Imposed Temperature Difference Research Articles

Side-heated Rayleigh–Bénard convection

Unlike in solids, heat transfer in fluids can be greatly enhanced due to the presence of convection. Under gravity, an unevenly distributed temperature field results in differences in buoyancy, driving fluid motion that is seen in Rayleigh–Bénard convection (RBC). In RBC, the overall heat flux is found to have a power-law dependence on the imposed temperature difference, with enhanced heat transfer much beyond thermal conduction. In a bounded domain of fluid such as a cube, how RBC responds to thermal perturbations from the vertical sidewall is not clear. Will sidewall heating or cooling modify flow circulation and heat transfer? We address these questions experimentally by adding heat to one side of the RBC. Through careful flow, temperature and heat flux measurements, the effects of adding side heating to RBC are examined and analysed, where a further enhancement of flow circulation and heat transfer is observed. Our results also point to a direct and simple control of the classical RBC system, allowing further manipulation and control of thermal convection through sidewall conditions.

Enhancement of power density and hydrogen productivity of the reverse electrodialysis process by optimizing the temperature gradient between the working solutions

Reverse electrodialysis (RED) is an emerging technology that converts the salinity gradient energy (SGE) embedded in two solutions of different concentrations into electricity or hydrogen productivity. Given that two solutions containing SGE in nature are often at different temperatures, this study experimentally investigated the effects of the temperature and salinity ratio of the working solutions on the performance of RED systems. The results show that the power density and hydrogen productivity of RED systems could be markedly improved by adjusting the imposed temperature difference on the basis of the salinity gradient. The degree of improvement mainly depends on the magnitude and direction of the temperature difference between the high- and low-concentration solutions. The enhancement effect was more pronounced under a negative temperature difference (NTD) than under a positive temperature difference (PTD). Additionally, the improvement in system performance owing to the temperature difference was not affected by the endothermic or exothermic effects of different solutes in the water. At a NTD of 25℃ and a concentration of 0.1 M aqueous NaOH as the electrode solution, a RED system based on LiBr solution with a salinity ratio of 4.5 M/0.02 M as the working solution achieved a maximum power density of 0.754 ± 0.006 W∙m−2 and the highest hydrogen productivity of 1.128 ± 0.019 mol∙m−2∙h−1, which was 15.1% and 5.5% higher than the hydrogen productivity under LiCl and NaCl solution conditions, respectively. These results show the great potential of LiBr solution in the field of RED hydrogen productivity.

Thermal rectification in novel two-dimensional hybrid graphene/BCN sheets: A molecular dynamics simulation

The graphene-like monolayer of carbon, boron and nitrogen that maintains the native hexagonal atomic lattice (BCN), is a novel semiconductor with special thermal properties. Herein, with the aid of a non-equilibrium molecular dynamics approach (NEMD), we study phonon thermal rectification in a hybrid system of pure graphene and BCN (G-BCN) in various configurations under a series of positive and negative temperature gradients. We begin by investigating the relation of thermal rectification to sample’s mean temperature, T, and the imposed temperature difference, ΔT, between the two heat baths at its ends. We then move to explore the effect of varying strain levels of our sample on thermal rectification, followed by Kapitza resistance calculations at the G-BCN interface, which shed light on the interface effects on thermal rectification. Our simulation results reveal a BCN-configuration-dependent behavior of thermal rectification. Finally, the underlying mechanism leading to a preferred direction for phonons is studied using phonon density of states (DOS) on both sides of the G-BCN interface.

Thermally enhanced osmotic power generation from salinity difference

Recently, membrane-based power generation from salinity difference has been in the spotlight as a blue energy harvesting, but achieving a high power density and conversion efficiency still remains as a major challenge in this approach. Instead of developing new membranes, regulating the thermal condition within the membrane has been proposed as a way to enhance the power generation by several numerical studies, but this concept has rarely been explored through the systematic experimental studies due to the difficulty of imposing a controlled temperature gradient within the membrane. In this work, we experimentally and systematically study how the temperature difference can influence osmotic power generation using a commercial polycarbonate membrane and demonstrate that even when a thermal gradient is negligibly small within the nanoporous membrane, it is still possible to achieve a significant enhancement of the power generation. We propose that the effective ion concentration at the interfacial region between the reservoir and the membrane varies with the direction of the imposed temperature difference, such that the opposite direction of salinity and temperature differences can lead up to 5.3 times power enhancement as a result of the increase of the effective ion concentration ratio across the membrane. As an example of practical applications, we apply our findings to a floating type nanogenerator by incorporating a solar absorber to generate the temperature difference spontaneously under solar radiation conditions, and the results with the nanogenerator show that the power generation is indeed enhanced under both simulated and actual solar radiation conditions. We believe that our approach can be applied to any nanoporous membrane regardless of its thermal property, and therefore would provide a practical path to the power enhancement of reverse electrodialysis systems.

Scaling of Turbulence and Microphysics in a Convection–Cloud Chamber of Varying Height

AbstractThe convection–cloud chamber enables measurement of aerosol and cloud microphysics, as well as their interactions, within a turbulent environment under steady‐state conditions. Increasing the size of a convection–cloud chamber, while holding the imposed temperature difference constant, leads to increased Rayleigh, Reynolds and Nusselt numbers. Large–eddy simulation coupled with a bin microphysics model allows the influence of increased velocity, time, and spatial scales on cloud microphysical properties to be explored. Simulations of a convection–cloud chamber, with fixed aspect ratio and increasing heights of H = 1, 2, 4, and (for dry conditions only) 8 m are performed. The key findings are: Velocity fluctuations scale as H1/3, consistent with the Deardorff expression for convective velocity, and implying that the turbulence correlation time scales as H2/3. Temperature and other scalar fluctuations scale as H−3/7. Droplet size distributions from chambers of different sizes can be matched by adjusting the total aerosol injection rate as the horizontal cross‐sectional area (i.e., as H2 for constant aspect ratio). Injection of aerosols at a point versus distributed throughout the volume makes no difference for polluted conditions, but can lead to cloud droplet size distribution broadening in clean conditions. Cloud droplet growth by collision and coalescence leads to a broader right tail of the distribution compared to condensation growth alone, and this tail increases in magnitude and extent monotonically as the increase of chamber height. These results also have implications for scaling within turbulent, cloudy mixed‐layers in the atmosphere, such as fog layers.

Rayleigh–Bénard thermal convection perturbed by a horizontal heat flux

In Rayleigh–Bénard convection, it has been found that the amount of heat passing through the fluid has a power-law dependence on the imposed temperature difference. Modifying this dependence, either enhancing or reducing the heat transfer capability of fluids, is important in many scientific and practical applications. Here, we present a simple means to control the vertical heat transfer in Rayleigh–Bénard convection by injecting heat through one lateral side of the fluid domain and extracting the same amount of heat from the opposite side. This horizontal heat flux regulates the large-scale circulation, and increases the heat transfer rate in the vertical direction. Our numerical and theoretical studies demonstrate how a classical Rayleigh–Bénard convection responds to such a perturbation when the system is near or well above the onset of convection.

Patterning Behavior of Hybrid Buoyancy-Marangoni Convection in Inclined Layers Heated from Below

Alongside classical effects driven by gravity or surface tension in non-isothermal fluids, the present experimental study concentrates on other exotic (poorly known) modes of convection, which are enabled in a fluid layer delimited from below by a hot plate and unbounded from above when its container is inclined to the horizontal direction. By means of a concerted approach based on the application of a thermographic visualization technique, multiple temperature measurements at different points and a posteriori computer-based reconstruction of the spatial distribution of wavelengths, it is shown that this fluid-dynamic system is prone to develop a rich set of patterns. These include (but are not limited to), spatially localized (compact) cells, longitudinal wavy rolls, various defects produced by other instabilities and finger-like structures resulting from an interesting roll pinching mechanism (by which a single longitudinal roll can be split into two neighboring rolls with smaller wavelength). Through parametric variation of the tilt angle, the imposed temperature difference and the volume of liquid employed, it is inferred that the observable dynamics are driven by the ability of gravity-induced shear flow to break the in-plane isotropy of the system, the relative importance of surface-tension-driven and buoyancy effects, and the spatially varying depth of the layer. Some effort is provided to identify universality classes and similarities with other out-of-equilibrium thermal systems, which have attracted significant attention in the literature.

On the Rarefied Thermally-Driven Flows in Cavities and Bends

This study examined rarefied thermally-driven flow in a square cavity (Case 1) and rectangular bend (Case 2), with various uniform wall temperatures in two dimensions. We employed the direct simulation Monte Carlo (DSMC) to solve problems with a wide range of Knudsen numbers Kn = 0.01 to 10, and the discrete unified gas kinetic scheme (DUGKS) solver was used at Kn = 0.01. The scenario was that, in case 1, the bottom side and its opposite were set hot, and the other sides were set cold. Diffuse reflector boundary conditions were set for all walls. The imposed temperature differences created four primary vortices. The results of the continuum set of equations of the slow non-isothermal flow (SNIT) solver proved that the primary vortices in the square cavity were caused by nonlinear thermal stress effects, and other smaller vortices appearing at Kn = 0.01, 0.1 were brought about by thermal creep processes. As the Kn increased, vortices generated by thermal creep disappeared, and eddies created by nonlinear thermal stress occupied the cavity. In case 2, i.e., a rectangular bend, two sides were set cold, and the others were hot. Two primary vortices were formed, which were caused by nonlinear thermal stress effects. The direction of streamlines in the two main vortices was opposite, from the warm to the cold zone, as some eddies on the left were counterclockwise, and others were clockwise.

Taylor-Couette flow with mixed convection heat transfer and variable properties in a horizontal annular pipe

Taylor-Couette flows in a horizontal annular gap between finite coaxial cylinders in rotor-stator configuration are numerically investigated. The inner cylinder (rotor) rotates at a constant angular velocity while the outer cylinder (stator) is at rest. They are limited at their extremities by two fixed walls that prevent axial fluid-flow. In addition, a heat transfer is generated by an imposed temperature difference, with the rotor hotter than the stator while the end-walls are adiabatic. The fluid physical properties are temperature dependent. This non-linear physics problem, with a strong coupling of the conservation equations and boundary conditions, is solved by a finite volume method with numerical schemes of second order space and time accuracies. The radius and aspect ratios and the Taylor, Grashof, and Prandtl numbers are the control parameters. The developed numerical code has been tested for different meshes and perfectly validated. Extensive calculations have been made in large ranges of the Taylor and Grashof numbers to analyze the Taylor-Couette flow in convection modes. The results highlight the dynamic and thermal instabilities generated in the Taylor-Couette flow from the appearance of Ekman cells to the Taylor vortex propagation in the entire annulus. The combined effect of these vortices with the secondary flow improves the heat transfer. Furthermore, the influence of the physical properties in the radial direction is more marked in the vicinity of the walls. Finally, we propose an empirical correlation of the Nusselt number in the studied parameter ranges.

An Investigation into the Behavior of Non-Isodense Particles in Chaotic Thermovibrational Flow

The ability to control the distribution of particles in a fluid is generally regarded as a factor of great importance in a variety of fields (manufacturing processes, biomedical applications, materials engineering and various particle separation processes, to cite a few). The present study considers the hitherto not yet addressed situation in which solid spherical particles are dispersed in a non-isothermal fluid undergoing turbulent vibrationally-induced convection (chaotic thermovibrational flow in a square cavity due to vibrations perpendicular to the imposed temperature difference). Although the possibility to use laminar thermovibrational flows (in microgravity) and turbulent flows of various types (in normal gravity conditions) to induce the accumulation of solid mass dispersed in a non-isodense fluid is already known, the interplay of finite-size finite-mass particles with chaotic flow in weightlessness conditions has never been considered. In the present study this subject is tackled through direct numerical solution of the fluid and particle tracking equations in the framework of a one-way coupling approach. Results are presented for relatively wide ranges of vibrational amplitude, particle size and density.

Analytical modeling of conjugate heat transfer between a bed of phase change material and laminar convective flow

Latent heat of a phase change material (PCM) is commonly used for energy storage and thermal management. A flat bed configuration is commonly used in latent energy storage systems, in which, fluid flow over a PCM bed results in heat transfer to/from and melting/solidification of the PCM. Understanding the nature of heat transfer in such systems is critical for design and performance optimization. This paper presents an analytical model for conjugate heat transfer in a flat bed latent energy storage system. The model solves the governing energy conservation equations by utilizing the interfacial conditions to iterate between the PCM and fluid flow. Good convergence of the iterative technique is demonstrated. The model is used to study phase change propagation and energy storage as functions of various problem parameters such as imposed temperature difference, flow velocity and thermal properties of PCM and fluid flow. It is found that the convective heat transfer coefficient between the PCM and fluid flow is not constant, but, rather, is a function of space and time, and is affected by flow speed and properties, as well as thermal properties of the PCM. The general methodology presented here can be adapted for annular and other latent energy storage configurations. This work contributes towards a fundamental understanding of heat transfer processes in an important energy storage system that is commonly encountered in practical applications.

The Zoo of Modes of Convection in Liquids Vibrated along the Direction of the Temperature Gradient

Thermovibrational flow can be seen as a variant of standard thermogravitational convection where steady gravity is replaced by a time-periodic acceleration. As in the parent phenomena, this type of thermal flow is extremely sensitive to the relative directions of the acceleration and the prevailing temperature gradient. Starting from the realization that the overwhelming majority of research has focused on circumstances where the directions of vibrations and of the imposed temperature difference are perpendicular, we concentrate on the companion case in which they are parallel. The increased complexity of this situation essentially stems from the properties that are inherited from the corresponding case with steady gravity, i.e., the standard Rayleigh–Bénard convection. The need to overcome a threshold to induce convection from an initial quiescent state, together with the opposite tendency of acceleration to damp fluid motion when its sign is reversed, causes a variety of possible solutions that can display synchronous, non-synchronous, time-periodic, and multi-frequency responses. Assuming a square cavity as a reference case and a fluid with Pr = 15, we tackle the problem in a numerical framework based on the solution of the governing time-dependent and non-linear equations considering different amplitudes and frequencies of the applied vibrations. The corresponding vibrational Rayleigh number spans the interval from Raω = 104 to Raω = 106. It is shown that a kaleidoscope of possible variants exist whose nature and variety calls for the simultaneous analysis of their temporal and spatial behavior, thermofluid-dynamic (TFD) distortions, and the Nusselt number, in synergy with existing theories on the effect of periodic accelerations on fluid systems.

Invariant states in inclined layer convection. Part 1. Temporal transitions along dynamical connections between invariant states

Thermal convection in an inclined layer between two parallel walls kept at different fixed temperatures is studied for fixed Prandtl number Pr=1.07. Depending on the angle of inclination and the imposed temperature difference, the flow exhibits a large variety of self-organized spatio-temporal convection patterns. Close to onset, these patterns have been explained in terms of linear stability analysis of primary and secondary flow states. At larger temperature difference, far beyond onset, experiments and simulations show complex, dynamically evolving patterns that are not described by stability analysis and remain to be explained. Here we employ a dynamical systems approach. We construct stable and unstable exact invariant states, including equilibria and periodic orbits of the fully nonlinear three-dimensional Oberbeck-Boussinesq equations. These invariant states underlie the observed convection patterns beyond their onset. We identify state-space trajectories that, starting from the unstable laminar flow, follow a sequence of dynamical connections between unstable invariant states until the dynamics approaches a stable attractor. Together, the network of dynamically connected invariant states mediates temporal transitions between coexisting invariant states and thereby supports the observed complex time-dependent dynamics in inclined layer convection.

A new process for the determination of the Soret coefficient of a binary mixture under microgravity

In the presence of a gravity field or under microgravity, pure thermo-diffusion leads to very weak species separation in binary mixtures. To increase the species separation in the presence of gravity, many authors use thermo-gravitational diffusion in vertical columns (TGC). For a given binary mixture, the species separation between the top and the bottom of these columns depends on the temperature difference, ΔT, imposed between the two vertical walls facing each other, and the thickness, H, between these two walls (annular or parallelepipedic column). These studies show that, for a fixed temperature difference, the species separation is optimal for a thickness, Hopt, much smaller than one millimetre. The species separation decreases sharply when the thickness H decreases with respect to this optimum value. It decreases progressively as H increases with respect to Hopt. In addition, for mixtures with a negative thermo-diffusion coefficient, the heaviest component migrates towards the upper part of the column and the lightest one towards the lower part. The loss of stability of the configuration thus obtained leads to a brutal homogeneity of the binary solution.The objective of this study in microgravity was to increase the optimum of species separation. For this purpose, the binary fluid motion was provided by uniform velocities imposed on the two walls of the cavity facing each other. This forced flow led to species separation between the two motionless walls of the cavity. In this case, the fluid motion generated in the cavity was not dependent on the imposed temperature difference, ΔT contrarily to the case of thermogravitational column. Under these conditions and for a given column of thickness H, there are three independent control parameters: ΔT and the two velocities of the walls facing each other. Using the parallel flow approximation for a cell of large aspect ratio, the velocity, temperature and mass fraction fields within the cavity were determined analytically. Thus the parameters leading to optimal species separation were calculated. The analytical results were corroborated by direct numerical simulations. The present paper thus proposes a new process for the determination of the Soret coefficient, the thermodiffusion coefficient and mass-diffusion coefficient.

Effect of low temperature boundary on fuel distribution of pool fires on an immiscible sub-layer

A series of experiments were carried out to study the effect of low boundary temperature on the fuel distribution and consequent flame shapes in small-scale pool fires. M-xylene on an immiscible sublayer was burned in a circular tray, while the boundary temperature was varied from −30 °C to 20 °C. The results show that the imposed temperature difference on the fuel layer was predominant in altering the fuel distribution. The fuel was pushed towards the boundary to be divided into a uniform distribution stage and an annular distribution stage under the effect of surface tension. As a result, the ring fire appeared with the annular distribution stage. A decrease in boundary temperature resulted in a growth of critical thickness of fuel layer at the onset of ring fire. The findings reveal that low boundary temperatures is potentially of significant importance for oil spill cleanup by in-situ burning in ice-infested waters.

Numerical study of heat transfer from a synthetic impinging jet with a detailed model of the actuator membrane

Synthetic jets are produced by the oscillatory movement of a membrane inside a cavity, causing the fluid to enter and leave through a small orifice. The present study is focused on investigating the cooling capabilities of a synthetic jet enclosed between two parallel isothermal plates with an imposed temperature difference. The unsteady three-dimensional Navier-Stokes equations have been solved for a range of Reynolds numbers from 50 to 1000 using time-accurate numerical simulations. A detailed model based on an Arbitrary Lagrangian-Eulerian (ALE) formulation is used to account for the movement of the actuator membrane. All the resulting flows are inherently three-dimensional and dominated by two major vortices, which find their counterparts inside the actuator cavity. A new structure, which is not found in open cavities, appears as an interaction of the synthetic jet flow with the bottom wall and results in a change on the jet's heat transfer mechanisms. Analysis of the outlet temperature has shown that assuming a uniform profile is reasonable if the Reynolds number is high enough, however, the outlet jet temperature is significantly higher than the cold plate temperature. Finally, this study proposes correlations for the heat transfer at the hot wall and the outlet temperature with the Reynolds number, which can be used to account for the cavity effects without the computationally expensive ALE model.

Laboratory model of tropical cyclone with controlled forcing

The paper presents laboratory model of tropical cyclone with controlled forcing and describes the technology to integrate a measurement system and a supercomputer. Procedures of real-time data acquisition, storing and processing, specifics of PIV and heating control systems integration are described. A series of experiments with laboratory analogue of tropical cyclone using feedback between velocity and a heating is carried out. It is found that imposed temperature difference defines the mean radial velocity and intensity of the vortex. It is shown that the relationship between velocity and heat release is of crucial importance for the cyclonic vortex formation.

Linear and nonlinear aspects of Marangoni and evaporative instabilities

Surface-tension gradient driven instability or Marangoni instability and phase-change driven evaporative instability are two classical examples of interfacial instability in fluid layers subject to a vertical temperature gradient. We review the physics of these two problems, comparing their behavior when the interfaces are taken to be either non-deformable or deformable. First, a full linear stability analysis is carried out. Conditions for neutral stability as well as linear growth rates are computed. In both problems, when the interface is non-deformable, a finite short-wavelength instability is seen beyond a critical threshold temperature difference. The presence of a deformable interface renders both problems unstable to long-wave instability with significant deformation of the interface. In this case, both problems become unstable for any non-zero value of imposed temperature difference. The velocity and temperature perturbation profiles at the onset of instability is shown to be different and as a result the differing role of thermal convection in the two problems is explained. Next, we look at the nonlinear evolution of the long-wave instability in these problems. Using the method of weighted residual integral boundary layer, nonlinear evolution equations retaining thermal inertia are derived to track the interface position. The approach to interfacial rupture is shown to be different in both problems. Evaporation results in sharp troughs due to an increased rate of evaporative mass flux near the wall, while the Marangoni problem results in a cascade of buckling events due to a symmetric drainage about the troughs. The presence of thermal inertia in evaporation is destabilizing and is shown to result in a shorter rupture time, while it results in lower linear growth rates in Marangoni instability and therefore a larger rupture time.

Heat and mass transfer in a cylindrical heat pipe with a circular-capillary wick under small imposed temperature differences

Heat pipes are efficient in transferring heat and have been applied in various thermal systems. Previous models of heat pipes use the heat rate through the pipe as an input parameter, and therefore lack predictive capabilities. Here, we demonstrate, using a simple heat pipe, that if the evaporation and condensing kinetics are properly modeled, then the heat rate is predicted. We consider a cylindrical heat pipe with the inner wall lined with a circular-capillary wick. The capillaries are filled with a partially wetting liquid, and the center of the pipe is filled with its vapor. Initially, the heat pipe is at temperature T0 and the system is under thermodynamic equilibrium. Then, one end of the pipe is heated to T0 + ΔT, while the other end cooled to T0 − ΔT, and the system reaches a steady state. The equilibrium vapor pressure at the hot end is higher than that at the cold end, and this pressure difference drives a vapor flow. As the vapor moves, the vapor pressure at the hot end drops below the equilibrium vapor pressure which induces continuous evaporation from circular pores on the wick surface. At the cold end, the vapor pressure exceeds the equilibrium vapor pressure so that the vapor condenses and releases the latent heat. The condensate moves back to the hot end through the capillaries in the wick to complete a cycle. We assume that the pore size is infinitesimal compared with the pipe dimensions. Thus, pore-level events can be treated separately from pipe-level events. The evaporation rate in each pore is solved in the limit the evaporation number E→∞, and an analytic leading-order solution is obtained, assuming ΔT/T0 ≪ 1. The evaporation rate is incorporated into vapor-flow and energy-balance equations along the pipe. Two dimensionless numbers emerge from these equations: the heat pipe number, H, which is the ratio of heat transfer by vapor flow to conductive heat transfer in the liquid and wall, and the evaporation exponent, S, which controls the evaporation gradient along the pipe. We find that vapor-flow heat transfer dominates in heat pipes and H ≫ S ≫ 1. Under these conditions, the non-dimensionalized heat rate through the insulated pipe is found to be simply S. Analytic solutions are also obtained for the pipe temperature and all the other variables. For maximum evaporative heat transfer, we find an optimal pipe length for fixed pipe cross-sectional dimensions, and an optimal wick thickness for a fixed pipe length. These optimal pipe length and wick thickness can help to improve the design of heat pipes and are found for the first time.

A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part III. Turbulent Rayleigh–Bénard convection

A Lagrangian fluctuation–dissipation relation has been derived in a previous work to describe the dissipation rate of advected scalars, both passive and active, in wall-bounded flows. We apply this relation here to develop a Lagrangian description of thermal dissipation in turbulent Rayleigh–Bénard convection in a right-cylindrical cell of arbitrary cross-section, with either imposed temperature difference or imposed heat flux at the top and bottom walls. We obtain an exact relation between the steady-state thermal dissipation rate and the time $\unicode[STIX]{x1D70F}_{mix}$ for passive tracer particles released at the top or bottom wall to mix to their final uniform value near those walls. We show that an ‘ultimate regime’ with the Nusselt number scaling predicted by Spiegel (Annu. Rev. Astron., vol. 9, 1971, p. 323) or, with a log correction, by Kraichnan (Phys. Fluids, vol. 5 (11), 1962, pp. 1374–1389) will occur at high Rayleigh numbers, unless this near-wall mixing time is asymptotically much longer than the free-fall time $\unicode[STIX]{x1D70F}_{free}$. Precisely, we show that $\unicode[STIX]{x1D70F}_{mix}/\unicode[STIX]{x1D70F}_{free}=(RaPr)^{1/2}/Nu,$ with $Ra$ the Rayleigh number, $Pr$ the Prandtl number, and $Nu$ the Nusselt number. We suggest a new criterion for an ultimate regime in terms of transition to turbulence of a thermal ‘mixing zone’, which is much wider than the standard thermal boundary layer. Kraichnan–Spiegel scaling may, however, not hold if the intensity and volume of thermal plumes decrease sufficiently rapidly with increasing Rayleigh number. To help resolve this issue, we suggest a program to measure the near-wall mixing time $\unicode[STIX]{x1D70F}_{mix}$, which is precisely defined in the paper and which we argue is accessible both by laboratory experiment and by numerical simulation.