In the present work, the tangential (swirl) velocity component is superimposed at the intake of a narrow fluidic cylindrical pipe to achieve the desired mixing of inelastic non-Newtonian fluids/solutes at the outlet. We discuss an analytical method for obtaining the swirl velocity profile, considering the nonlinear viscous effects for both shear-thinning and shear-thickening fluids, represented by the power-law model. We numerically solve the species transport equation, coupled with the analytically derived swirl velocity, using our in-house developed code for the concentration distribution in the flow field. The results show that the inlet swirl and an increase in the shear-thinning fluid property improve advection-dominated mixing. Additionally, higher Reynolds numbers significantly enhance advection's dominance, as more rotation leads to the generation of vortices, resulting in an engulfment flow (chaotic convection) based mixing. We demonstrate that considering the increase in the shear-thinning fluid property with swirl intake reduces the amount of mixing time required in the convective regime.

In this study, we analyse ‘magneto-Stokes’ flow, a fundamental magnetohydrodynamic (MHD) flow that shares the cylindrical-annular geometry of the Taylor–Couette cell but uses applied electromagnetic forces to circulate a free-surface layer of electrolyte at low Reynolds numbers. The first complete, analytical solution for time-dependent magneto-Stokes flow is presented and validated with coupled laboratory and numerical experiments. Three regimes are distinguished (shallow-layer, transitional and deep-layer flow regimes), and their influence on the efficiency of microscale mixing is clarified. The solution in the shallow-layer limit belongs to a newly identified class of MHD potential flows, and thus induces mixing without the aid of axial vorticity. We show that these shallow-layer magneto-Stokes flows can still augment mixing in distinct Taylor dispersion and advection-dominated mixing regimes. The existence of enhanced mixing across all three distinguished flow regimes is predicted by asymptotic scaling laws and supported by three-dimensional numerical simulations. Mixing enhancement is initiated with the least electromagnetic forcing in channels with order-unity depth-to-gap-width ratios. If the strength of the electromagnetic forcing is not a constraint, then shallow-layer flows can still yield the shortest mixing times in the advection-dominated limit. Our robust description of momentum evolution and mixing of passive tracers makes the annular magneto-Stokes system fit for use as an MHD reference flow.

We optimize jet mixing using large eddy simulations (LES) at a Reynolds number of $3000$. Key methodological enablers consist of Bayesian optimization, a surrogate model enhanced by deep learning and persistent data topology for physical interpretation. The mixing performance is characterized by an equivalent jet radius ($R_{eq}$) derived from the streamwise velocity in a plane located $8$ diameters downstream. The optimization is performed in a 22-dimensional actuation space that comprises most known excitations. This search space parameterizes the distributed actuation imposed on the bulk flow and at the periphery of the nozzle in the streamwise and radial directions. The momentum flux measures the energy input of the actuation. The optimization quadruples the jet radius $R_{eq}$ with a $7$-armed blooming jet after around $570$ evaluations. The control input requires $2\,\%$ momentum flux of the main flow, which is one order of magnitude lower than an ad hoc dual-mode excitation. Intriguingly, a pronounced suboptimum in the search space is associated with a double-helix jet, a new flow pattern. This jet pattern results in a mixing improvement comparable to the blooming jet. A state-of-the-art Bayesian optimization converges towards this double-helix solution. The learning is accelerated and converges to another better optimum by including a deep-learning-enhanced surrogate model trained along the optimization. Persistent data topology extracts the global and many local minima in the actuation space. These minima can be identified with flow patterns beneficial to the mixing.

This study presents a dual-channel vortex generator (VG) that leverages the snap-through behaviour of flexible sheets. The VG outperforms a similar-sized rigid VG in generating vortices within dual-channel flows while minimizing pressure loss. Numerical simulations using the immersed boundary-lattice Boltzmann method analyse the dynamics and vortex generation performance of the sheet under various system parameters. Two distinct modes are identified for the elastic sheet: a sustained snap-through mode (SSTM) and a dormant mode (DM). The sheet's mode is predominantly influenced by its length ratio (L*), bending stiffness $(K_b^\ast )$ and flow strength, with the mass ratio having a minimal impact. The sheet exhibiting regular SSTM can effectively generate vortices in both channels and the vortex generation performance can be conveniently tuned by altering the sheet's initial buckling (i.e. L*). An increase in $K_b^\ast $ results in a higher critical Reynolds number (Rec) required for mode transition. An increase in L*, however, initially raises Rec and then lowers it, suggesting an optimal length ratio (approximately 0.7 for our considered system) for minimizing the Rec necessary to trigger SSTM. Furthermore, a disparity in the flow strength between channels is found to suppress the snap-through of the sheet; a greater disparity, however, is permissible to induce the SSTM of more compliant sheets. These findings underscore the potential of snap-through behaviour for enhanced flow manipulation in dual-channel systems.

Turbulent flows over porous lattices consisting of rectangular cuboid pores are investigated using scale-resolving direct numerical simulations. Beyond a certain threshold which is primarily determined by the wall-normal Darcy permeability, ${{\mathsf{K}}_y}$, near-wall turbulence transitions from its canonical regime, marked by the presence of streak-like structures, to another marked by the presence of Kelvin–Helmholtz-like (K–H-like) spanwise-coherent structures. The threshold agrees well with that previously established in studies where permeable-wall boundary conditions had been used as surrogates for a porous substrate (Gómez-de Segura & García-Mayoral, J. Fluid Mech., vol. 875, 2019, pp. 124–172). In the smooth-wall-like regime, none of the investigated substrates demonstrate any reduction in drag relative to a smooth-wall flow. At the permeable surface, a notable component of the flow is that which adheres to the pore geometry and undergoes modulation by the turbulent scales of motions due to the interaction mechanism described by Abderrahaman-Elena et al. (J. Fluid Mech., vol. 865, 2019, pp. 1042–1071). Its resulting effect can be quantified in terms of an amplitude modulation (AM) using the approach of Mathis et al. (J. Fluid Mech., vol. 628, 2009, pp. 311–337). This pore-coherent flow component persists throughout the porous substrate, highlighting the importance of a given substrate's microstructure in the presence of an overlying turbulent flow. This geometry-related aspect of the flow is not accounted for when continuum-based models for a porous medium or effective representations of them, such as wall boundary conditions, are used. The intensity of the AM effect is enhanced in the K–H-like regime and becomes strengthened with larger permeability. As a result, structured porous materials may be designed to exploit or mitigate these flow features depending upon the intended application.

The understanding of the entrainment mechanism of synthetic jets can help optimise the synthetic jet actuators in engineering applications. It is generally believed that vortex rings or strong velocity fluctuations in the near field of the synthetic jet are responsible for its enhanced entrainment. However, in recent years, it has been found that the enhanced entrainment of the synthetic jet may be caused by the instability or the vortex ring breakdown in the transition region. To shed new light on this issue, synthetic jets with different Reynolds numbers and dimensionless stroke lengths are investigated with time-resolved two-dimensional particle image velocimetry. Based on the analyses of velocity triple-decomposition, Fourier mode decomposition and phase-averaged $\lambda _{ci}D/U_0$ field, the streamwise positions of the vortex ring breakdown are determined for the synthetic jets, and the entrainment coefficient can be divided into three components, i.e. the coherent turbulent kinetic energy production, the random turbulent kinetic energy production and the shape of the velocity profile. It is found that the entrainment coefficient is dominated by the component related to the random turbulent kinetic energy production, and reaches its peak value at the position of vortex ring breakdown. The results obtained in different cases show a strong correlation between vortex ring breakdown and entrainment enhancement. From the perspective of instantaneous snapshot, the mechanism of vortex ring breakdown enhanced entrainment is revealed, that is, vortex ring breakdown enhanced the small-scale vortex near the turbulent/non-turbulent interface, resulting in an increase of enstrophy production, and thus enhanced local entrainment.

Based on the characteristics of large-scale plume currents in rotating plane Poiseuille flows (RPPF), an injection/suction control strategy is introduced to augment the intensity of plume currents and improve the turbulent transport of passive scalar. The control strategy maintains the conventional non-penetrative condition on the stable side, and applies a wall-normal velocity that varies in the spanwise direction on the unstable side. For comparison, the control with non-penetrative condition on the unstable side and injection/suction on the stable side is also examined. The RPPF at $Re_\tau$ ranging from $180$ to $300$ and $Ro_\tau$ ranging from $0$ to $30$ are studied. Injection/suction with fixed root-mean-squared wall-normal velocity (below $1\,\%$ of the bulk mean velocity) at the wall is considered. Direct numerical simulations reveal that the injection/suction on the stable side has minor effect for all cases studied, whereas the injection/suction with properly distributed slots on the unstable side can significantly enhance the large-scale plume currents and the turbulent transport efficiency at small or moderate rotation numbers. This is attributed to the fact that plumes are generated on the unstable side, and controlling the origin of plumes is more effective. A proper strategy for maximum enhancement of turbulent transport is limiting the distance between injection slots equal to or slightly larger than the intrinsic distance between plume currents. This strategy is valid for all physical and computational parameters currently considered, and is potentially applicable over a much wider parameter range.

In microfluidic systems, analyte/reagent mixing is essential to achieve rapid chemical/biochemical reactions. The low Reynolds number (Re) flow, however, makes passive mixing difficult and necessitates the use of some external stimulus to cause disturbance in the system. Here, we report how a periodic thermocapillary effect may interact with strategically patterned wall wettability and boost the mixing dynamics in a micro-confined ternary-liquid film system. Approximate, yet without compromise on the physics involved, analytical solutions to the energy and Navier–Stokes equations under creeping flow conditions are obtained to comprehend the thermal and hydrodynamic characteristics of the thermo-capillarity. In the binary fluid limit, our model is validated with the findings of Pendse & Esmaeeli (Intl J. Therm. Sci., vol. 49, 2010, pp. 1147–1155), and good agreement is found. The flow characteristics in the ternary-liquid system due to discrete wall temperatures is also demonstrated via finite-element-based numerical simulations, and agreement with the semi-analytical solution is noted. The qualitative study of the flow pattern reveals that enhanced mixing is obtained as a result of vortical motion created by the interplay of the periodic thermo-capillarity-driven interfacial flow and wall slip. The observation is further consolidated from the numerical simulation of the species transport equation. The species distribution so obtained is compared with fully developed laminar flow for the same inlet concentration. Our study further investigates the effects of the fluids’ relative thermal conductivity, wall slip length, relative film thickness and thermal (or slip) phase differences on the mixing efficiency of the proposed arrangement. While wall slip, relative film thickness and relative thermal conductivity regulate mixing in the top and bottom layers, phase difference determines how well the middle fluid layer mixes. Slip at the walls creates vortex distortions to get a strong churning effect. The lower thermal conductivity of the top and bottom fluids weakens the thermocapillary flow; however, the dominance of the patterned slip improves mixing in such cases.

The collision of binary droplets plays a key role in several industrial, chemical and biological processes. In these processes, the quality of the desired outcome is strongly dependent on the mixing of the liquid droplets as they collide in mid-air. In this work, multiphase direct numerical simulations based on the volume-of-fluid method have been used to investigate the process of mixing and analyse the effects of parameters such as injection velocity, timing and collision angles. The evolution of mixing due to convection and irreversible diffusive processes has been quantified by means of the segregation parameter. To synthesise the outcome of a collision, the impact parameter has been redefined to account for the collision of non-spherical droplets. It has been found that the optimal mixing does not occur for symmetric head-on collisions, but rather at moderately asymmetrical configurations. This behaviour has been explained by analysing the velocity gradient tensor. It has been demonstrated that by breaking the symmetry, the local topology of the flow is altered and the resulting convective flows increase the contact area between the liquids, thereby augmenting the mixing process. However, it was also observed that lateral misalignment transforms the initial kinetic energy into the spinning of the merged droplets, thus preventing an enhanced mixing.

Turbulent shear flows are abundant in geophysical and astrophysical systems and in engineering-technology applications. They are often riddled with large-scale secondary flows that drastically modify the characteristics of the primary stream, preventing or enhancing mixing, mass and heat transfer. Using experiments and numerical simulations, we study the possibility of modifying these secondary flows by using superhydrophobic surface treatments that reduce the local shear. We focus on the canonical problem of Taylor–Couette flow, the flow between two coaxial and independently rotating cylinders, which has robust secondary structures called Taylor rolls that persist even at significant levels of turbulence. We generate these structures by rotating only the inner cylinder of the system, and show that an axially spaced superhydrophobic treatment can weaken the rolls through a mismatching surface heterogeneity, as long as the roll size can be fixed. The minimum hydrophobicity of the treatment required for this flow control is rationalized, and its effectiveness beyond the Reynolds numbers studied here is also discussed.

The helix approach is a new individual pitch control method to mitigate wake effects of wind turbines. Its name is derived from the helical shape of the wake caused by a rotating radial force exerted by the turbine. While its potential to increase power production has been shown in previous studies, the physics of the helical wake are not well understood to date. Open questions include whether the increased momentum in the wake stems from an enhanced wake mixing or from the wake deflection. Furthermore, its application to a row of more than two turbines has not been examined before. We study this approach in depth from both an analytical and numerical perspective. We examine large-eddy simulations (LES) of the wake of a single turbine and find that the helix approach exhibits both higher entrainment and notable deflection. As for the application to a row of turbines, we show that the phase difference between two helical wakes is independent of ambient turbulence. Examination of LES of a row of three turbines shows that power gains greatly depend on the phase difference between the helices. We find a maximum increase in the total power of approximately 10 % at a phase difference of $270^\circ$. However, we do not optimise the phase difference any further. In summary, we provide a set of analytical tools for the examination of helical wakes, show why the helix approach is able to increase power production, and provide a method to extend it to a wind farm.

Synthetic jets have received extensive attention due to their superior mixing property. However, its mechanisms have not been investigated from the perspective of the turbulent/non-turbulent interface (TNTI). To shed new light on this issue, the entrainment and TNTI properties of a synthetic jet are experimentally investigated and compared with a continuous jet at $Re_j = 3150$. The fuzzy clustering method is applied to select an appropriate vorticity threshold to detect the TNTI. Statistically, it is revealed that the entrainment coefficients of the two jets significantly differ in the near field, while they become almost identical in the far field. Instead of the vortex ring, the ‘breakdown of the vortex ring’ enhances the entrainment in the present synthetic jet. Instantaneously, the TNTI more violently fluctuates in the near field of the synthetic jet, which leads to a larger fluctuation in the TNTI radial position, a higher fractal dimension and enhanced local entrainment. Moreover, the transition of the probability density function of the TNTI orientation from the unimodal distribution in the near field to the bimodal distribution in the far field is found in both jets. The multi-scale analysis reveals a new mechanism for the bimodal distribution caused by the TNTI-thickness-scale structures.

The pulsed jet is a novel and effective active mixing enhancement approach. For the transverse pulsed jet in the supersonic crossflow, the frequency influence is investigated using the three-dimensional Reynolds-averaged Navier–Stokes (RANS) equations coupled with the SST k-ω turbulence model. The averaged flow field properties of the pulsed jet are better than those of the steady jet when considering mixing efficiency and jet penetration depth, especially for the case with the pulsed frequency being 50kHz. The flow field structures of the pulsed jet are connected with the time, with periodic wave structures generating in the flow field and moving downstream. The size of the wave structures and its distance are related to the frequency, namely the size and flow distance are relatively small at 50kHz, and it takes some time for the pulsed jet to establish its influence in the full flow field. At low frequencies, the flow field produces large fluctuations, and this may be detrimental to the stable operation of the engine.

Passive heat transfer enhancement by spanwise rollers associated with the Kelvin–Helmholtz instability was studied through direct numerical simulations of high-aspect-ratio longitudinal ribs at the friction Reynolds number $300$. The temperature was treated as a passive scalar with Prandtl number unity to discuss the similarity between the heat and momentum transfer. The results reveal that the high-aspect-ratio longitudinal ribs lead to a favourable breakdown of the Reynolds analogy, that is, the enhancement of the heat transfer rate surpasses that of the frictional resistance. The favourable breakdown of the Reynolds analogy can be attributed to the enhanced turbulent heat flux compared with the Reynolds shear stress, whereas the rib-induced secondary flow plays a role in reducing the favourable breakdown of the Reynolds analogy. The conditional average statistics reveal that the high-pressure region accompanied by the spanwise rollers suppresses the spanwise roller-induced sweep and ejection motions, leading to smaller Reynolds shear stress than for the turbulent heat flux.

This work aims to investigate experimentally the effect of the Reynolds number Re, based on the nozzle diameter D, on jet mixing manipulation using an unsteady radial minijet. A novel artificial intelligence (AI) control system has been developed to manipulate the jet over Re = 5800–40 000. The system may optimize simultaneously the control law and a time-independent parameter, which dictate the actuation ON/OFF states and amplitude, respectively. The control parameters include the mass flow rate, excitation frequency and diameter ratios (Cm, fe/f0 and d/D) of the minijet to the main jet as well as the duty cycle (α) of minijet injection. Jet mixing is quantified using Ke and K0, where K is the decay rate of the jet centreline mean velocity, and subscripts e and 0 denote the manipulated and unforced jets, respectively. It has been found that the maximum Ke achievable does not vary with Re. Scaling analysis of the huge volume of experimental data obtained from the AI system reveals that the relationship Ke = g1 (Cm, fe/f0, α, d/D, Re, K0) may be reduced to Ke/K0 = g2 $(\zeta )$, where g1 and g2 are different functions and the scaling factor $\zeta = ({C_m}/\alpha ){(D/d)^{1 - n}}(1/Re){({f_e}/{f_{e,opt}})^m}$, m and 1 − n are the power indices, and subscript opt denotes the value at which Ke is maximum. The scaling law is discussed in detail, along with the physical meanings of the dimensionless parameters Ke/K0, ζ, $({C_m}/\alpha ){(D/d)^{1 - n}}(1/Re)$ and ${({f_e}/{f_{e,opt}})^m}$.

Liquid-infused surfaces can reduce friction drag in both laminar and turbulent flows. However, the heat transfer properties of such multi-phase surfaces have still not been investigated to a large extent. We use numerical simulations to study conjugate heat transfer of liquid-filled grooves. It is shown that heat transfer can increase for both laminar and turbulent liquid flows due to recirculation in the surface texture. Laminar flow simulations show that for the increase to be substantial, the thermal conductivity of the solid must be similar to the thermal conductivity of the fluids, and the recirculation in the grooves must be sufficiently strong (Péclet number larger than 1). The ratio of the surface cavity to the system height is an upper limit of the direct contribution from the recirculation. While this ratio can be significant for laminar flows in microchannels, it is limited for turbulent flows, where the system scale (e.g. channel height) usually is much larger than the texture height. However, heat transfer enhancement of the order of $10\,\%$ is observed (with a net drag reduction) in a turbulent channel flow at a friction Reynolds number ${{Re}}_\tau \approx 180$. It is shown that the turbulent convection in the bulk can be enhanced indirectly from the recirculation in the grooves.

We conduct a well-controlled model experiment for a wide variety of canopy flows. Examples of these include engineering flows such as wind flow, dispersion of scalars through and over urban areas, and the convective heat transfer in many heat exchangers, as well as natural canopies such as flows through terrestrial or aquatic vegetation. We aim to shed the light on fundamental flow and transport phenomena common to these applications. Specifically, the characteristics of mean flow and scalar concentration characteristics of a turbulent boundary layer flow impinging on a canopy, which comprises a cluster of tall obstacles (this can also be interpreted as a porous obstruction). The cluster is created with a group of cylinders of diameter $d$ and height $h$ arranged in a circular patch of diameter $D$. The solidity of the patch/obstruction is defined by $\phi$ (the total planar area covered by cylinders), which is systematically varied ($0.098 \leq \phi \leq 1$) by increasing the number of cylinders in a patch ($N_c$). A point source is placed at ground level upstream of the patch and its transport over and around the patch is examined. Time-averaged velocity and scalar fields, obtained from simultaneous planar particle image velocimetry-planar laser-induced fluorescence (PIV-PLIF) measurements, reveal that the characteristics of wake and flow above porous patches are heavily influenced by $\phi$. In particular, we observe that the horizontal and vertical extent of the wake and scalar concentration downstream of the patches decreases and increases with $\phi$, respectively. Here, the recirculation bubble is shifted closer to the trailing edge (TE) of the patches as $\phi$ increases, limiting the flow from convecting downstream, decreasing the scalar concentration and virtually ‘extending’ the patch in the streamwise direction. As the bubble forms in the TE, vertical bleeding increases and hence the concentration increases above the patch where the cylinders appear to ‘extend’ vertically towards the freestream.

To investigate the intrinsic mechanism for mixing enhancement by variable-density (VD) behaviour, a canonical VD mixing extracted from a supersonic streamwise vortex protocol, a shock–bubble interaction (SBI), is numerically studied and compared with a counterpart of passive-scalar (PS) mixing. It is meaningful to observe that the maximum concentration decays much faster in a VD SBI than in a PS SBI regardless of the shock Mach number ($Ma=1.22 - 4$). The quasi-Lamb–Oseen-type velocity distribution in the PS SBI is found by analysing the azimuthal velocity that stretches the bubble. Meanwhile, for the VD SBI, an additional stretching enhanced by the secondary baroclinic vorticity (SBV) production contributes to the faster-mixing decay. The underlying mechanism of the SBV-enhanced stretching is further revealed through the density and velocity difference between the light shocked bubble and the heavy ambient air. By combining the SBV-accelerated stretching model and the initial shock compression, a novel mixing time estimation for VD SBI is theoretically proposed by solving the advection–diffusion equation under a deformation field of an axisymmetric vortex with the additional SBV-induced azimuthal velocity. Based on the mixing time model, a mixing enhancement number, defined by the ratio of VD and PS mixing time further, reveals the contribution from the VD effect, which implies a better control of the density distribution for mixing enhancement in a supersonic streamwise vortex.

Direct numerical simulations have been performed for heat and momentum transfer in internally heated turbulent shear flow with constant bulk mean velocity and temperature, $u_{b}$ and $\theta _{b}$, between parallel, isothermal, no-slip and permeable walls. The wall-normal transpiration velocity on the walls $y=\pm h$ is assumed to be proportional to the local pressure fluctuations, i.e. $v=\pm \beta p/\rho$ (Jiménez et al., J. Fluid Mech., vol. 442, 2001, pp. 89–117). The temperature is supposed to be a passive scalar, and the Prandtl number is set to unity. Turbulent heat and momentum transfer in permeable-channel flow for the dimensionless permeability parameter $\beta u_b=0.5$ has been found to exhibit distinct states depending on the Reynolds number $Re_b=2h u_b/\nu$. At $Re_{b}\lesssim 10^4$, the classical Blasius law of the friction coefficient and its similarity to the Stanton number, $St\approx c_{f}\sim Re_{b}^{-1/4}$, are observed, whereas at $Re_{b}\gtrsim 10^4$, the so-called ultimate scaling, $St\sim Re_b^0$ and $c_{f}\sim Re_b^0$, is found. The ultimate state is attributed to the appearance of large-scale intense spanwise rolls with the length scale of $O(h)$ arising from the Kelvin–Helmholtz type of shear-layer instability over the permeable walls. The large-scale rolls can induce large-amplitude velocity fluctuations of $O(u_b)$ as in free shear layers, so that the Taylor dissipation law $\epsilon \sim u_{b}^{3}/h$ (or equivalently $c_{f}\sim Re_b^0$) holds. In spite of strong turbulence promotion there is no flow separation, and thus large-amplitude temperature fluctuations of $O(\theta _b)$ can also be induced similarly. As a consequence, the ultimate heat transfer is achieved, i.e. a wall heat flux scales with $u_{b}\theta _{b}$ (or equivalently $St\sim Re_b^0$) independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction.

Thermally driven flows in fractures play a key role in enhancing the heat transfer and fluid mixing across the Earth's lithosphere. Yet the energy pathways in such confined environments have not been characterised. Building on Letelier et al. (J. Fluid Mech., vol. 864, 2019, pp. 746–767), we introduce novel expressions for energy transfer rates – energetics – of geometrically constrained Rayleigh–Bénard convection in Hele-Shaw cells (HS-RBC) based on two different conceptual frameworks. First, we derived the energetics following the well-established framework introduced by Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115–128), in which the gravitational potential energy, $E_{\textit {p}}$, is decomposed into its available, $E_{\textit {ap}}$, and background, $E_{\textit {bp}}$, components. Secondly, we derived the energetics considering a new decomposition for $E_{\textit {p}}$, named dynamic, $E_{\textit {dp}}$, and reference, $E_{\textit {rp}}$, potential energies; $E_{\textit {dp}}$ is defined as the departure of the system's potential energy from the reference state $E_{\textit {rp}}$, determined by the ‘energy’ of the scalar fluctuations. For HS-RBC, both frameworks lead to the same energy transfer rates at a steady state, satisfying the relationship $\langle E_{\textit {ap}} \rangle _{\tau } = \langle E_{\textit {dp}} \rangle _{\tau } + 1/6$. Consistent with the work by Hughes et al. (J. Fluid Mech., vol. 729, 2013) on three-dimensional Rayleigh–Bénard convection, we report analytical expressions for the energetics and efficiencies of HS-RBC in terms of the Rayleigh number and the global Nusselt number. Additionally, we performed numerical experiments to illustrate the application of the energetics for the analysis of HS-RBC. Finally, we discuss the impact of the thermal forcing and the geometrical control exerted by Hele-Shaw cells on the development of boundary layers, protoplumes and the self-organisation of large-scale flows.