Three-dimensional knot convection in a layer heated from below

Abstract

Steady three-dimensional convection flows induced by the knot instability of two-dimensional convection rolls are studied numerically for various Prandtl numbers. The Galerkin method is used to obtain the three-dimensional solutions of the basic equations in the case of rigid, infinitely conducting boundaries. These solutions exhibit the typical knot-like structure superimposed onto the basic rolls. The Nusselt number and kinetic energy of motion do not differ much for two- and three-dimensional solutions and the toroidal part of the kinetic energy associated with vertical vorticity always remains a small fraction of the total in the case of the knot solution. The analysis of the steady solutions is complemented by a stability analysis with respect to disturbances that fit the same horizontal periodicity interval as the knot solution. All instabilities correspond to Hopf bifurcations. Some example of finite-amplitude oscillatory knot convection are presented.