## 1 Introduction

Rayleigh (1916) proposed a model for Bénard’s turn of the 20th century experiments consisting of the Boussinesq approximation to the Navier–Stokes equations of motion, wherein the density
$\unicode[STIX]{x1D70C}$
is fixed in all but the temperature-dependent buoyancy force term and the fluid’s material properties (viscosity
$\unicode[STIX]{x1D708}$
, specific heat
$c$
and thermal diffusion and expansion coefficients
$\unicode[STIX]{x1D705}$
and
$\unicode[STIX]{x1D6FC}$
) are assumed constant, applied to a layer of fluid confined between parallel horizontal iso-thermal no-penetration plates. Even though the actual thermo-capillary mechanism underlying Bénard’s observations – the Bénard–Marangoni instability – was only properly formulated as a hydrodynamic problem decades later by Pearson (Reference Pearson1958), Rayleigh’s model for buoyancy-driven thermal convection is still called the Rayleigh–Bénard problem.

A key feature of interest is the relation between the temperature drop
$\unicode[STIX]{x0394}T$
across a fluid layer of thickness
$h$
and the resulting heat flux
$Q$
. In Rayleigh’s model the temperature drop is controlled at the boundaries, and the first result of his analysis was to identify the non-dimensional group determining instability of the quiescent conduction state:
$g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}Th^{3}/\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}$
where
$g$
is the acceleration of gravity. Today we call this the Rayleigh number
$Ra$
, leaving the fluid’s Prandtl number
$Pr=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}$
as the other non-dimensional parameter characterizing the particular system at hand. Heat transport is traditionally measured in units of the conductive heat flux via the Nusselt number
$Nu=Qh/\unicode[STIX]{x1D70C}c\unicode[STIX]{x1D705}\unicode[STIX]{x0394}T$
, and the challenge is, for a given domain, to determine
$Nu$
as a function of
$Ra$
and
$Pr$
thereby quantifying the effective thermal conductivity of the convecting layer. The strongly nonlinear regime of paramount importance for geophysical and astrophysical applications corresponds to large Rayleigh numbers and two distinct theories, conventionally referred to as the ‘classical’ and ‘ultimate’ theories, remain in contention for the asymptotic behaviour of
$Nu$
as
$Ra\rightarrow \infty$
. Mathematical analysis has failed to prove, and experimental studies have failed to definitively rule out, either of these asymptotic theories for
$O(1)$
values of
$Pr$
.

The work of Bouillaut *et al.* (Reference Bouillaut, Lepot, Aumaître and Gallet2019), based on the novel experimental approach of Lepot, Aumaître & Gallet (Reference Lepot, Aumaître and Gallet2018) in which the fluid is radiatively heated rather than relying on conduction through the rigid boundaries, sheds new light on the conundrum. Boundaries and boundary layers play key roles in the alternative ‘classical’ and ‘ultimate’ theories for the extreme limit of Rayleigh–Bénard convection and direct internal heating allows for the disentanglement of velocity and temperature boundary layers. The experimental data suggest that, depending on details of the structure and strength of the thermal forcing, the predictions of either competing theory may be observed.

## 2 Overview

The ‘classical’ theory asserts that
$Nu\sim Ra^{1/3}$
. It was simultaneously proposed by Priestly (Reference Priestly1954), who argued that
$Q$
should be independent of
$h$
for turbulent convection in large aspect ratio domains, and Malkus (Reference Malkus1954), whose complicated maximal dissipation theory also predicted that the scaling is uniform in
$Pr$
. A decade later Howard (Reference Howard and Görtler1964) reinterpreted the uniform-in-
$Pr$
prediction as a marginally stable thermal boundary layer argument, explicitly neglecting the potential effect of a velocity boundary layer.

The ‘ultimate’ theory asserts
$Nu\sim Pr^{1/2}Ra^{1/2}$
for
$Pr\leqslant O(1)$
. It was originally proposed by Spiegel (Reference Spiegel1963) based on the idea that buoyant fluid elements transport heat at the free-fall velocity
$U\sim (g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}Th)^{1/2}$
so that
$Q$
becomes independent of the material transport coefficients
$\unicode[STIX]{x1D708}$
and
$\unicode[STIX]{x1D705}$
. (Spiegel’s theory actually predates this reference as evidenced in the testimony of Batchelor Reference Batchelor and Thomas1961.) Spiegel referred to this approach as a ‘mixing length’ theory without reference to the nature of boundaries or boundary layers. But soon thereafter his postdoc mentor Kraichnan (Reference Kraichnan1962) considered ultra-high
$Ra$
situations when velocity boundary layers at the rigid plates presumably transition to shear turbulence, postulating logarithmic corrections to the asymptotic
$Ra^{1/2}$
scaling. The moniker ‘ultimate’ was first used by Chavanne *et al.* (Reference Chavanne, Chillà, Castaing, Chabaud and Chaussy1997) referring to Kraichnan’s modification of Spiegel’s theory and has since been adopted by the community for an asymptotic state with predominant
$1/2$
scaling. Moreover,
$Nu\sim Ra^{1/2}$
scaling, albeit uniform in
$Pr$
, is mathematically ‘ultimate’ in the sense that it is a rigorous upper bound for heat transport in Rayleigh’s model (Howard Reference Howard1963; Doering & Constantin Reference Doering and Constantin1996).

Rather than fixing the temperature and inserting and removing heat at the top and bottom boundaries, Bouillaut *et al.* (Reference Bouillaut, Lepot, Aumaître and Gallet2019) heat the bottom of the layer by illuminating the fluid, which is treated with optically absorbing dye, from below. The optical absorption length
$\ell$
can be varied by adjusting the dye concentration to control the spatial profile of heat injection into the container. The container is thermally insulated on all boundaries so the bulk temperature increases linearly with time as the experiment proceeds. As long as the Boussinesq approximation remains valid, however, the difference between the local temperature and the linearly increasing-in-time bulk temperature behaves as if the system is heated near the bottom and cooled in the bulk above. In this set-up, distinct from the set-up of Rayleigh’s model, the heat flux
$Q$
is controlled and temperature drop
$\unicode[STIX]{x0394}T$
across the layer must be measured. That is, both the Nusselt and Rayleigh numbers are emergent quantities.

Controlling the heat flux through, rather than the temperatures at, the rigid boundaries has been considered before. While this variation of Rayleigh’s model profoundly changes the nature of the bifurcation at onset (Hurle, Jakeman & Pike Reference Hurle, Jakeman and Pike1967) it does not apparently affect the
$Nu$
–
$Ra$
relation for high
$Ra$
turbulent convection (Johnston & Doering Reference Johnston and Doering2009). Internal heating or cooling – regulating heat flux into or out of the system in the bulk – while correspondingly extracting or injecting heat via conduction at at least one rigid boundary has also been considered (Goluskin Reference Goluskin2016). But in all these cases the interplay of thermal and velocity boundary layers cannot be separated somewhere in the system.

Bouillaut *et al.* (Reference Bouillaut, Lepot, Aumaître and Gallet2019) control the thermal injection length scale directly via the optical absorption length, introducing the fundamentally new dimensionless parameter
$\ell /h$
into the problem. The limit of small
$\ell /h$
corresponds to fixing the heat flux at the rigid bottom boundary while finite
$\ell /h$
maintains heat input in a portion of the domain well outside any potentially shrinking velocity boundary layer. Their experimental data for various values of
$Nu$
,
$Ra$
and
$\ell /h$
collapse in the form
$(\ell /h)^{2}Nu=f[(\ell /h)^{6}Ra]$
with scaling function
$f$
satisfying
$f[x]\sim x^{1/3}$
for
$x\ll 1$
and
$f[x]\sim x^{1/2}$
for
$x\gg 1$
. (Impressively, the data collapse appears over nearly twenty decades of
$(\ell /h)^{6}Ra$
.) This implies ‘classical’
$Nu\sim Ra^{1/3}$
scaling when the absorption length
$\ell \ll \sqrt{h\unicode[STIX]{x1D6FF}}$
, where
$\unicode[STIX]{x1D6FF}\equiv h/2Nu$
is the apparent thermal boundary layer thickness, and ‘ultimate’
$Nu\sim Ra^{1/2}$
scaling for fixed non-zero
$\ell /h$
as
$Ra\rightarrow \infty$
.

## 3 Future

The work of Bouillaut *et al.* (Reference Bouillaut, Lepot, Aumaître and Gallet2019) opens new directions for Rayleigh–Bénard research. It should stimulate new experimental investigations aimed both at independent confirmation and at understanding Prandtl number influence on the empirical scaling function. The corresponding class of mathematical models for Rayleigh–Bénard convection, with internal heat sources and sinks rather than conduction boundaries, presents new challenges for computation, theory and analysis.

The most obvious theoretical approach to bypass complicating boundary layer effects is to consider the Boussinesq equations with an imposed thermal gradient in a fully periodic domain (Borue & Orszag Reference Borue and Orszag1997), but that turns out to be an unphysical idealization: there is no limit to the resulting flows’ energy due to unbounded runaway solutions that inevitably pollute simulations and obviate analysis (Calzavarini *et al.*
Reference Calzavarini, Doering, Gibbon, Lohse, Tanabe and Toschi2006). The generalized models proposed by Lepot *et al.* (Reference Lepot, Aumaître and Gallet2018) and Bouillaut *et al.* (Reference Bouillaut, Lepot, Aumaître and Gallet2019), however, are physically well defined in the sense that energy in all solutions remains uniformly bounded for all times (Fantuzzi & Doering Reference Fantuzzi and Doering2018; Lepot Reference Lepot2018), even in idealized fully periodic domains (Muite *et al.*
Reference Muite, Whitehead and Doering2017). Perhaps surprisingly these studies also show that, in terms of sensible definitions of
$Nu$
and
$Ra$
in this setting, asymptotic heat transport scaling as high as
$Nu\sim Ra$
[sic] may be realized. That is, the ‘mixing-length’ theory
$Nu\sim Ra^{1/2}$
scaling is no longer ‘ultimate’ in the sense of ‘maximal’ for these internally thermally driven systems.

## 1 Introduction

Rayleigh (1916) proposed a model for Bénard’s turn of the 20th century experiments consisting of the Boussinesq approximation to the Navier–Stokes equations of motion, wherein the density $\unicode[STIX]{x1D70C}$ is fixed in all but the temperature-dependent buoyancy force term and the fluid’s material properties (viscosity $\unicode[STIX]{x1D708}$ , specific heat $c$ and thermal diffusion and expansion coefficients $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D6FC}$ ) are assumed constant, applied to a layer of fluid confined between parallel horizontal iso-thermal no-penetration plates. Even though the actual thermo-capillary mechanism underlying Bénard’s observations – the Bénard–Marangoni instability – was only properly formulated as a hydrodynamic problem decades later by Pearson (Reference Pearson1958), Rayleigh’s model for buoyancy-driven thermal convection is still called the Rayleigh–Bénard problem.

A key feature of interest is the relation between the temperature drop $\unicode[STIX]{x0394}T$ across a fluid layer of thickness $h$ and the resulting heat flux $Q$ . In Rayleigh’s model the temperature drop is controlled at the boundaries, and the first result of his analysis was to identify the non-dimensional group determining instability of the quiescent conduction state: $g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}Th^{3}/\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}$ where $g$ is the acceleration of gravity. Today we call this the Rayleigh number $Ra$ , leaving the fluid’s Prandtl number $Pr=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}$ as the other non-dimensional parameter characterizing the particular system at hand. Heat transport is traditionally measured in units of the conductive heat flux via the Nusselt number $Nu=Qh/\unicode[STIX]{x1D70C}c\unicode[STIX]{x1D705}\unicode[STIX]{x0394}T$ , and the challenge is, for a given domain, to determine $Nu$ as a function of $Ra$ and $Pr$ thereby quantifying the effective thermal conductivity of the convecting layer. The strongly nonlinear regime of paramount importance for geophysical and astrophysical applications corresponds to large Rayleigh numbers and two distinct theories, conventionally referred to as the ‘classical’ and ‘ultimate’ theories, remain in contention for the asymptotic behaviour of $Nu$ as $Ra\rightarrow \infty$ . Mathematical analysis has failed to prove, and experimental studies have failed to definitively rule out, either of these asymptotic theories for $O(1)$ values of $Pr$ .

The work of Bouillaut

et al.(Reference Bouillaut, Lepot, Aumaître and Gallet2019), based on the novel experimental approach of Lepot, Aumaître & Gallet (Reference Lepot, Aumaître and Gallet2018) in which the fluid is radiatively heated rather than relying on conduction through the rigid boundaries, sheds new light on the conundrum. Boundaries and boundary layers play key roles in the alternative ‘classical’ and ‘ultimate’ theories for the extreme limit of Rayleigh–Bénard convection and direct internal heating allows for the disentanglement of velocity and temperature boundary layers. The experimental data suggest that, depending on details of the structure and strength of the thermal forcing, the predictions of either competing theory may be observed.## 2 Overview

The ‘classical’ theory asserts that $Nu\sim Ra^{1/3}$ . It was simultaneously proposed by Priestly (Reference Priestly1954), who argued that $Q$ should be independent of $h$ for turbulent convection in large aspect ratio domains, and Malkus (Reference Malkus1954), whose complicated maximal dissipation theory also predicted that the scaling is uniform in $Pr$ . A decade later Howard (Reference Howard and Görtler1964) reinterpreted the uniform-in- $Pr$ prediction as a marginally stable thermal boundary layer argument, explicitly neglecting the potential effect of a velocity boundary layer.

The ‘ultimate’ theory asserts $Nu\sim Pr^{1/2}Ra^{1/2}$ for $Pr\leqslant O(1)$ . It was originally proposed by Spiegel (Reference Spiegel1963) based on the idea that buoyant fluid elements transport heat at the free-fall velocity $U\sim (g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}Th)^{1/2}$ so that $Q$ becomes independent of the material transport coefficients $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D705}$ . (Spiegel’s theory actually predates this reference as evidenced in the testimony of Batchelor Reference Batchelor and Thomas1961.) Spiegel referred to this approach as a ‘mixing length’ theory without reference to the nature of boundaries or boundary layers. But soon thereafter his postdoc mentor Kraichnan (Reference Kraichnan1962) considered ultra-high $Ra$ situations when velocity boundary layers at the rigid plates presumably transition to shear turbulence, postulating logarithmic corrections to the asymptotic $Ra^{1/2}$ scaling. The moniker ‘ultimate’ was first used by Chavanne

et al.(Reference Chavanne, Chillà, Castaing, Chabaud and Chaussy1997) referring to Kraichnan’s modification of Spiegel’s theory and has since been adopted by the community for an asymptotic state with predominant $1/2$ scaling. Moreover, $Nu\sim Ra^{1/2}$ scaling, albeit uniform in $Pr$ , is mathematically ‘ultimate’ in the sense that it is a rigorous upper bound for heat transport in Rayleigh’s model (Howard Reference Howard1963; Doering & Constantin Reference Doering and Constantin1996).Rather than fixing the temperature and inserting and removing heat at the top and bottom boundaries, Bouillaut

et al.(Reference Bouillaut, Lepot, Aumaître and Gallet2019) heat the bottom of the layer by illuminating the fluid, which is treated with optically absorbing dye, from below. The optical absorption length $\ell$ can be varied by adjusting the dye concentration to control the spatial profile of heat injection into the container. The container is thermally insulated on all boundaries so the bulk temperature increases linearly with time as the experiment proceeds. As long as the Boussinesq approximation remains valid, however, the difference between the local temperature and the linearly increasing-in-time bulk temperature behaves as if the system is heated near the bottom and cooled in the bulk above. In this set-up, distinct from the set-up of Rayleigh’s model, the heat flux $Q$ is controlled and temperature drop $\unicode[STIX]{x0394}T$ across the layer must be measured. That is, both the Nusselt and Rayleigh numbers are emergent quantities.Controlling the heat flux through, rather than the temperatures at, the rigid boundaries has been considered before. While this variation of Rayleigh’s model profoundly changes the nature of the bifurcation at onset (Hurle, Jakeman & Pike Reference Hurle, Jakeman and Pike1967) it does not apparently affect the $Nu$ – $Ra$ relation for high $Ra$ turbulent convection (Johnston & Doering Reference Johnston and Doering2009). Internal heating or cooling – regulating heat flux into or out of the system in the bulk – while correspondingly extracting or injecting heat via conduction at at least one rigid boundary has also been considered (Goluskin Reference Goluskin2016). But in all these cases the interplay of thermal and velocity boundary layers cannot be separated somewhere in the system.

Bouillaut

et al.(Reference Bouillaut, Lepot, Aumaître and Gallet2019) control the thermal injection length scale directly via the optical absorption length, introducing the fundamentally new dimensionless parameter $\ell /h$ into the problem. The limit of small $\ell /h$ corresponds to fixing the heat flux at the rigid bottom boundary while finite $\ell /h$ maintains heat input in a portion of the domain well outside any potentially shrinking velocity boundary layer. Their experimental data for various values of $Nu$ , $Ra$ and $\ell /h$ collapse in the form $(\ell /h)^{2}Nu=f[(\ell /h)^{6}Ra]$ with scaling function $f$ satisfying $f[x]\sim x^{1/3}$ for $x\ll 1$ and $f[x]\sim x^{1/2}$ for $x\gg 1$ . (Impressively, the data collapse appears over nearly twenty decades of $(\ell /h)^{6}Ra$ .) This implies ‘classical’ $Nu\sim Ra^{1/3}$ scaling when the absorption length $\ell \ll \sqrt{h\unicode[STIX]{x1D6FF}}$ , where $\unicode[STIX]{x1D6FF}\equiv h/2Nu$ is the apparent thermal boundary layer thickness, and ‘ultimate’ $Nu\sim Ra^{1/2}$ scaling for fixed non-zero $\ell /h$ as $Ra\rightarrow \infty$ .## 3 Future

The work of Bouillaut

et al.(Reference Bouillaut, Lepot, Aumaître and Gallet2019) opens new directions for Rayleigh–Bénard research. It should stimulate new experimental investigations aimed both at independent confirmation and at understanding Prandtl number influence on the empirical scaling function. The corresponding class of mathematical models for Rayleigh–Bénard convection, with internal heat sources and sinks rather than conduction boundaries, presents new challenges for computation, theory and analysis.The most obvious theoretical approach to bypass complicating boundary layer effects is to consider the Boussinesq equations with an imposed thermal gradient in a fully periodic domain (Borue & Orszag Reference Borue and Orszag1997), but that turns out to be an unphysical idealization: there is no limit to the resulting flows’ energy due to unbounded runaway solutions that inevitably pollute simulations and obviate analysis (Calzavarini

et al.Reference Calzavarini, Doering, Gibbon, Lohse, Tanabe and Toschi2006). The generalized models proposed by Lepotet al.(Reference Lepot, Aumaître and Gallet2018) and Bouillautet al.(Reference Bouillaut, Lepot, Aumaître and Gallet2019), however, are physically well defined in the sense that energy in all solutions remains uniformly bounded for all times (Fantuzzi & Doering Reference Fantuzzi and Doering2018; Lepot Reference Lepot2018), even in idealized fully periodic domains (Muiteet al.Reference Muite, Whitehead and Doering2017). Perhaps surprisingly these studies also show that, in terms of sensible definitions of $Nu$ and $Ra$ in this setting, asymptotic heat transport scaling as high as $Nu\sim Ra$ [sic] may be realized. That is, the ‘mixing-length’ theory $Nu\sim Ra^{1/2}$ scaling is no longer ‘ultimate’ in the sense of ‘maximal’ for these internally thermally driven systems.## Acknowledgement

Author supported in part by National Science Foundation Award DMS-1813003.