Onset of Küppers–Lortz-like dynamics in finite rotating thermal convection

Abstract

The onset of thermal convection in a finite rotating cylinder is investigated using direct numerical simulations of the Navier–Stokes equations with the Boussinesq approximation in a regime in which spatio-temporal complexity is observed directly after onset. The system is examined in the non-physical limit of zero centrifugal force as well as with an experimentally realizable centrifugal force, leading to two different paths to Küppers–Lortz-like spatio-temporal chaos. In the idealized case, neglecting centrifugal force, the onset of convection occurs directly from a conduction state, resulting in square patterns with slow roll switching, followed at higher thermal driving by straight roll patterns with faster roll switching. The case with a centrifugal force typical of laboratory experiments exhibits target patterns near the theoretically predicted onset of convection, followed by a rotating wave that emerges via a Hopf bifurcation. A subsequent Hopf bifurcation leads to ratcheting states with sixfold symmetry near the axis. With increasing thermal driving, roll switching is observed within the ratcheting lattice before Küppers–Lortz-like spatio-temporal chaos is observed with the dissolution of the lattice at a slightly stronger thermal driving. For both cases, all of these states are observed within a 2% variation in the thermal driving.

Modulated waves in a periodically driven annular cavity

“…For example, in periodically modulated TaylorCouette flow (Marques & Lopez 1997Weisberg, Kevrekidis & Smits 1997), Sinha, Kevrekidis & Smits (2006) reported on experimental results that indicated a direct transition from the periodic axisymmetric basic state to complicated spatiotemporal flow in regimes where linear Floquet analysis and nonlinear subspacerestricted simulations predicted pure mode solutions, but full nonlinear simulations revealed that the complex spatio-temporal behaviour was a result of a cascade of symmetry-breaking bifurcations in an extremely narrow range of Reynolds numbers, much finer than could be resolved in the experiments (Avila et al 2007). Similar cascades to spatio-temporal complexity have been found in rotating thermal convection, where the onset of Küppers-Lortz dynamics is reported to be a 'direct' transition from the basic state in experiments (Bodenschatz, Pesch & Ahlers 2000), whereas full DNS shows that the complex dynamics also result from a cascade of symmetry-breaking bifurcations over an extremely narrow range of parameter space (Rubio, Lopez & Marques 2010).…”

Confined rotating convection with large Prandtl number: Centrifugal effects on wall modes

“…When there is bulk convection at low glycerine concentrations, the time evolution of the patterns is of Küppers-Lortz type [33], with rolls changing their orientation in time (see Refs. [13,19] for similar simulations with Fr = 0). However, this dynamic is destroyed for high glycerine concentrations as centrifugal buoyancy effects become more dominant.…”

“…For Prandtl numbers greater than 100, there are few studies in rotating convection. There exist some theoretical studies in the limit of infinite Prandtl number [6][7][8], but these neglect two aspects of rotating convection which can be dominant, especially in a realistic physical setting, namely, confinement and centrifugal buoyancy [9][10][11][12][13][14]. Experiments are often designed to operate in parameter regimes where the Froude number Fr is small so that centrifugal effects can be neglected [15].…”

Heat transport and flow structure in rotating Rayleigh–Bénard convection

“…Experimental and numerical investigations concerning the onset of convective heat transfer and the pattern formation in cylindrical cells just above the onset under the influence of rotation have been performed by many authors, see e.g. [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45].…”