The two-dimensional Navier-Stokes equations with a large-scale instability of the Kuramoto-Sivashinsky type: Numerical exploration on the Connection Machine

Abstract

The two-dimensional Navier-Stokes equations with a large-scale instability of the Kuramoto-Sivashinsky type, describing marginally negative eddy-viscosity situations, is simulated on a Connection Machine CM-2. Up to millions of time steps at the resolution 2562 and tens of thousands at the resolution 10242 are performed. Advantage is taken of a novel complex variable form of the two-dimensional Navier-Stokes equations, which requires only two complex FFTs per time step. A linear growth phase, a disorganized inverse cascade phase, and a structured vortical phase are successively observed. In the vortical phase monopolar and multipolar structures are proliferating and display strongly depleted nonlinearities.