Kelvin-Helmholtz Instability
Shear flow between fluid layers — exponential growth then vortex roll-up
Vortex Sheet Evolution (Point Vortex Method)
Interface Amplitude vs Time
Dispersion Relation σ(k)
Kelvin-Helmholtz instability: At the interface between two fluid layers moving at different velocities U₁ and U₂, any perturbation grows exponentially at rate σ = k|ΔU|/2 · √(ρ₁ρ₂)/(ρ₁+ρ₂) where k is the wavenumber. All wavelengths are unstable! (in the absence of surface tension or gravity).
Vortex sheet method: The interface is represented as N point vortices (Birkhoff-Rott equation). Each vortex moves under the induced velocity of all others. The Biot-Savart kernel is regularized: v ∝ Γ/(2π) · ẑ × r/(r²+δ²).
Roll-up: Initially exponential growth saturates nonlinearly as vorticity concentrates into discrete "cat's eyes" / billows. This is the classic KH roll-up seen in clouds (cirrus), ocean mixing layers, and Jupiter's bands.
Atwood number: A = (ρ₂-ρ₁)/(ρ₂+ρ₁). When A≠0 (density stratification), gravity stabilizes (Kelvin's criterion). Growth rate: σ = k√[(ρ₁U₁²+ρ₂U₂²)/(ρ₁+ρ₂) - ρ₁ρ₂(U₁-U₂)²/(ρ₁+ρ₂)² - gA/k].