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

Significance

Symmetry-breaking bifurcations are omnipresent in nature and in artificial settings, from quantum physics to dynamic systems, and can also be found in many fluid–structure interactions. In this study, we investigate the interplay between a freely rotatable body and thermal convective flows in a table-top experiment. We find a surprising and persistent corotation of the free body together with the main flow structures. The two corotational solutions, one clockwise and the other counterclockwise, result from a symmetry-breaking bifurcation of the coupled system, powered merely by the turbulent convective flow. We successfully interprets the mechanism of the fluid–structure interaction and explains the stochastic reversals between two corotational directions.

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.

Rayleigh-Bénard convection (RBC) has been widely studied as a paradigm for the understanding of turbulence, pattern formation, and dynamic systems (1–3). There, a fluid of depth L is subject to heating from below and cooling from the top, with a temperature difference ΔT. The imbalance of fluid densities, denser near the top and lighter at the bottom, drives the fluid into motion. A large-scale circulation (LSC), also called the “wind” of convection, arises from the background turbulent flows as the Rayleigh number Ra = βgΔTL³/νκ is sufficiently high (4–9). Here, g is the gravitational acceleration, β, ν, and κ are the thermal expansion coefficient, kinematic viscosity, and thermal diffusivity of the fluid, respectively. Number Ra is a measure of how strongly the fluid system is being forced. For a cylindrical convection cavity or cell with width-to-height aspect ratio not far from one, the LSC takes the form of a single vertical roll, which breaks the axial symmetry of the cell and takes a random orientation. The dynamics and fluctuation of this LSC have been well studied, including its azimuthally diffusive motion, reorientation, and sloshing (10–17).

So far, most of the studies on RBC have dealt with fixed boundary conditions; that is, the fluid-confining walls are fixed and rigid, and they do not respond to fluid forces. However, many real-world phenomena that involve thermal convection consist of movable boundaries or objects, which react to and also modify the flow structures. One well-known example is the geophysical problem of continental drift or plate tectonic (18–24), in that continents are carried around by the convective mantle, which is in turn modified by the continents. More recently, the superrotation of the Earth’s solid core has been discovered through seismic-wave analyses (25–29), but its rotation-sustaining mechanism is yet under investigations. Gravitational and electromagnetic couplings have been proposed by researchers (30–32). Given the fact that the solid core is suspended within the liquid core (33–37), thermal convection could also play an important role in affecting the solid core’s dynamics.

Experimental Design

In this work, we investigate the interplay between a passive and freely rotatable body, termed “rotor,” and the convective flows in a cylindrical RBC system. One of our motivations is to understand how the rotor, which could play a double role as a “windvane” to the LSC and a flow-guide to the flow, will possibly respond to and change the convective flow. Within the range of Ra explored in our experiments, we find that both the rotor and the LSC can persistently rotate together at the same rate, and their corotational direction can occasionally and spontaneously reverse. The observed directional corotation is found to be the result of the mutual interaction between the rotor and the LSC.

As shown in Fig. 1A, a cylindrical RBC cell, of inner diameter 25 cm and depth L = 25 cm, is heated uniformly from below by a heater working at a constant power, cooled at the top with a circulating water bath set at a constant temperature. The transparent wall of the cylinder allows flow inspection. The cell is filled with deionized and degassed water. The temperatures within the top and bottom plates are measured by thermistors embedded in the plates. For the experiments explored here, ΔT ranges from 1.3 °C to 38.0 °C, yielding a Ra that spans over an order of magnitude, from 5.1 × 10⁸ to 1.5 × 10¹⁰. Regardless of ΔT, the bulk temperature is maintained at T_m = 32.0  ±  1.5 °C, yielding a Prandtl number Pr = ν/κ ≈ 5.3.

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.

The rotor is a square plate, 23 by 23 cm in size, consisting of two rigid PVC plastic sheets that are firmly glued to a glass tube, which serves as the central axis. The rotor suspends in the fluid, centers along the axis of the convection cell, and is free to rotate with extremely low friction torque, around 1.5 dyne⋅cm. Each gap between the rotor’s edges and the sidewalls or plates measures 1 cm wide, allowing convective flows to cross. Additional experimental details are available in Materials and Methods and SI Appendix.

Results and Interpretations

At sufficiently high Ra, a surprising, persistent rotation of the rotor emerges. Fig. 1B shows the time series of the cumulative azimuthal orientations θ(t) of the rotor at various ΔT. At short times of several minutes, the motion of the rotor seems to fluctuate with small amplitudes. Over longer timescales, however, the rotational motion is persistent, and the directional speed is steady, yielding a well-defined angular speed $\overline{\omega }$. As ΔT is increased from 3.9 °C to 38.0 °C, $\overline{\omega }$ changes from 0.02 to 1.1 revolutions per hour. The corresponding Reynolds number associated with the flow speed U of the LSC, Re = UL/ν, ranges from around 3,000 to 6,500. The rotor shows roughly equal probability of rotating clockwise (CW) or counterclockwise (CCW). A remarkable feature of the persistent rotation is that its direction can be spontaneously reversed, followed by another persistent run in the opposite direction until the next reversal, as shown by the zigzag-shaped curves in Fig. 1B. At a fixed ΔT, the averaged speed $\overline{|\omega |}$ during a persistent run gives the same value, within ±4.6%, in either direction. This is also evident from Fig. 1B, as the positive and negative slopes have the same magnitude. The reversal events are more likely to occur at higher ΔT. When ΔT is below 12 °C (Ra <  4.7 × 10⁹), no reversal is observed within our experimental run time, which typically lasts for several days.

The rotation speed of the rotor is seen to monotonically increase with ΔT, as shown in Fig. 1C. If reversals happen, the values of $\overline{|\omega |}$ are the weighted average speeds obtained from segments uninterrupted by reversals, with the segments’ duration as the weights. When ΔT <  3.9 °C (Ra <  1.5 × 10⁹, blue region), the motion of the rotor is no longer directional but wanders around an arbitrary orientation with an averaged speed $\overline{\omega }\approx 0$.

While the orientation of the rotor is recorded, we use eight precalibrated thermistors in the bottom plate to determine the orientation of the LSC (14). These thermistors have an equal azimuthal separation and a distance of 8.5 cm from the center, shown by the inset of Fig. 2A. Their resistances are measured every 10 seconds and converted into temperatures. Due to the LSC, the temperatures at the warmer region are noticeably higher than the others, especially the one in the opposite cold side (10, 14). We identify the orientation of LSC by fitting 8 temperatures with function T_i = T₀ + Acos[(iπ/4)−φ], i = 1, 2, ..., 8, where φ(t) is the azimuthal position of the warmest region, defined as the direction of the LSC.

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. 2 A and D show how LSC rotates together with the rotor at the same average speed. Indeed, the rotation of the LSC is also evident in the time series of T₁ to T₈ in Fig. 2 B and E. The negative spikes of each curve are the manifestations of cold flows descending around that position, diametrically opposite to the warmer region.

The LSC is never seen to move along the surface of the rotor, seldom perpendicular to the surface, but forms an offset angle α. This angle, defined as α = θ − φ + π/2, can be as large as ±35°. If α is positive (measured from the LSC vector to the rotor’s bisector, CCW positive), the corotation is CW. If α is negative, however, the corotation is CCW. The corotation of the LSC and the rotor is not only reflected by the parallelism between θ(t) and φ(t) shown in Fig. 2 A and D but also in the narrow distributions of α shown in Fig. 2 C and F. There, the time series of α(t) are shown on the left and its histogram or probability distribution function (p.d.f.) from longer duration (around 24 h) on the right. After a reversal, the bisector switches to the other side of the LSC, i.e., flips the sign of α, as shown by the time series of α(t) and the double-peaked histogram in Fig. 2F. The corotation and the reversals of the rotor and the LSC are also shown in Movies S1 and S2.

To understand the rotor–LSC interaction and the origin of their corotation, we carefully examine the flow structures in the convection cell. Seeding with flow tracers and illuminating them with laser sheets allow direct observation of the flow fields in different vertical and horizontal cross-sections. We find that the planar rotor changes the typical single-rolled LSC into one mainstream and two tributaries, as shown in Fig. 3 A and B, one in 3D depiction and the other as a Top view. The mainstream retains a vertical flywheel structure, whose horizontal streams cross the rotor from the top and bottom gaps, and vertical streams along the sidewall. The two tributary flows (each remains within the half cell divided by the rotor) join the mainstream near the sidewall. In Fig. 3 A and B, the tributary flows sink along MM’ and rise along N’N, respectively, before joining the mainstream. Two similar flow structures exist when the corotation takes two different directions and are mirror-symmetric about the plane of the rotor’s bisector (SI Appendix, Fig. S2).

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).

The rotor acts as a divider for the RBC cell and also a guide plate to the convective flows. The cylindrical cell is divided into two less connected half cells, which allows the formation of a weaker circulation within each space. When α is negative, as shown in Fig. 3A, the rotor guides the flow, making regions near points N’ and N warmer and regions near M and M’ cooler. Therefore, at the bottom plate, a cooler area at M’ effectively repels the LSC’s warmest area W, while the warmer area N’ pulls it. The combined effect causes the area W to turn CCW. Similarly, at the top plate, the LSC’s coldest area C is pulled by the cold area M and pushed by area N. Overall, the whole LSC is turned CCW, diagonally at the bottom and top plates. If α is positive, however, the flow structure establishes a pattern that is mirror-symmetric about the plane of the rotor’s bisector, as depicted in SI Appendix, Fig. S2. The above process will take place in a symmetric fashion, resulting in a CW rotation of the LSC.

Meanwhile, the dominant horizontal streams of the LSC tend to reorient the planar rotor perpendicular to itself, or α → 0, so that the rotor is turned consequently. Similar effects have been demonstrated in earlier studies (38–42). Generally, when a flow encounters a tilted plate, the center of action, which is close to the stagnation point where the flow splits into two branches around the plate, is not at the center of the plate but closer to the leading end. For both the cold stream at the Top and the warm stream at the Bottom, torques of the same sign will turn the rotor in the direction reducing |α|. In addition, as shown by the flow diagram in Fig. 3B, the tributary flows along MM’ and N’N create low-pressure regions, due to the Bernoulli effect, on opposite sides of the rotor. The imbalanced forces lead to a net torque that also drives the rotor and decreases |α|. These two mechanisms are likely present at the same time, but their relative contribution is yet to be investigated.

While the rotor responds to the above two types of torques from the LSC, the latter is being modified by the rotor through its flow-guide effect. This two-way feedback mechanism makes the rotor constantly follow the LSC, which is in turn modified and pushed by the rotor.

The above reasoning is depicted in our “accompanying potential energy well” picture to elucidate the corotation of the rotor and LSC. As shown by the schematic in Fig. 3C, the rotor (its bisector is represented as a sphere) tends to fall toward the bottom of the potential well generated by the LSC. At the same time, since the rotor actively guides the thermal flow and drives the LSC to rotate, this process is represented by the small person below the sphere pushing the potential well forward. The falling of the rotor to the basin (where α = 0) and the advancing of the potential well create a situation in which the rotor remains on one side of the well and maintains a nonzero |α|. As a result, the corotation solution for the rotor and LSC emerges.

The potential well discussed above can be unveiled experimentally. We measure the static torques τ(α) experienced by the rotor in a fixed LSC, by the minute deflection of an elastic beam outside of the fluid. Fig. 3D shows that the torque τ is roughly linear with α, suggesting that the potential well E_p, the integral of −τ over α, is quadratic with α, as pictured in Fig. 3C. More details about the torque measurements are available in Materials and Methods and SI Appendix.

Next, we study the mechanism of the spontaneous reversals. Fig. 4A shows the time series of θ(t) over 130 h at ΔT ≈ 35.0 °C, with all 47 reversals marked by red circles. Each reversal is identified if the rotational direction flips and maintains for at least 1 rad. Examining the time series of θ(t), one notices that each reversal is preceded by an abrupt increase of angular speed, likely due to the fluctuation in thermal turbulence, making the rotor’s bisector rapidly rotate forward and switch the sign of α. The corotation is then reversed. This process is shown by the expanded view of two consecutive reversals in Fig. 4B and also pictured by the inset of Fig. 2D. To confirm the above mechanism, we aim to manually reverse the corotation at a lower ΔT where the spontaneous reversal is infrequent. As shown in Fig. 4C, at ΔT ≈ 12.0 °C, the rotor is quickly turned forward by hand, for about 60° (enough to change the sign of α), and the rotational direction is successfully reversed. With our potential well picture, a burst of acceleration, either by the thermal fluctuation or external forcing, pushes the rotor’s bisector to the other side of the potential well, flips the signs of α and τ, and then reverses the rotational direction. Also shown in SI Appendix, Fig. S2, during the reversal, the overall flow structure undergoes a configuration change, switches between two symmetric and bifurcated states. This is likely caused by a cascade of events that lead to LSC reconfiguration and reversal.

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.

Long time series of the corotation also reveal statistical properties of the reversals. We define the time interval between successive reversals as the residence time t_r, which is found to follow an exponential distribution P(t_r)=λ ⋅ exp(−λt_r). This is evident by its cumulative distribution function F(τ_r)=P(t_r ≤ τ_r) with its fitting shown in Fig. 4D, when ΔT = 35.0 °C and 27.8 °C. The fitting function is F_fit(τ_r)=1 − exp(−λτ_r), and the coefficient λ is calculated by maximum likelihood estimation. The exponential distribution suggests that the reversals occur randomly and follow a Poisson process. The inverse of the coefficient λ is the mean residence time ${\overline{t}}_r$. For instance, ${\overline{t}}_r\approx 2.79$ and 4.94 h when ΔT = 35.0 °C and 27.8 °C, respectively. The Poissonian nature of reversals allows us to estimate the mean residence time at a lower ΔT, even though the reversals are rare. The inset of Fig. 4D shows how ${\overline{t}}_r$ decreases with increasing ΔT, as the reversal events become more frequent.

Conclusions

In summary, we report an experimental finding on the corotation between a free body and the LSC in an axial symmetric RBC system. The rotational rate increases monotonically with the flow intensity, measured by ΔT or Ra. The corotational direction can spontaneously and randomly reverse, and the reversal rate increases with ΔT. Such a dynamical state emerges as a result of spontaneous symmetry breaking, powered by turbulent thermal convection. We find that the mutual interactions between the free body and the LSC (each plays passive and active roles in affecting the other) lead to the sustained rotation. Turbulence does cause reversals, but it is often dictated by the feedback mechanisms at play: The LSC drives the free body to rotation, and the latter guides the LSC to new orientations.

Materials and Methods

Details on Experimental Setup.

The cylindrical container of the Rayleigh–Bénard convection system is made of a transparent acrylic tube of inner diameter 25 cm, height 25 cm, and thickness 0.5 cm. Its axis of symmetry is placed vertically. The fluid within the convection cell is heated uniformly from below by a heater that works at a constant power H and cooled from above with a circulating water bath set at a desired temperature. The cell is filled with deionized and degassed water. The top and bottom plates are made of aluminum alloy. To visualize the flows, we seed the water with microparticles and illuminate them with planar laser light. The temperatures within the top and bottom plates are measured by thermistors embedded in the plates merely 1 mm away from the surfaces that touch the fluid. Their resistances are measured by a multimeter at 10-s intervals and converted to temperatures with precalibrated relations. The bulk temperature is measured by a thermistor near the center of the cell.

The Assembling and Positioning of the Rotor.

A side view of the rotor, together with the convection cell, is schematically shown in SI Appendix, Fig. S1. The rotor is a square plate, 23 by 23 cm in size, consisting of two rigid PVC plastic sheets of 0.4 mm in thickness that are firmly glued to an air-filled glass tube of diameter 5 mm, which serves as the central axis. The overall buoyant rotor suspends and centers at the vertical axis of the convection cell by methods detailed below, leaving gaps of 1 cm between its edges and the convection cell’s sidewalls/plates.

To keep the assembled rotor stably and vertically afloat, a low-density Styrofoam cylinder and an extra cylindrical weight are attached to the top and bottom of the glass tube, respectively, to make the center of buoyancy high and the center of mass low. When fully submerged, the assembly is slightly buoyant, and it is forced down by a cantilever from outside of the convection cell, through an opening in the center of the top plate. The cantilever, mounted with a cap jewel bearing, presses the rotor down, maintains its height, and aligns its axis with that of the convection cell by adjusting an X–Y translation stage. Most importantly, the suspension mechanism permits the free rotation of the rotor with extremely low-friction torque, around 1.5 dyne⋅cm. Over the top cooling plate, a horizontal pointer is glued to the axis of the rotor. A marker is placed at the tip of the pointer. The orientation of the rotor, represented by the azimuthal position of the marker, is recorded by a camera over the setup.

Methods of Static Torque Measurement.

The static flow-induced torque τ(α) is measured using an elastic arm deflected by the rotor. The top view of the experimental apparatus is schematically shown in SI Appendix, Fig. S3. First, the orientation of the LSC is locked by adding localized heating and cooling to the bottom and top plates, respectively, in addition to the homogeneous heating and cooling used in the experiments. The two regions are diagonally or diametrically opposite. To achieve localized extra heating, a small circular foil heater, powered by an additional DC source working at a constant power, is attached to the lateral surface of the bottom plate. For the top plate, extra side-cooling is applied by passing colder water from another water circulator to a small region in direct contact with the lateral surface of the plate.

Once the LSC is fixed, an elastic beam is clamped at various positions and orientations, outside and above the convection cell, so that α is changed incrementally. This elastic beam is a thin PVC sheet, whose flexural bending rigidity is experimentally measured. The free end of the beam is in touch with the top pointer that is rigidly connected to the axis of the rotor. The flow-induced torque experienced by the rotor slightly deflects the beam through the tip of the pointer and is resisted by a force f exerted by the beam. The static torque is measured by the minute deflection of the free end of the beam. A camera over the setup records the deflection, which gives the force measurement. The flow-induced torque applied on the rotor equals the force f multiplied by the length of the pointer l.

Further details of the experimental methods can be found in SI Appendix.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

The co-rotation of the rotor and the LSC when viewed from above, at ΔT = 10.5°C and Ra = 4.3×10⁹. The color ring shows the temperature distribution at the bottom plate, obtained and interpolated from the measurements of eight thermistors, from which the LSC is determined (shown by the red arrow, which points to the warmest region of the bottom plate). The length of the red arrow represents the relative magnitude of the LSC. The solid black line and dotted gray line are the orientations of the rotor and its bisector, respectively. The co-rotation of the rotor and the LSC is persistently counter-clockwise. The orientation and magnitude of the LSC fluctuate due to the turbulent flows in the convective system.

Movie S2.

The co-rotation of the rotor and the LSC when viewed from above, at ΔT = 26.5°C and Ra = 1.1×10¹⁰. Several reversals are recorded in this time series, with the LSC switching between two opposing sides of the rotor’s bisector.

Data, Materials, and Software Availability

Detailed experimental design, protocol, as well as key instrument information, are available in SI Appendix. MATLAB figures data have been deposited in Figshare (https://doi.org/10.6084/m9.figshare.22730036.v1) (43).