Navier-Stokes and Turbulence#
The general-purpose engine behind this package’s instability simulations: a doubly periodic, pseudo-spectral solver for the full 2D incompressible vorticity-transport equation, exact up to machine precision and aliasing in space and 4th-order accurate in time. The first example shows its most basic signature – a vortex patch’s peak vorticity bleeding away under viscous diffusion, at a rate no inviscid simulation could reproduce. The second pushes the same solver into a genuinely turbulent regime: many vortices of mixed sign interacting nonlinearly, cascading kinetic energy from the large scales they were seeded at down to small scales where viscosity finally dissipates it. Watch, in the energy spectrum plot, how a real inertial range lines up with Kolmogorov’s predicted -5/3 power law over more than a decade of wavenumber – one of the most-tested predictions in classical physics, reproduced here from nothing but the bare incompressible equations.
The Kolmogorov -5/3 cascade in decaying 2D turbulence