels. Each carries an unknown distribution of vorticity (our solver uses a linearly varying vortex strength along each panel, the classic method of Kuethe & Chow). The unknowns are the strengths at the panel nodes.

The conditions that pin down the answer

Two physical requirements make the unknown strengths solvable:

• Flow tangency (a Neumann boundary condition): nothing flows through the solid surface, so the velocity normal to each panel is zero at its midpoint (the “control point”) — one equation per panel:
  $$\left(\vec{V}_\infty + \vec{V}_{\text{induced}}\right)\cdot \hat{n} = 0$$
• The Kutta condition: the flow must leave the sharp trailing edge smoothly. This supplies the one extra equation that fixes the circulation — without it, the lift would be undetermined (Article 3).

Together these are $N + 1$ linear equations in $N + 1$ unknown strengths — a matrix system:

where $A$ holds the geometric influence of every panel on every control point, $\mathbf{b}$ comes from the freestream, and $\mathbf{x}$ is the list of strengths. The solver builds $A$ and solves it by Gaussian elimination every time you move a slider.

From strengths to everything you see

Once $\mathbf{x}$ is known, the surface velocity at each panel follows immediately; Bernoulli (as $C_p = 1 - (V/V_\infty)^2$) gives the pressure coefficient; integrating pressure around the surface gives $C_L$; and adding the building-block velocities at any point in the field lets us trace the streamlines and advect the particles. Every visual in the demo is a view of this one solution.