Fluid Vorticity & Kelvin–Helmholtz Instability
N-vortex Biot–Savart · 2D inviscid flow · Vortex sheet rollup · Shear instability
🌊 Kelvin–Helmholtz Instability
When two fluid layers move at different velocities, any perturbation at their interface grows exponentially — this is the Kelvin–Helmholtz instability (KHI). It is one of the most important mechanisms driving turbulence in nature: cloud billows, ocean surface waves, Jupiter's Great Red Spot, solar corona jets, and the magnetopause all show characteristic KH rolls.
🔬 Physics
A continuous vortex sheet (sheet of vorticity separating two uniform streams) is approximated by N discrete point vortices of circulation Γ. Each vortex moves with the velocity induced by all others via the inviscid Biot–Savart law: v = Σ Γ_j/(2π) × r̂⊥/|r|² (regularised with a finite core ε to prevent numerical singularities). Small sinusoidal perturbations grow exponentially at rate σ = |Γ|k/4π (k = wavenumber) — the Kaden–Birkhoff rollup. Eventually distinct spiral vortex structures emerge and merge (vortex pairing).
🎮 How to Use
Press Pause to freeze. Reset re-initialises with current parameters. Use the Vortices N slider to change the resolution of the vortex sheet. Increase Perturbation to trigger faster rollup. Double Layer adds a counter-rotating vortex row 0.08 units above — this creates a vortex street (classic Kármán configuration). Random Vortices seeds chaotic initial conditions.
💡 Occurrences in Nature
KHI creates the characteristic curling wave-cloud patterns (Kelvin–Helmholtz clouds), ocean internal waves visible from satellite, mixing layers in stellar interiors, plasma instabilities at Earth's magnetopause, Rayleigh–Taylor fingers in supernova remnants, and shear instabilities in accretion discs around black holes. The same mathematics governs vortex shedding behind bluff bodies.