Building a reduced model for nonlinear dynamics in Rayleigh-Bénard convection with counter-rotating disks

Abstract

A reduced model to decrease the number of degrees of freedom of the discretized Navier-Stokes equations to a small set that nevertheless captures the essential dynamics of the flow is proposed. The Rayleigh-Bénard convection problem in a cylinder of aspect ratio one where the lower and upper disks, maintained at hot and cold temperatures, respectively, rotate at equal and opposite angular velocities has been chosen to test the technique. The nonlinear dynamics is rich and complex when the temperature difference between disks and their angular velocity is varied. Representatives states--stationary, periodic near sinusoidal, and near heteroclinic--are presented. In each case, the reduced model is compared with temporal integration, and we show that 41 degrees of freedom are sufficient to reproduce the signal. We discuss the strengths and weaknesses of the algorithm by which we build our reduced model.