Simulations & Interactive Learning
Four live widgets, each aimed at a place where the algebra of PC316 hides the physics. All are self-contained and run in your browser; the point-vortex sandbox integrates real equations of motion, and the other three evaluate the exact analytic expressions rather than cartoons of them.
Live
- Potential-flow builder — superpose uniform flow, sources, vortices and doublets; watch streamlines assemble a cylinder or a lifting body in real time
- Point vortex sandbox — RK4 on the exact point-vortex equations: orbiting and translating pairs, wall glide by the method of images, leapfrogging rings, chaos
- Kelvin–Helmholtz billow tank — the exact interface dispersion relation, with the wind-over-water onset at 6.60 m/s falling out of it
- Energy cascade sandbox — the K41 spectrum, the inertial range widening as \(Re^{3/4}\), and the DNS cost exploding as \(Re^{9/4}\)
What each is checked against
| Widget | Benchmark | Agreement |
|---|---|---|
| Point vortices | co-rotating period \(T = 2\pi^2d^2/\Gamma\) | 0.001% |
| Point vortices | counter-rotating speed \(\Gamma/2\pi d\); wall glide \(\Gamma/4\pi h\) | exact |
| Point vortices | Hamiltonian and impulse invariants | 1 part in \(10^{13}\) |
| KH tank | critical shear and wavelength, air over water | 6.604 m/s, 1.71 cm |
| KH tank | onset threshold behaviour | switches at \(\Delta U_c\) exactly |
| Cascade | \(L/\eta = Re^{3/4}\), \(Re_\eta = 1\), slope \(-5/3\) | exact / 0.003% |
Where each belongs in the course
| Chapter | Topic | Widget |
|---|---|---|
| 4 | Potential flow, conformal maps | Potential-flow builder |
| 8 | Vortex dynamics | Point vortex sandbox |
| 9 | Hydrodynamic instabilities | KH billow tank |
| 10 | Turbulence and the cascade | Energy cascade sandbox |
Read the caveats
Each page has a section saying what its model cannot do — point vortices never merge, the KH tank is linear theory, the cascade spectrum is a model form rather than a solution. Those sections are not disclaimers; they are where the physics of the approximation lives, and they are the part most likely to appear on an exam.