Prograde, retrograde, and oscillatory modes in rotating Rayleigh–Bénard convection

Susanne Horn, Peter J. Schmid

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Rotating Rayleigh–B´enard convection is typified by a variety of regimes with very distinct flow morphologies that originate from several instability mechanisms. Here we present results from direct numerical simulations of three representative setups: First, a fluid with Pr “ 6.4, corresponding to water, in a cylinder with a diameter-to-height aspect ratio of Γ “ 2, secondly, a fluid with Pr “ 0.8, corresponding to SF6 or air, confined in a slender cylinder with Γ “ 0.5, and thirdly, the main focus of this paper, a fluid with Pr “ 0.025, corresponding to a liquid metal, in a cylinder with Γ “ 1.87. The obtained flow fields are analysed using the sparsity-promoting variant of the dynamic mode decomposition
(DMD). By means of this technique, we extract the coherent structures that govern the dynamics of the flow, as well as their associated frequencies. In addition, we follow the temporal evolution of single modes and present a criterion to identify their direction of travel, i.e. whether they are precessing prograde or retrograde. We show that for moderate Pr a few dynamic modes suffice to accurately describe the flow. For large aspect ratios, these are wall-localised waves that travel retrograde along the periphery of the cylinder. Their DMD frequencies agree with the predictions of linear stability theory. With increasing Rayleigh number Ra, the interior gradually fills with columnar vortices, and eventually a regular pattern of convective Taylor columns
prevails. For small aspect ratios and close enough to onset, the dominant flow structures are body modes that can precess either prograde or retrograde. For Pr “ 0.8, DMD additionally unveiled the existence of so far unobserved low-amplitude oscillatory modes. Furthermore, we elucidate the multi-modal character of oscillatory convection in low Prandtl number fluids. Generally, more dynamic modes must be retained to accurately approximate the flow. Close to onset, the flow is purely oscillatory and the DMD reveals that these high-frequency modes are a superposition of oscillatory columns and cylinderscale inertial waves. We find that there are co-existing prograde and retrograde modes, as well as quasi-axisymmetric torsional modes. For higher Ra, the flow also becomes unstable to wall modes. These low-frequency modes can both co-exist with the oscillatory modes, and also couple to them. However, the typical flow feature of rotating convection at moderate Pr , the quasi-steady Taylor vortices, is entirely absent in low Pr flows.
Original languageEnglish
Pages (from-to)182-211
Number of pages30
JournalJournal of Fluid Mechanics
Early online date13 Oct 2017
Publication statusPublished - 25 Nov 2017
Externally publishedYes


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