1. Introduction
The ocean plays a central role in regulating the global climate by storing and redistributing vast amounts of heat, salinity, freshwater and dissolved
$\mathrm{CO}_2$
(Gayen & Klocker Reference Gayen and Klocker2024). Density variations driven by temperature and salinity gradients sustain the large-scale thermohaline circulation (Ghil & Lucarini Reference Ghil and Lucarini2020; Gayen & Griffiths Reference Gayen and Griffiths2022), whose most studied manifestation is the Atlantic meridional overturning circulation, a system of currents responsible for the mild climate of northern Europe and for significant poleward heat and saline transport (Buckley & Marshall Reference Buckley and Marshall2016; Soons et al. Reference Soons, Grafke and Dijkstra2025). Since buoyancy exchange with the atmosphere occurs predominantly at the upper surface (Whitehead Reference Whitehead1995; Cushman-Roisin & Beckers Reference Cushman-Roisin and Beckers2011), horizontal convection, in which a non-uniform buoyancy flux is imposed along a single horizontal boundary, provides a canonical minimal model for this class of flows. Beyond the ocean, another geophysically relevant flow displaying horizontal convection is the flow in subglacial lakes (Couston Reference Couston2021; Couston et al. Reference Couston, Nandaha and Favier2022), which controls the subglacial melting. More broadly, horizontal convection (HC) captures the essential physics of buoyancy-driven transport in large-scale geophysical systems (Beardsley & Festa Reference Beardsley and Festa1972; Hughes & Griffiths Reference Hughes and Griffiths2008; Scotti & White Reference Scotti and White2011), including rotating configurations (Barkan et al. Reference Barkan, Winters and Smith2013; Vreugdenhil et al. Reference Vreugdenhil, Griffiths and Gayen2017), wind-stress-driven configurations (Cessi Reference Cessi2007; Hazewinkel et al. Reference Hazewinkel, Paparella and Young2012) and configurations in which temperature and salinity act jointly to determine the surface buoyancy forcing (Li & Yang Reference Li and Yang2021; Vacca et al. Reference Vacca, Yang, Howland, Verzicco and Lohse2026).
Since the pioneering study of Rossby (Reference Rossby1965), HC has been investigated extensively in laboratory experiments (Wang & Huang Reference Wang and Huang2005; Griffiths et al. Reference Griffiths, Hughes and Gayen2013; Passaggia et al. Reference Passaggia, Cohen and Scotti2024) and numerical simulations (Shishkina Reference Shishkina2017; Reiter & Shishkina Reference Reiter and Shishkina2020; Rocha et al. Reference Rocha, Constantinou, Smith and Young2020; Rein et al. Reference Rein, Smith and Young2026). Much of this work has focused on how the global heat and momentum transport, quantified by the Nusselt number
$\textit{Nu}$
and Reynolds number
$\textit{Re}$
, depend on the Rayleigh (
$\textit{Ra}$
) and Prandtl (
$\textit{Pr}$
) numbers. Definitions of the dimensionless numbers are given in the next section. One of the earliest scaling hypotheses for HC is the Rossby-type scaling
$\textit{Nu} \sim \textit{Ra}^{1/5}$
for laminar HC (Rossby Reference Rossby1965). Although this scaling has received considerable support from both numerical simulations and experiments (Mullarney et al. Reference Mullarney, Griffiths and Hughes2004), its range of validity and universality over broad parameter ranges remain actively debated.
A major step towards a unified description of HC was the adaptation of the Grossmann & Lohse (Reference Grossmann and Lohse2000, Reference Grossmann and Lohse2001) theory to HC. Using a decomposition of the volume and time-averaged thermal
$(\langle \varepsilon _{T} \rangle _V )$
and kinetic
$(\langle \varepsilon _u \rangle _V )$
dissipation rates into boundary-layer and bulk contributions, Shishkina, Grossmann & Lohse (Reference Shishkina, Grossmann and Lohse2016) (hereafter SGL) derived scaling relations for
$\textit{Nu}$
and
$\textit{Re}$
across the laminar, turbulent and ultimate (Lohse & Shishkina Reference Lohse and Shishkina2024; Shishkina & Lohse Reference Shishkina and Lohse2026) regimes. An important result of this theory is that different scaling behaviours can arise even within laminar HC, depending on the Prandtl number. Here,
$\lambda _u$
denotes the kinetic boundary-layer thickness. At relatively low
$\textit{Pr}$
, the kinetic boundary layer remains free to grow and follows the classical Prandtl–Blasius scaling
$\lambda _u/L \sim \textit{Re}^{-1/2}$
, leading to
$\langle \varepsilon _u \rangle _V \sim (\nu ^3/L^4)\textit{Re}^{5/2}$
and a Rossby-type scaling
$\textit{Nu} \sim \textit{Pr}^{1/10} \textit{Ra}^{1/5}$
(called the
$I_l$
regime in SGL). In contrast, at sufficiently large
$\textit{Pr}$
, the kinetic boundary-layer thickness saturates, so that
$\langle \varepsilon _u \rangle _V \sim (\nu ^3/L^4)\textit{Re}^2$
and the steeper
$\textit{Nu} \sim \textit{Pr}^{0} \textit{Ra}^{1/4}$
scaling emerges (called the
$I_l^*$
regime in SGL) (Shishkina & Wagner Reference Shishkina and Wagner2016).
Although the scaling of
$\textit{Nu}$
with
$\textit{Ra}$
and
$\textit{Pr}$
in HC has been studied extensively (Shishkina et al. Reference Shishkina, Grossmann and Lohse2016; Passaggia et al. Reference Passaggia, Cohen and Scotti2024), the role of geometry remains much less explored. Sheard & King (Reference Sheard and King2011) investigated aspect-ratio effects in HC driven by a linear temperature profile imposed along the horizontal boundary. This configuration was further investigated by Hossain et al. (Reference Hossain, Vo and Sheard2019), who extended the analysis to very shallow domains. They showed that, at sufficiently low Rayleigh numbers, the dynamics of shallow HC are more naturally characterised using the domain height rather than its horizontal length. Aspect-ratio effects in the very-large- or infinite-Prandtl-number limit have also been investigated by Chiu-Webster et al. (Reference Chiu-Webster, Hinch and Lister2008).
Beyond geometric effects, the imposed thermal forcing profile can itself affect the observed heat-transport scaling. At fixed aspect ratio, Ramme & Hansen (Reference Ramme and Hansen2019) compared different thermal forcing profiles and found behaviour consistent with
$\textit{Nu}\sim \textit{Ra}^{1/5}$
for a linear temperature profile, whereas a steeper effective scaling, approaching
$\textit{Nu}\sim \textit{Ra}^{1/4}$
, was observed for step-like forcing. These results highlight that both geometry and the imposed thermal forcing can influence the heat transport in HC.
In the present work, we address this issue by a systematic series of three-dimensional direct numerical simulations (DNSs) spanning a broad range of Rayleigh numbers and aspect ratios, at fixed Prandtl number (
$\textit{Pr}=10$
). We show that the aspect ratio is an active control parameter governing the formation of large-scale circulation, the diffusive reference state, the global
$\textit{Nu}$
vs
$\textit{Ra}$
relation and the gradual cross-over from behaviour compatible with an effective
$1/5$
exponent towards behaviour associated with the
$I_l^*$
regime. In particular, we show that the progressive change in the effective scaling exponent towards behaviour associated with the
$I_l^*$
regime occurs together with the saturation of the kinetic boundary-layer thickness with increasing aspect ratio. The present study thus places aspect-ratio effects directly within the SGL theory.
2. Governing equations
The classical HC configuration considered here is shown in figure 1(d). The computational domain has length
$L=\varGamma H$
in the streamwise direction
$x$
, height
$H$
in the vertical direction
$z$
and depth
$H$
in the spanwise direction
$y$
, with gravity acting in the negative
$z$
direction. Thermal forcing is applied at the bottom boundary through two isothermal plates, each extending over a distance
$H$
along the
$x$
direction and maintained at temperatures
$T_{+}$
and
$T_{-}$
. The section of the lower boundary between the two plates is adiabatic, and all remaining boundaries are also adiabatic. No-slip boundary conditions are imposed at all walls.
(a–d) Volume renderings of the mean temperature field, together with streamlines shown on a streamwise–wall-normal plane, for
$\textit{Ra}=5\times 10^{8}$
and increasing aspect ratio
$\varGamma$
. The cases shown (and marked as purple symbols in (e)) are (a)
$\varGamma =3$
, (b)
$\varGamma =5$
, (c)
$\varGamma =7$
and (d)
$\varGamma =12$
. Panel (d) also indicates the main geometrical features of the considered configuration;
$\hat {g}_z$
denotes the gravity vector. (e) Explored parameter space in the
$(\textit{Ra},\varGamma )$
plane. Symbols indicate the numerical grid resolutions: circles,
$(N_x,N_y,N_z)=(256,32,50)$
; squares,
$(256,64,128)$
; upward triangles,
$(512,64,96)$
; diamonds,
$(512,96,180)$
; downward triangles,
$(1024,100,96)$
; filled plus symbols,
$(1024,100,192)$
; left-pointing triangles,
$(936,184,272)$
.
Figure 1. Long description
Panels (a)-(d) show mean temperature fields and streamlines at Rayleigh number Ra = 5 × 10^8 for aspect ratios Γ = 3, 5, 7 and 12, respectively. Magenta indicates warmer fluid and blue indicates colder fluid. The boxes become progressively longer relative to their height, and the streamline patterns change with their geometry. Panel (d) labels the domain length L = ΓH, height H, spanwise depth H, coordinate directions and downward gravity. Heated and cooled plates, each of length H, occupy opposite ends of the bottom boundary. Panel (e) shows the simulated cases in the Rayleigh-number/aspect-ratio plane. Marker shapes identify numerical grid resolutions, and purple markers identify the four cases illustrated in panels (a)-(d).
Under the Oberbeck–Boussinesq approximation, the dimensionless governing equations for the velocity
$\boldsymbol{u}$
, temperature
$T$
and pressure
$p$
are
\begin{equation} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}=0, \end{equation}
\begin{equation} \frac {\partial \boldsymbol{u}}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(\boldsymbol{u}\boldsymbol{u}) =-\boldsymbol{\nabla }\!p+T\,\hat {\boldsymbol{e}}_z+\sqrt {\frac {\textit{Pr}}{\textit{Ra}}}\,{\nabla} ^2\boldsymbol{u}, \end{equation}
\begin{equation} \frac {\partial T}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(\boldsymbol{u}T) =\sqrt {\frac {1}{\textit{Ra}\,\textit{Pr}}}\,{\nabla} ^2 T. \end{equation}
The equations have been non-dimensionalised using the characteristic length scale
$L$
, the free-fall velocity
$u_{ff}\equiv (\alpha g \Delta L)^{1/2}$
, the free-fall time
$t_{ff}\equiv L/u_{ff}$
and the imposed temperature difference
$\varDelta \equiv T_{+}-T_{-}$
between the heated and cooled plates. Here,
$\nu$
denotes the kinematic viscosity,
$\kappa$
the thermal diffusivity,
$\alpha$
the thermal expansion coefficient and
$g$
the gravitational acceleration. The resulting dimensionless control parameters are the Rayleigh number
$\textit{Ra}$
, the Prandtl number
$\textit{Pr}$
and the aspect ratio
$\varGamma$
, defined as
\begin{equation} \textit{Ra}=\frac {\alpha g \Delta L^3}{\nu \kappa }, \quad \textit{Pr}=\frac {\nu }{\kappa }, \quad \varGamma =\frac {L}{H}. \end{equation}
The main global response parameter of the system is the Nusselt number, which quantifies the non-dimensional heat transport between the heated and cooled plates
\begin{equation} \textit{Nu} = -\frac {\langle \partial T/\partial z \rangle _{S+}}{\varDelta /L}= \frac {\langle \partial T/\partial z \rangle _{S-}}{\varDelta /L}. \qquad \end{equation}
For comparison, we also introduce the
$\textit{Nu}_d$
, the purely diffusive solution. Here,
$\langle \boldsymbol{\cdot }\rangle _{S+}$
and
$\langle \boldsymbol{\cdot }\rangle _{S-}$
denote averages in time and over the heated and cooled plates, respectively.
The simulations are performed using the finite-volume code Goldfish (Shishkina et al. Reference Shishkina, Horn, Wagner and Ching2015; Kooij et al. Reference Kooij, Botchev, Frederix, Geurts, Horn, Lohse, van der Poel, Shishkina, Stevens and Verzicco2018; Reiter et al. Reference Reiter, Zhang, Stepanov and Shishkina2021, Reference Reiter, Zhang, Stepanov and Shishkina2022), which is fourth-order accurate in space and employs a third-order Runge–Kutta scheme for time advancement. The computations are carried out on non-uniform grids in all three spatial directions, with grid clustering near the walls in order to provide sufficient resolution within the boundary layers. To assess the effect of the aspect ratio on the heat transfer in HC, we fixed the Prandtl number
$\textit{Pr}=10$
and performed simulations over the ranges
$3 \leqslant \varGamma \leqslant 14$
and
$3\times 10^{6}\leqslant \textit{Ra} \leqslant 9 \times 10^{10}$
. The corresponding parameter space is shown in figure 1(e), where the different symbols identify the grid resolutions adopted for the various cases; in all cases, the grids are sufficiently fine to resolve both the Kolmogorov and Batchelor microscales. All simulations are initialised from a quiescent state, with the velocity field set to zero and the temperature field taken to be uniform and equal to the mean value
$T_m=(T_{+}+T_{-})/2$
. The simulations are advanced for up to
$8000$
free-fall time units. Time-averaged statistics are collected only after the flow has reached a statistically stationary state, over intervals of at least
$4000$
free-fall time units.
Log–log plot of the normalised Nusselt number,
$\textit{Nu}/\textit{Nu}_d$
, as a function of (a)
$\textit{Ra}$
and (b)
$\textit{Ra}_H=\textit{Ra}/\varGamma ^3$
. Colours indicate the aspect ratio
$\varGamma$
, according to the common colour map. Crosses denote the corresponding data from Reiter & Shishkina (Reference Reiter and Shishkina2020) for
$\varGamma =10$
.
Figure 2. Long description
Two log-log plots show the normalised Nusselt number Nu/Nu_d, where Nu_d is the purely diffusive reference value. Panel (a) uses the Rayleigh number Ra based on the horizontal domain length. Panel (b) uses Ra_H = Ra/Γ^3, based on the domain height. Colours distinguish aspect ratios from Γ = 3 to 14. Heat transport is close to the diffusive value at low forcing and increases as forcing strengthens. At a fixed Ra, larger aspect ratios delay the departure from the diffusive state. Using Ra_H brings the data closer together in the diffusive and early transitional ranges. Reference lines indicate one-quarter-power scaling. Crosses show data from Reiter and Shishkina (2020) for Γ = 10.
3. Results
We begin by considering the Nusselt number defined in (2.5), which quantifies the total heat transfer normalised by the diffusive reference transport
$\textit{Nu}_d$
. The diffusive reference state was determined independently via an asymptotic expansion in the limit
$\textit{Ra}\to 0$
and by numerically solving the corresponding Laplace problem with the imposed mixed thermal boundary conditions. The resulting solutions were found to be in excellent agreement. In HC,
$\textit{Nu}_d$
is not a universal constant but depends on the thermal boundary conditions and on the aspect ratio
$\varGamma$
; for the present configuration, it decreases monotonically from
$\textit{Nu}_d(\varGamma =3)\approx 1.55$
towards unity as
$\varGamma$
increases; in particular,
$\textit{Nu}_d(\varGamma =5)\approx 1.27$
,
$\textit{Nu}_d(\varGamma =7)\approx 1.18$
,
$\textit{Nu}_d(\varGamma =10)\approx 1.12$
,
$\textit{Nu}_d(\varGamma =12)\approx 1.10$
and
$\textit{Nu}_d(\varGamma =14)\approx 1.08$
. Figure 2(a) shows that at low Rayleigh number the system remains close to the diffusive state, as expected, while increasing
$\textit{Ra}$
leads to a progressive enhancement of the heat transport by convection. This change strongly depends on the aspect ratio: at fixed
$\textit{Ra}$
, cases with larger
$\varGamma$
remain closer to the diffusive reference, whereas smaller
$\varGamma$
cases already exhibit substantially enhanced heat transfer. A clearer picture emerges when the data are plotted against the rescaled Rayleigh number
$\textit{Ra}_H = \textit{Ra}/\varGamma ^3$
(i.e. a Rayleigh number defined with the vertical length scale
$H$
, rather than with the horizontal length
$L$
), as shown in figure 2(b). In this representation, the curves collapse well in the diffusive and early transitional regimes, indicating that
$\textit{Ra}_H$
captures the leading aspect-ratio dependence of the departure from the diffusive reference case, and that in the diffusive state, the relevant system length scale is the domain height,
$H$
, whereas the effect of the domain length,
$L$
, becomes pronounced only at larger forcing, namely for
$\textit{Ra}_H \gtrsim 10^5$
.
Compensated plot (with the Rossby Reference Rossby1965 scaling
$\textit{Nu} \sim \textit{Ra}^{1/5}$
) of the normalised Nusselt number. Panel (a) shows
$(\textit{Nu}/\textit{Nu}_d)/\textit{Ra}^{1/5}$
versus
$\textit{Ra}$
, while panel (b) shows
$(\textit{Nu}/\textit{Nu}_d)/\textit{Ra}_H^{1/5}$
versus
$\textit{Ra}_H=\textit{Ra}/\varGamma ^3$
. The solid lines indicate the
$1/4$
scaling associated with the
$I_l^*$
SGL regime
$\textit{Nu} \sim \textit{Ra}^{1/4}$
. Colours indicate the aspect ratio
$\varGamma$
, according to the common colour map. Crosses denote the corresponding data from Reiter & Shishkina (Reference Reiter and Shishkina2020) for
$\varGamma =10$
.
Figure 3. Long description
Two log-log plots show the normalised Nusselt number compensated by the Rossby one-fifth-power scaling. Panel (a) plots (Nu/Nu_d)/Ra^(1/5) against Ra. Panel (b) plots (Nu/Nu_d)/Ra_H^(1/5) against Ra_H = Ra/Γ^3. Colours indicate aspect ratios from Γ = 3 to 14. Crosses show reference data from Reiter and Shishkina (2020) at Γ = 10. Approximately horizontal portions indicate behaviour compatible with one-fifth-power scaling. These portions become less evident as the aspect ratio increases. Inclined reference lines indicate one-quarter-power scaling of the uncompensated heat transport.
Further insight is provided by the compensated Nusselt plots in figure 3(a,b), where the data are normalised according to the Rossby-type scaling, namely with
$\textit{Nu}/\textit{Nu}_d \sim \textit{Ra}^{1/5}$
in figure 3(a) and with
$\textit{Nu}/\textit{Nu}_d \sim \textit{Ra}_H^{1/5}$
in figure 3(b). Figure 3(a) already suggests that the apparent
$1/5$
behaviour is not equally developed across all aspect ratios. For the largest values of
$\varGamma$
, the data move rather directly towards a
$1/4$
behaviour, whereas for smaller aspect ratios only a finite intermediate interval remains approximately compatible with the Rossby-type compensation. We identify this range as the interval over which
$(\textit{Nu}/\textit{Nu}_d)/\textit{Ra}^{1/5}$
remains approximately constant. Its bounds are not abrupt but gradual, reflecting the gradual transitions, since the compensated data depart gradually from the plateau. The limited extent of this interval becomes more evident in figure 3(b), where the use of the rescaled control parameter
$\textit{Ra}_H$
reveals more distinctly the limited extent of this intermediate interval. This limited range approximately compatible with a
$1/5$
scaling can be clearly identified only for small aspect ratios, but it systematically narrows as
$\varGamma$
increases and tends to disappear for the largest considered aspect ratios. This supports the interpretation that, in the present configuration, the apparent
$1/5$
exponent should not be regarded as a robust asymptotic regime over the whole parameter space. Rather, it may be understood as an intermediate signature associated with the departure from the diffusive state and the progressive approach to the confined
$I_l^*$
regime,
$\textit{Nu} \sim \textit{Ra}^{1/4}$
of the SGL model. The more robust trend at large
$\varGamma$
is instead associated with the steeper
$1/4$
scaling of the SGL regime. To rationalise and understand this change in scaling, we next examine the mean kinetic energy and temperature profiles, together with the associated boundary-layer thicknesses. Following Shishkina (Reference Shishkina2017), we calculate the vertical profiles of the normalised double kinetic energy,
$E_{kin} \equiv \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{u}$
. Figure 4(a–c) shows the corresponding mean kinetic-energy profiles, while figure 4(d–f) shows the corresponding mean temperature profiles, for
$\textit{Ra}=10^8$
,
$5\times 10^8$
and
$10^9$
, respectively. The profiles are averaged in time, in the spanwise direction and over the horizontal extent of the heated plate. As before, moving from darker to lighter shades corresponds to increasing aspect ratio. DNS results reveal that the vertical temperature profiles are non-monotonic functions of
$z$
, with a minimum located at a finite distance above the bottom boundary. The kinetic-energy profiles display a similar non-monotonic behaviour, with a local maximum away from the wall. For both quantities, the amplitudes of these extrema vary systematically with the aspect ratio: as
$\varGamma$
decreases, the local maximum in the kinetic energy and the local minimum in the temperature become more pronounced. This behaviour also clarifies the structure of the flow. The streamline plots in figure 1(a–d) show that, at small aspect ratio, the convective cell is strongly asymmetric. It remains confined near the lower boundary, while the upper part of the domain is only weakly active. As the aspect ratio increases, the circulation becomes more symmetric. It progressively fills the full domain height and approaches the classical structure of laminar HC. The mean profiles are consistent with this picture. At fixed Rayleigh number, smaller aspect ratios produce more asymmetric kinetic-energy profiles, reflecting the stronger confinement of the motion near the bottom boundary. Overall, reducing the aspect ratio weakens the vertical extension of the large-scale circulation and makes the flow more asymmetric. We define characteristic thermal and kinetic boundary-layer thicknesses from the vertical mean profiles. Specifically, the thermal boundary-layer thickness
$\delta _T$
is taken as the wall-normal location of the first local minimum of the temperature profile, whereas the kinetic boundary-layer thickness
$\delta _u$
is defined as the wall-normal location of the first local maximum of the kinetic-energy profile.
Vertical profiles of the normalised kinetic energy in panels (a–c) and temperature in panels (d-f), for
$\textit{Ra}=10^8$
in (a,d),
$\textit{Ra}=5\times 10^8$
in (b,e) and
$\textit{Ra}=10^9$
in (c,f). The profiles are averaged in time, in spanwise direction and additionally over the region above the heated plate. Colours indicate the different aspect ratios
$\varGamma$
(see the colour map).
Figure 4. Long description
Six panels show vertical profiles against normalised height z/H. The upper row, panels (a)-(c), shows E_kin, defined as the velocity dotted with itself. The lower row, panels (d)-(f), shows temperature. The three columns correspond to Rayleigh numbers Ra = 10^8, 5 × 10^8 and 10^9, from left to right. Profiles are averaged in time, in the spanwise direction and over the horizontal extent of the heated plate. Colours indicate aspect ratios from Γ = 3 to 14. The kinetic-energy profiles have a first local maximum above the bottom wall, while the temperature profiles have a first local minimum. Smaller aspect ratios generally produce a more pronounced first kinetic-energy maximum and a deeper temperature minimum. The positions of these extrema define the boundary-layer thicknesses examined in figure 5.
Dependence on aspect ratio
$\varGamma$
of the normalised, time-averaged kinetic (a) and thermal (b) boundary-layer thicknesses. Colours indicate the different Rayleigh numbers, following the colour map. Crosses denote the corresponding data from Shishkina (Reference Shishkina2017).
Figure 5. Long description
Two plots show boundary-layer thicknesses divided by the domain height H as functions of aspect ratio Γ. Panel (a) shows the kinetic thickness δ_u/H and panel (b) the thermal thickness δ_T/H. Both vertical axes are logarithmic. Colours distinguish Rayleigh numbers from 3 × 10^6 to 9 × 10^10, progressing from blue at lower values to red at higher values. At a fixed aspect ratio, both thicknesses decrease as the Rayleigh number increases. For sufficiently large Rayleigh numbers, the kinetic thickness approaches a plateau at aspect ratios around 7 and above. Crosses show comparison data from Shishkina (2017) at Γ = 10.
The resulting normalised boundary-layer thicknesses are shown in figure 5, where panel (a) reports
$\delta _u$
and panel (b) reports
$\delta _T$
, both as functions of the aspect ratio
$\varGamma$
. The colours denote different Rayleigh numbers (blue for smaller
$\textit{Ra}$
and red for larger
$\textit{Ra}$
). Data extracted from Shishkina (Reference Shishkina2017), for
$\varGamma = 10$
, are also included for comparison. For fixed aspect ratio, both
$\delta _T$
and
$\delta _u$
decrease with increasing Rayleigh number, as expected. The more important result, however, concerns the aspect-ratio dependence of the kinetic boundary layer. Figure 5(a) indicates that
$\delta _u$
approaches a (Ra-dependent) saturated value for
$\varGamma \gtrsim 7$
for large enough
$\textit{Ra}$
. This provides a physically consistent interpretation of the compensated Nusselt plots in figure 3(a,b): low-aspect-ratio cases retain a finite range broadly compatible with the Rossby-type
$\textit{Nu} \sim \textit{Ra}^{1/5}$
scaling, whereas larger aspect ratios evolve more directly towards the
$\textit{Nu} \sim \textit{Ra}^{1/4}$
behaviour associated with the
$I_l^*$
regime of SGL. This distinction is consistent with the SGL theory. The
$I_l^*$
regime requires the kinetic boundary layer to be effectively constrained, so that its thickness no longer grows freely with increasing
$\textit{Ra}$
. When the kinetic boundary layer is not yet effectively constrained, the flow can still display behaviour compatible with the
$I_l$
regime
$\textit{Nu} \sim \textit{Ra}^{1/5}$
of Rossby (Reference Rossby1965). Once
$\delta _u$
becomes bounded by the finite vertical extent of the domain, however, the assumptions underlying the
$I_l^*$
regime of SGL become applicable.
4. Conclusions
In summary, we have systematically examined the role of the aspect ratio in HC over the ranges
$3 \leqslant \varGamma \leqslant 14$
and
$3 \times 10^{6} \leqslant \textit{Ra} \leqslant 9 \times 10^{10}$
at fixed
$\textit{Pr}=10$
. Our results show that the aspect ratio, whose influence has rarely been examined, has a direct and significant influence on the heat-transport scaling. More specifically, the aspect ratio not only influences the Rayleigh number at which convective enhancement becomes appreciable, but also modifies the effective scaling of the global heat transport. Within the SGL theory, this behaviour can be interpreted in terms of the vertical confinement of the kinetic boundary layer. When the boundary layer remains free to grow, the flow retains a short intermediate range compatible with Rossby-type scaling behaviour. When its growth is limited by confinement, the kinetic boundary layer saturates and the flow exhibits a gradual cross-over towards the confined
$I_l^*$
regime
$\textit{Nu} \sim \textit{Ra}^{1/4}$
of SGL. This interpretation is consistent with the observed cross-over from a finite
$\textit{Nu} \sim \textit{Ra}^{1/5}$
a short intermediate range compatible towards the steeper
$\textit{Nu} \sim \textit{Ra}^{1/4}$
scaling. In the present configuration, however, the apparent
$1/5$
exponent observed at small aspect ratio is not a robust asymptotic scaling, but only a finite-range behaviour that progressively disappears as the aspect ratio increases. Within the SGL interpretation adopted here, we do not expect this
$1/5$
scaling to re-emerge at larger
$\textit{Ra}$
, although the present simulations do not test this high-
$\textit{Ra}$
expectation directly. This conclusion is supported first by the global response parameter and then by the mean velocity and temperature profiles. Taken collectively, the present results quantitatively show that the growth of the kinetic boundary layer is arrested for
$\varGamma \gtrsim 7$
. The present study therefore demonstrates that the aspect ratio is a fundamental control parameter in HC and that the geometry can produce apparent scaling behaviour over restricted Rayleigh-number intervals, while the asymptotic behaviour is more consistently interpreted within the SGL theory. Besides the streamwise aspect ratio considered here, several additional geometric parameters may also influence HC. For example, the ratio between the length of the isothermal plates and the domain height is expected to affect the overall heat transfer. In addition, the imposed thermal forcing profile can affect the observed heat-transport scaling, with linear and step-like temperature profiles potentially leading to different
$\textit{Nu}$
–
$\textit{Ra}$
behaviours. In contrast, the spanwise aspect ratio, provided it is large enough, will only have a limited effect on the heat transfer.
Funding
This work was financially supported by the European Union (ERC, MultiMelt, 101094492) and the German Research Foundation (DFG, Sh405/20, So2399/2). Numerical resources were provided by Dutch Research Council (NWO), by the Gauss Centre for Supercomputing through project ID pr74sa on SuperMUC-NG and by the EuroHPC Joint Undertaking through project ID EHPC-EXT-2024E02122 on MareNostrum 5. Open access funding provided by University of Twente.
Declaration of interests
The authors report no conflict of interest.
Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Ra=5×108
Γ
Γ=3
Γ=5
Γ=7
Γ=12
g^z
(Ra,Γ)
(Nx,Ny,Nz)=(256,32,50)
(256,64,128)
(512,64,96)
(512,96,180)
(1024,100,96)
(1024,100,192)
(936,184,272)
Nu/Nud
Ra
RaH=Ra/Γ3
Γ
Γ=10
Nu∼Ra1/5
(Nu/Nud)/Ra1/5
Ra
(Nu/Nud)/RaH1/5
RaH=Ra/Γ3
1/4
Il∗
Nu∼Ra1/4
Γ
Γ=10
Ra=108
Ra=5×108
Ra=109
Γ
Γ
Open access
