Abstract
We have investigated invariant solutions (time-independent solutions) to the Boussinesq equations which are the simplified model for Rayleigh–Bénard convection. In this work, steady and unsteady flows are obtained for three-dimensional domains between horizontal parallel plates with no-slip boundary conditions, which are held at constant temperature difference. It is well known that Rayleigh–Bénard convection is characterized by the Rayleigh number Ra and the Prandtl number Pr. For Pr=1, an unstable three-dimensional steady solution, which bifurcates from a thermal conductive solution at Ra~103, is obtained up to Ra~107 by using Newton-Krylov iteration. At high Ra, the three-dimensional solution consists of large-scale convection cells and small-scale vortex structures. The multi-scale invariant solution exhibits a scaling of the Nusselt number Nu with Ra, Nu~Ra0.31, observed turbulent Rayleigh–Bénard convection, and represents well the spatial structure and statistics in a turbulent state.
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