Persistent corotation of the large-scale flow of thermal convection and an immersed free body

Abstract

Inspired by the superrotation of the Earth's solid core, we investigate the dynamics of a free-rotating body as it interacts with the large-scale circulation (LSC) of the Rayleigh-Bénard thermal convection in a cylindrical container. A surprising and persistent corotation of both the free body and the LSC emerges, breaking the axial symmetry of the system. The corotational speed increases monotonically with the intensity of thermal convection, measured by the Rayleigh number Ra, which is proportional to the temperature difference between the heated bottom and cooled top. The rotational direction occasionally and spontaneously reverses, occurring more frequently at higher Ra. The reversal events follow a Poisson process; it is feasible that flow fluctuations randomly interrupt and reestablish the rotation-sustaining mechanism. This corotation is powered by thermal convection alone and promoted by the addition of a free body, enriching the classical dynamical system.

Keywords: dynamic system; fluid–structure interaction; symmetry breaking; thermal convection.

Fig. 1.

The RBC-rotor system and their corotation. (A) A cylindrical convection cell, filled with water, is heated from below and cooled at the top. A square planar rotor suspends vertically at the mid-height and centered along the RBC cell’s axis of symmetry, can rotate freely in response to fluid torques. (B) The rotor’s cumulative orientation θ versus time for various temperature differences ΔT. The orientation is defined positive when rotating counterclockwise, CCW, when viewed from above. (C) The magnitude of the mean rotation speed $\overline{|\omega |}$ increases with ΔT or the Rayleigh number Ra.

Fig. 2.

Corotation of the rotor and the LSC. (A) A unidirectional rotation of the rotor (solid line) and the LSC (dotted line) at a moderate ΔT = 10.5 °C. Inset: Eight thermistors are embedded in the bottom plate of RBC. (B) Time series of temperatures measured by the thermistors. Fitting the temperature data gives the LSC’s direction φ(t). For clarity on the plot, each T_i is shifted up by (i − 1)°C, i = 1, ..., 8. Every depression in the time series shows a moment when the cold flow of the LSC swipes by. Combined, eight time series of T_i show how LSC corotates with the rotor. (C) The offset angle α is measured from LSC to the rotor’s bisector, which remains negative during this unidirectional CCW rotation. The histogram of α is shown on the right. (D) At a higher ΔT = 28.6 °C, one of the many rotational reversals is shown. Insets show the orientation of the LSC (red arrow), the rotor (solid black line), and its bisector (dotted gray line). If rotating in the opposite direction, the offset angle α changes its sign. (E) The reversal of LSC is also evident from temperature signals. (F) With reversals, the histogram of α becomes double-peaked.

Fig. 3.

Flow structures and the mechanism for corotation. (A) Flow visualization reveals that the LSC has a tributary structure: one main flow across the rotor but a weaker circulation within each half cell. The ribbons represent the flow structures; red and blue colors show warm and cold streams, respectively. (B) In this top view, the main flow of the LSC forms an offset angle α with the rotor’s bisector (dotted gray line). The arrows show the direction of streams near the top (solid blue lines) and bottom plates (dashed red lines). The vertical streams of two tributaries (into the paper near point M and out near N) cause low-pressure regions near edges MM’ and NN’, leading to a net torque that drives the rotor CCW. Concurrently, the rotor acts as a flow guide, cooling areas near M and M’, heating areas near N and N’, and causing the LSC to turn CCW. (C) The interplay between the rotor and the LSC is depicted as a mass falling inside a potential well E_p, which is pushed forward by the rotor in the angular space. The angular distance between the mass and the bottom is the offset angle α. (D) When ΔT = 16.6 °C, the experimentally measured torque τ (circles) depends linearly on α. Integrating the linear fit (dotted line) to the experimental data gives a quadratic potential well E_p(α).

Fig. 4.

Reversal events and their statistical property. (A) Long time series of the rotor’s azimuthal orientation θ(t) at ΔT = 35.0 °C, with 47 reversals marked by red circles. (B) Zoom in to a short θ(t) segment in the gray dashed box of (A): two spontaneous reversals are seen. Right before each reversal, the rotor shows a burst of acceleration. (C) At ΔT = 12.0 °C, where spontaneous reversal rarely occurs, manually speeding up the rotor by turning it forward about 60° (two blue arrows) in a short time can successfully reverse the rotational direction. (D) The cumulative distribution function F(τ_r) =P(t_r≤ τ_r) for the residence time t_r, on log-linear scales. The dashed lines are fitted with the function F(τ_r)=1 − exp(−λτ_r), where the coefficient λ is calculated by maximum likelihood estimation. Inset: The mean residence time ${\overline{t}}_r$ decreases with ΔT.