Real-time 2D shear-layer instability simulation via the vorticity-streamfunction method, showing perturbation growth, vortex rollup, and pairing dynamics
This visualization models a free shear layer in the 2D incompressible Navier-Stokes equations. A thin tanh-profile vorticity sheet is seeded with a sinusoidal inlet perturbation, then evolved to show the classic Kelvin-Helmholtz pathway: linear growth, vortex rollup, pairing, and increasingly complex mixing.
The solver advances the vorticity transport equation ∂ω/∂t + (u·∇)ω = ν∇²ω on a uniform grid. The streamfunction satisfies ∇²ψ = -ω, solved approximately each step with SOR iterations. Velocities are recovered from u = ∂ψ/∂y and v = -∂ψ/∂x. Advection uses a first-order upwind stencil and the time step is limited adaptively by advection and diffusion stability constraints.
The simulation demonstrates a canonical shear-layer transition process: (1) a thin shear layer with weak forcing; (2) exponential perturbation growth from Kelvin-Helmholtz instability; (3) rollup into coherent vortices; (4) vortex pairing and merger; (5) progressively finer-scale mixing. Reynolds number controls how strongly viscosity damps each stage.
Vortex sheet dynamics are crucial in engineering: leading-edge separation and stall on airfoils, noise generation in jet mixing layers, vortex-flame interaction in combustion chambers, shear instabilities in atmospheric boundary layers, and ocean internal waves. Understanding vortex sheet instability and rollup mechanisms is essential for predicting flow transition, optimizing mixing efficiency, and designing flow control strategies.
The main panel shows the vorticity field: red = positive vorticity (counter-clockwise), blue = negative vorticity (clockwise). Watch the perturbed shear layer amplify, curl into billows, then undergo pairing and filamentation. The history plot tracks both peak vorticity and enstrophy against simulation time. The lower-right panel is a centerline vorticity power spectrum, useful for seeing how dominant wavelengths shift during rollup.
1) Start with 'Laminar Shear' to observe slow, nearly linear perturbation growth. 2) Switch to 'Kelvin-Helmholtz' for the classic billow instability. 3) Try 'Vortex Rollup' to emphasize coherent vortex formation and pairing. 4) Use 'Turbulent Transition' for a more rapidly mixing shear layer. 5) Adjust Reynolds number to compare viscous damping against instability growth. 6) Reduce perturbation amplitude to linger longer in the linear regime.