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. Another follows Richardson’s qualitative cascade – a few large eddies folding into ever finer filaments until viscosity takes over. The last adds steady random forcing and a large-scale drag, so that the flow reaches a statistically steady state with a constant energy flux through the scales. Its time-averaged energy spectrum follows Kolmogorov’s predicted -5/3 power law over the range between the forcing scale and the drag scale.
Richardson’s cascade: big whorls have little whorls
The Kolmogorov -5/3 cascade in forced 2D turbulence