Invariant solutions in homogeneous Rayleigh-Bénard convection

Thermal convection with boundary conditions in both horizontal and vertical directions, known as the homogeneous Rayleigh-Bénard convection, is a simple model for the bulk of convection cells, representing the ultimate turbulent state in which the vertical heat flux is independent of the thermal conductivity of fluids. In this system, however, exact solutions exponentially grow in time, leading to high intermittency in heat transfer and difficulties in stability and bifurcation analysis of the nonlinear dynamical system. We have found three-dimensional steady solutions to the Boussinesq equations for the homogeneous Rayleigh-Bénard convection, which bifurcate from a two-dimensional steady solution with a mirror symmetry about the horizontal plane, using a Newton-Krylov iteration. The nonlinear invariant solutions have hierarchical multiscale vortex structures and exhibit significantly higher heat flux than the two-dimensional steady solution. These solutions will help us better understand the homogeneous Rayleigh-Bénard convection and the conventional Rayleigh-Bénard convection.