# Thermal boundary condition studies in large aspect ratio Rayleigh-Bénard
convection

Theo Käufer1, Philipp P. Vieweg1, Jörg Schumacher1,2, and Christian Cierpka1 1
Institute of Thermodynamics and Fluid Mechanics, Technische Universität
Ilmenau, D-98684 Ilmenau, Germany,
2 Tandon School of Engineering, New York University, New York City, NY 11201,
USA

###### Abstract

We study the influence of thermal boundary conditions on large aspect ratio
Rayleigh-Bénard convection by a joint analysis of experimental and numerical
data sets for a Prandl number $\mathrm{Pr}=7$ and Rayleigh numbers
$\mathrm{Ra}=10^{5}-10^{6}$. The spatio-temporal experimental data are
obtained by combined Particle Image Velocimetry and Particle Image Thermometry
measurements in a cuboid cell filled with water at an aspect ratio
$\Gamma=25$. In addition, numerical data are generated by Direct Numerical
Simulations (DNS) in domains with $\Gamma=25$ as well as $\Gamma=60$ subject
to different thermal boundary conditions. Our experimental data show an
increased characteristic horizontal extension scale of the flow structures,
$\tilde{\lambda}$, for increasing $\mathrm{Ra}$, which is coupled with a raise
of the Biot number $\mathrm{Bi}$ in particular at the cooling plate. However,
we find the experimental flow structure size to range in any case between the
ones observed for the idealized thermal conditions captured by the
simulations. On the one hand, they are larger than in the numerical case with
applied uniform temperatures at the plates, but, on the other hand, smaller
than in the case of an applied constant heat flux, the latter of which leads
to a structure that grows gradually up to the horizontal domain size. We are
able to link this observation qualitatively to theoretical predictions for the
onset of convection. Furthermore, we study the effect of the asymmetric
boundary conditions on the heat transfer. Contrasting experimental and
numerical data reveals an increased probability of far-tail events of reversed
heat transfer. The successive decomposition of the local Nusselt number
$\mathrm{Nu}_{\textrm{loc}}$ traces this effect back to the sign of the
temperature deviation $\tilde{\Theta}$, eventually revealing asymmetries of
the heating and cooling plate on the thermal variance of the generated thermal
plumes.

##  I Introduction

Thermal convection drives many natural flow phenomena such as convection in
the Earth’s outer core [1, 2], mantle [3, 4], ocean [5, 6], or atmosphere [7,
8]. Beside its presence throughout the layers of Earth, convection can also be
found, e.g., in the Sun and other planets of our solar system [9, 10, 11, 12].
Characteristic to all of these systems is a typically large aspect ratio
$\Gamma=W/H$ [13, 14], i.e., the height of the flow domain $H$ in the
direction of gravity is typically much smaller compared to the width $W$
perpendicular to it.

Albeit thermal convection has meanwhile been studied for more than 100 years
[15, 16], the exploration of convection in large aspect ratios intensified
just recently [17, 18, 19, 20]. Thereby, the formation of long-living large-
scale flow structures – in this case, the so-called turbulent superstructures,
see also Section II – was discovered by numerical investigations of large
aspect ratio Rayleigh-Bénard convection (RBC) [17, 18]. Since their discovery,
these flow structures are subject of intense research due to their clear
distinction from small-scale turbulence, which acts on much smaller time
scales. Besides the effect of the thermal boundary condition, which we discuss
in a general sense in Section II and specifically for the case of mixed and
asymmetric boundary conditions in the remainder of this paper, further aspects
of the turbulent superstructures have been investigated. For example, the
coherence of velocity and temperature structures was shown by analysis of the
temperature and velocity co-spectra. [19]. Other studies focused on the
interaction between small-scale fluctuations and turbulent superstructures
[21, 22], underlining the richness of large aspect ratio convection.

As a complement to these studies in the Eulerian frame of reference, the role
of large-scale flow structures on the Lagrangian or material transport has
been revealed by means of spectral clustering of Lagrangian trajectories [23,
24, 25]. Theoretical and numerical studies are supported and extended by
experimental analyses of the long-term behavior of turbulent superstructures
together with their local convective heat transfer based on spatially and
temporally resolved temperature and velocity measurements [26, 27].

Contrasting experimental and numerical results, Moller et al. [27] observed
two striking disagreements: (i) the overall heat transport across the fluid
layer is decreased, and (ii) the flow structures are significantly larger in
the experimental setup compared to the numerical results. Interestingly, these
discrepancies increase once the Rayleigh number $\mathrm{Ra}$ – as a measure
of the strength of the thermal driving, see eq. (8) – is increased. As the
Prandtl number $\mathrm{Pr}$ – defining the working fluid, see again eq. (8) –
is constant and these two control parameters specify the entire dynamical
system, this observation suggests that these observations are a feature of the
experiment, which are not captured by idealized simulations. Considering the
experimental setup, Moller et al. [27] attributed these disagreements to the
non-ideal thermal boundary condition at the cooling plate made from glass. In
the present work, we pick up these observations and analyze the influence of
the present asymmetric thermal boundary conditions in more detail.

The impact of thermal boundary conditions on the heat transport across the
fluid layer – as can be quantified by the Nusselt number $\mathrm{Nu}$, see
eqs. (9), (10), and (12) – was investigated in several past studies [28, 29,
30, 31, 32, 20] and led, on the one hand, to suggestions of corrections for
the influence of the plates’ materials [28, 29], whereas, on the other hand,
deviations from isothermal conditions seemed to accelerate the convective heat
transport at least up to $\mathrm{Ra}\sim 10^{7}$ [31, 32, 20]. However,
except for [20], none of these studies provided a horizontally sufficiently
extended domain such that turbulent superstructures or equivalent structures
could form [33, 34, 35]. This circumstance becomes crucial, given the fact
that the large-scale flow structures transport most of the heat across the
fluid layer [19].

These reasons underline the relevance of further investigations of the
influence of mixed or asymmetric thermal boundary conditions on the heat
transfer and large-scale flow structure formation in large aspect ratio
domains. First, we contextualize the experimentally present conditions,
investigate the size and evolution of the large-scale flow structures by
extracting the characteristic wavelength from the power spectrum, and compare
our observation with theoretical predictions for the onset of convection.
Second, we analyze the local heat transfer and compare the corresponding
experimental and numerical probability density functions (PDFs). We observe
higher probabilities of a reversed heat transfer for experimental data. The
successive decomposition of the local Nusselt number
$\mathrm{Nu}_{\textrm{loc}}$ eventually unveils the underlying effect of the
asymmetric thermal boundary conditions.

The remainder of this manuscript is organized as follows. We start by
outlining large-scale flow structures and their formation in Section II and,
thereupon, briefly describe the experimental and numerical methods in Section
III. We then address the large-scale flow structure size in Section IV,
whereas the heat transfer is analyzed in Section V. We conclude with a
discussion of our results in Section VI.

##  II Large-scale flow structures in thermal convection

Figure 1: Thermal boundary conditions determine the large-scale flow structure
formation, the latter of which are termed supergranules or turbulent
superstructures in the case of Neumann or Dirichlet conditions, respectively.
The panels visualize the temperature fields in horizontal cross-sections at
aspect ratio $\Gamma=60$ from [20].

As briefly introduced above, Rayleigh-Bénard convection exhibits
characteristic long-living large-scale flow structures (LLFSs) that set
themselves apart from smaller-scale turbulence acting on significantly shorter
time scales. While the latter capture smaller eddies that might be as small as
the Kolmogorov scale $\eta_{\textrm{K}}\ll H$ and correspond to rapid temporal
fluctuations, the former offer characteristic horizontal extensions with
typical length scales of at least $\mathcal{O}\left(H\right)$ and time scales
$\tau_{\textrm{LLFSs}}\gg\tau_{\textrm{f}}$ (with $\tau_{\textrm{f}}$
representing the convective time unit, see also Section III.2). Due to this
clear scale separation, these different processes or flow structures can be
separated by a Reynolds decomposition, or temporal filter [17, 20].

Considering the simplest configuration of Rayleigh-Bénard convection with no
additional physics such as rotation [36, 37, 38], magnetohydrodynamic effects
[39, 40, 41], or complex fluid property dependencies [42, 43, 44], the nature
of these long-living large-scale flow structures have recently been found to
depend decisively on the aspect ratio [13, 41, 17, 18, 32] and thermal
boundary conditions [32, 20]. If the aspect ratio is small with $\Gamma\approx
1$, a so-called large-scale circulation or mean wind with a characteristic
horizontal extension of about $H$ forms [13] – in this case, the large-scale
flow structure is (independently of the thermal boundary conditions [32])
clearly dictated by the lateral boundaries. This changes once the flow is
released from the impact of the latter. Their importance decreases with
$\mathcal{O}\left(\Gamma^{-2}\right)$ [34, 35, 33], so domains with
$\Gamma\gtrsim 20\gg 1$ can be considered being “large” and fairly close
approximations of infinitely extended fluid layers [35, 18, 19]. As the impact
of the lateral boundaries vanishes, so arise other, or new dependencies that
govern the nature of the establishing long-living large-scale flow structures
– albeit discovered just recently [20], the thermal boundary conditions offer
the perhaps most striking effect.

In the classical Dirichlet case of applied uniform temperatures at the plates,
so-called turbulent superstructures form with time-independent characteristic
horizontal extensions of order $\mathcal{O}\left(H\right)$ [17, 20] – these
structures are in clear contrast to the time-dependent gradual supergranule
aggregation that takes place in the opposite Neumann case of an applied
uniform temperature gradient (or heat flux) at the plates [20]. More details
on the mathematical aspects of these choices are provided in Section III.2,
whereas Section IV will highlight that the particular choice is physically
related to the propagation or relaxation of thermal perturbations in the
plates and the fluid. Interestingly, this latter process leads (if not
controlled by rotation [38]) to a growth of the structures until the
horizontal domain size is reached, i.e., their characteristic length
$\tilde{\lambda}\sim\Gamma\gg\mathcal{O}\left(H\right)$, exceeding thus the
turbulent superstructures clearly in terms of their length scale. Figure 1 (a,
b) compares these turbulent long-living large-scale flow structures in a
$\Gamma=60$ domain for an equivalent heat transfer through the fluid layer.
Crucially, other aspects such as the Rayleigh number, Prandtl number, or
mechanical boundary conditions [17, 20] become secondary once compared to this
dominant effect of thermal boundary conditions. As these observations in the
turbulent regime are qualitatively in accordance with the behavior at and
slightly above the onset of convection, this highlights the importance of
primary and secondary instabilities for the dynamical system in general. We
will thus address the transition from Dirichlet to Neumann conditions in more
detail in Section IV.

Additional physical mechanisms, such as rotation around the vertical axis, may
additionally influence the long-living large-scale flow structures. Vieweg et
al. [38] showed that the impact of rotation depends quite differently on the
choice of the thermal boundary condition or nature of the latter. However,
such additional effects are beyond the scope of the present manuscript with
its focus on the thermal boundary conditions.

Experimental studies are the method of choice for long-time studies of
turbulent superstructures since the expenses for direct numerical simulations
in horizontally extended domains scale with $\sim\Gamma^{2}$ [45] and $\sim
t$, restricting numerical studies of the long-term behaviour of these long-
living large-scale flow structures seriously. Given the slow reorganisation of
the turbulent superstructures over time [17, 27], this represents a
significant limitation for their studies by numerical approaches. However, the
large aspect ratio requires the design of novel experimental setups and the
adaption and development of advanced temporally and spatially resolved
velocity and temperature measurements. These are briefly discussed in III.1.

##  III Methods

###  III.1 Experimental setup and measurements

All experiments that are considered in this report were conducted in a cuboid
Rayleigh-Bénard convection cell with aspect ratio $\Gamma=25$ and a width
$W=700$ mm and height $H=28$ mm. The cell is filled with water at a mean
temperature of $T_{\textrm{ref}}\approx 19.5^{\circ}\textrm{C}$ exhibiting a
Prandtl number $\mathrm{Pr}=7.1$. This cell design, with its large aspect
ratio, allows the turbulent superstructures to form and is sketched in Figure
2.

Figure 2: Sketch of the experimental setup for the simultaneous measurements
of all three velocity components and the temperature in a wide horizontal
plane through a transparent cooling plate.

The fluid layer is confined by glass side walls, a heating plate at the bottom
made of aluminum and a cooling plate assembly at the top made of two parallel
glass sheets with a horizontal gap in between to allow cooling water to flow
through. The temperature of both plates can be controlled independently by
thermostats.

Note here, in particular, that the transparent cooling plate arrangement is
required to grant optical access to the flow domain from the top, and enables
the large field of view (FOV) inevitable for the investigation of the large-
scale flow structures or turbulent superstructures. Moreover, a comparably
detailed study of the latter would not be possible in a vertical plane due to
the small cell height [46]. To measure the fluid velocity and temperature
simultaneously, a combined stereoscopic Particle Image Velocimetry (PIV) and
Particle Image Thermometry (PIT) approach with polymer-encapsulated
thermochromic liquid crystals (TLCs) as seeding particles is employed. The
unique property of a temperature-related wavelength change of the reflected
light of the TLCs is leveraged to quantify the temperature. Hence, a light
source with a continuous illumination spectrum is required since the TLCs can
reflect only spectral components of light that are part of the original
illumination spectrum. To achieve optimal illumination, a custom-made light
source was designed, which consists of an LED array combined with light sheet
optics to focus the light into a sheet of thickness $\approx$ 3-4 mm across
the entire FOV.

Three cameras observe the flow domain with an observation angle of
$\Phi\approx 65^{\circ}$. While two of those are monochrome cameras in a
stereoscopic arrangement and used to perform stereoscopic PIV – which measures
all velocity components within a plane [47, 48] –, the third one is a color
camera to simultaneously capture the color of the light that is reflected by
the TLCs. Subsequently, the temperature in the ligth sheet can be inferred by
means of PIT [49]. In more detail, the color image is sliced into sub-images
or interrogation windows, and the red, green and blue color values inside the
interrogation windows are averaged. Those averages are subsequently
transformed into the hue, saturation and value (HSV) color space. To determine
the temperature, the hue is compared to a calibration curve. As the TLC
reflection properties depend on the angle between the light source and
observation, calibration curves are obtained during the calibration process
for each interrogation window individually. Details on this technique can be
found in Moller et al. [50]. Alternatively, a neural network can be trained to
replace these local calibration functions [51, 52]. The extension towards
fully volumetric temperature and velocity measurements for wide observation
regions, currently only available for low-aspact ratio setups [53], is still
ongoing.

Ra | Raexact | $\mathrm{Nu}$ | $T_{\mathrm{h}}$ | $T_{\mathrm{c}}$ | $\tilde{x}\times\tilde{y}$ | $\tau_{\mathrm{f}}$ | $\tilde{t}_{\mathrm{total}}$
---|---|---|---|---|---|---|---
$2\times 10^{5}$ | $2.07\times 10^{5}$ | 4.00 | $19.78^{\circ}\mathrm{C}$ | $19.08^{\circ}\mathrm{C}$ | $16.1\times 16.7$ | $4.51\mathrm{~{}s}$ | $5.39\times 10^{3}$
$4\times 10^{5}$ | $4.34\times 10^{5}$ | 5.31 | $20.22^{\circ}\mathrm{C}$ | $18.76^{\circ}\mathrm{C}$ | $16.1\times 16.7$ | $3.12\mathrm{~{}s}$ | $7.79\times 10^{3}$
$7\times 10^{5}$ | $7.34\times 10^{5}$ | 5.58 | $20.87^{\circ}\mathrm{C}$ | $18.43^{\circ}\mathrm{C}$ | $16.2\times 16.6$ | $2.40\mathrm{~{}s}$ | $1.01\times 10^{4}$
Table 1: Overview of the global parameters for the different experimental
runs. Ra and Raexact denote the Ra we subsequently refer to and the exact Ra
and Nu the measured global Nusselt number. $T_{\mathrm{h}}$ and
$T_{\mathrm{c}}$ represent the heating and cooling plate temperature,
$\tilde{x}\times\tilde{y}$ the field of view, $\tau_{\mathrm{f}}$ the free-
fall time and $\tilde{t}_{\textrm{total}}$ the duration of the experimental
run in free-fall units. Data adapted from [54].

In this paper, we consider three experimental runs at $\mathrm{Ra}=2\times
10^{5}$, $4\times 10^{5}$, and $7\times 10^{5}$ with measurements performed in
the horizontal midplane. To reduce the amount of data and the required
processing time, the temperature and velocity fields are measured in bursts of
5 minutes and inter-burst-breaks of 15 minutes. For each experiment, 19 bursts
with 200 snapshots each are utilized. The dimensional data is subsequently
non-dimensionalized as explained in section III.2 for the Dirichlet case and
$\mathrm{Ra}\equiv\mathrm{Ra}_{\textrm{D}}$. Further details on the
experimental runs are provided in Table 1.

###  III.2 Numerical simulations

To compare the experimental setup with numerically obtained results, we
consider the simplest considerable turbulent convection configuration – three-
dimensional, turbulent Rayleigh-Bénard convection in the Oberbeck-Boussinesq
approximation. In this setup, buoyancy drives the flow by coupling the scalar
temperature field $T\left(\bm{x},t\right)$ with the incompressible velocity
vector field $\bm{u}\left(\bm{x},t\right)$ where $\bm{x}=(x,y,z)$ and
$\bm{u}=(u_{x},u_{y},u_{z})$. In more detail, the mass density varies in this
approximation only in the buoyancy forcing term, and there only at first order
with the temperature deviation from some reference value.

Similar to the experimental setup, we consider a Cartesian domain $V=W\times
W\times H$. The confining bottom and top planes at $z\in\left\\{0,H\right\\}$
obey either no-slip (ns) or free-slip (fs) mechanical boundary conditions
(mbcs),

$u_{x}=u_{y}=u_{z}=0\qquad\textrm{or}\qquad u_{z}=\frac{\partial
u_{x}}{\partial z}=\frac{\partial u_{y}}{\partial z}=0,$ (1)

respectively. The lateral boundaries (lbs) are either closed and insulating –
in the case of which they exhibit again no-slip boundary conditions –, or
periodic. In the case of the latter, any quantity $\Phi$ is repeated after the
periodic length $W=W_{x}=W_{y}$ such that

$\Phi\left(\bm{x},t\right)=\Phi\left(\bm{x}+i_{x}W_{x}\bm{e}_{x}+i_{y}W_{y}\bm{e}_{y},t\right)\quad\textrm{with
}i\in\mathbb{Z}.$ (2)

In accordance with the scientific objective of the present study, we vary the
thermal boundary conditions (tbcs) at the plates to correspond mathematically
either to a Dirichlet (D) or a Neumann (N) condition. In the case of Dirichlet
conditions, the top and bottom planes exhibit constant temperatures

$T\left(z=0\right)=T_{\textrm{bot}}\quad\textrm{and}\quad
T\left(z=H\right)=T_{\textrm{top}},$ (3)

whereas in the Neumann case the temperature is allowed to vary locally – all
that is fixed is the applied constant heat flux through the spatially constant
temperature gradient

$\left.\frac{\partial T}{\partial z}\right|_{z=0}=\left.\frac{\partial
T}{\partial z}\right|_{z=H}=-\beta\quad\textrm{with }\beta>0.$ (4)

The equations of motion are made dimensionless based on characteristic
quantities of the system. We adopt the layer height $H$ as unit of length and
the so-called free-fall time $\tau_{\textrm{f}}=H/U_{\textrm{f}}$ as the unit
of time with the free-fall velocity $U_{\textrm{f}}=\sqrt{\alpha
gT_{\textrm{char}}H}$. Here, $\alpha$ represents the isobaric expansion
coefficient and $g$ the acceleration due to gravity. $T_{\textrm{char}}$ is a
characteristic temperature which depends on the thermal boundary condition. In
the Dirichlet case $T_{\textrm{char}}=\Delta
T=T_{\textrm{bot}}-T_{\textrm{top}}>0$, whereas in the Neumann case
$T_{\textrm{char}}=\beta H$. The governing equations follow in their non-
dimensional description as

$\displaystyle\tilde{\nabla}\cdot\tilde{\bm{u}}$ $\displaystyle=0,$ (5)
$\displaystyle\frac{\partial\tilde{\bm{u}}}{\partial\tilde{t}}+(\tilde{\bm{u}}\cdot\tilde{\nabla})\thickspace\tilde{\bm{u}}$
$\displaystyle=-\tilde{\nabla}\tilde{p}+\sqrt{\frac{\mathrm{Pr}}{\mathrm{Ra}_{\textrm{D,N}}}}\tilde{\nabla}^{2}\tilde{\bm{u}}+\tilde{T}\bm{e}_{z},$
(6)
$\displaystyle\frac{\partial\tilde{T}}{\partial\tilde{t}}+(\tilde{\bm{u}}\cdot\tilde{\nabla})\thickspace\tilde{T}$
$\displaystyle=\frac{1}{\sqrt{\mathrm{Ra}_{\textrm{D,N}}\mathrm{Pr}}}\tilde{\nabla}^{2}\tilde{T},$
(7)

where $p$ is the pressure field, and tildes indicate dimensionless quantities.
Two dimensionless parameters – the Prandtl number $\mathrm{Pr}$ and the
Rayleigh number $\mathrm{Ra}$ – control the entire dynamics and summarise all
the dimensional coefficients and parameters. These numbers are defined as

$\mathrm{Pr}=\frac{\nu}{\kappa},\thickspace\thickspace\thickspace\mathrm{Ra}_{\textrm{D}}=\frac{g\alpha\Delta
TH^{3}}{\nu\kappa},\textrm{
and}\thickspace\thickspace\thickspace\mathrm{Ra}_{\textrm{N}}=\frac{g\alpha\beta
H^{4}}{\nu\kappa}$ (8)

with the kinematic viscosity and temperature diffusivity of the fluid, $\nu$
and $\kappa$, respectively. Note that due to the non-dimensionalization, e.g.,
the vertical coordinate $0\leq\tilde{z}\leq 1$.

We solve the equations of motion using the spectral element solver nek5000
[55, 56] and resolve all dynamically relevant scales down to the Kolmogorov
scale $\eta_{\textrm{K}}$. All simulations start with the fluid at rest and
the linear conduction temperature profile, which is randomly and
infinitesimally perturbed. The global mean temperature is thus independent of
the thermal boundary condition, $\langle\tilde{T}\rangle_{\tilde{V}}=0.5$.
More detailed information on numerical details can be found in our previous
works [56, 17, 20, 24, 57].

Finally, the global turbulent heat transfer across the fluid layer is
quantified by the Nusselt number $\mathrm{Nu}$, which is given for the
Dirichlet case by

$\mathrm{Nu}_{\textrm{D}}=-\Bigg{\langle}\left.\frac{\partial\tilde{T}}{\partial\tilde{z}}\right|_{\tilde{z}=0}\Bigg{\rangle}_{\tilde{A}}=-\Bigg{\langle}\left.\frac{\partial\tilde{T}}{\partial\tilde{z}}\right|_{\tilde{z}=1}\Bigg{\rangle}_{\tilde{A}}$
(9)

where $\tilde{A}$ is the non-dimensional horizontal cross section
$\tilde{A}=\Gamma\times\Gamma$, and for the Neumann case by

$\mathrm{Nu}_{\textrm{N}}=\frac{1}{\langle\tilde{T}\left(\tilde{z}=0\right)\rangle_{\tilde{A}}-\langle\tilde{T}\left(\tilde{z}=1\right)\rangle_{\tilde{A}}}.$
(10)

This allows eventually to connect the different definitions of the Rayleigh
numbers via
$\mathrm{Ra}_{\textrm{D}}=\mathrm{Ra}_{\textrm{N}}/\mathrm{Nu}_{\textrm{N}}$
[58]. In total, we consider three numerical data sets with various boundary
conditions – details are summarised in Table 2

Ra | Pr | $\Gamma$ | mbc | tbc | lb | $\tilde{\lambda}$
---|---|---|---|---|---|---
$1\times 10^{5}$ | 7 | 25 | ns | D | closed | 6.26
$1\times 10^{6}$ | 7 | 25 | ns | D | closed | 6.26
$2\times 10^{5}$ | 7 | 60 | fs | N | periodic |  $\leq$ 60
Table 2: Table of the parameters of the different numerical runs. Including
the Rayleigh number $\mathrm{Ra}$ , the Prandtl number $\mathrm{Pr}$ , the
aspect ratio $\Gamma$, the mechanical and thermal boundary condition, the
lateral boundaries and the characteristic large-scale flow structure size
$\tilde{\lambda}$.

##  IV The influence of the thermal boundary conditions on the large-scale
flow structure size

As highlighted in the introduction, one of the disagreements observed by
Moller et al. [27] concerned the size of the large-scale flow structures – in
particular, the experimental data showed larger structures compared to DNS
data with isothermal boundary conditions at similar $\mathrm{Ra}$. The authors
attributed this disagreement to the thermal boundary conditions at the cooling
plate. Due to the low thermal conductivity of glass the cooling plate cannot
be considered as perfectly isothermal.

Such a connection between the convective structure size and thermal boundary
condition is actually already supported by the linear stability at the onset
of convection. Figure 3 visualizes the neutral stability curves for various
thermal boundary conditions. While the Dirichlet case with
$\kappa_{\textrm{f}}/\kappa_{\textrm{p}}\rightarrow 0$ is the most stable one,
the opposing Neumann case in the limit
$\kappa_{\textrm{f}}/\kappa_{\textrm{p}}\rightarrow\infty$ is the least stable
one. Between these two extreme cases, many other neutral curves can be drawn
depending on the ratio of thermal diffusivities of the fluid and the solid
plate $\kappa_{\mathrm{f}}/\kappa_{\mathrm{p}}$. Interestingly, even the
critical convective structure size seems to change gradually from one limit to
the other, see Fig. 3.

Figure 3: Linear stability at varying thermal but mechanical no-slip boundary
conditions. As the ratio of thermal diffusivities deviates stronger from the
Dirichlet scenario, the critical structures at the onset of convection become
less stable and increasingly extended. The Dirichlet case is computed
according to [59], whereas the other curves are interpolated from [60].

As convection sets in, heat conduction is complemented by convective heat
transfer. While the latter’s strength is quantified by the Nusselt number
$\mathrm{Nu}$, a meaningful amendment to the ratio of thermal diffusivities is
provided by the effective ratio
$\kappa_{\textrm{f}}\mathrm{Nu}/\kappa_{\textrm{p}}$. Physically, the latter
controls the propagation or relaxation of thermal perturbations [60]. An
alternative measure to categorize the thermal boundary condition is the Biot
number $\mathrm{Bi}$, which describes the ratio of convective heat transfer at
the surface of a body to the heat transfer inside the body due to pure heat
conduction [61]. Translated to our experimental setup, this quantity is given
by

$\mathrm{Bi}=\mathrm{Nu}\frac{\lambda_{\mathrm{f}}d_{\mathrm{p}}}{\lambda_{\mathrm{p}}H}$
(11)

with the plate thickness $d_{\mathrm{p}}$.

Figure 4: Temperature field in the horizontal midplane obtained from
measurements at $\mathrm{Ra}$ = $2\times 10^{5}$ (top row) and from
simulations with thermal Neumann boundary conditions and mechanical free-slip
boundary conditions at $\mathrm{Ra}_{\textrm{N}}=2\times 10^{5}$ (bottom row)
at five exemplary time instances. To remove noise induced by the measurements,
the experimental temperature fields are time-averaged over
$33\tau_{\textrm{f}}$.

Differently to simulations, perfect thermal boundary conditions cannot be
achieved in any experiments. However, when the ratio of the effective thermal
diffusivities $\kappa_{\textrm{f}}\mathrm{Nu}/\kappa_{\textrm{p}}\ll 1$ and
$\mathrm{Bi}\ll 1$, isothermal conditions can be well approximated. We
summarise the corresponding ratios or values for all experiments presented in
this paper in Table 3. As can be seen, both characteristics depend crucially
on the choice of the plate. On the one hand, the heating plate can be
considered isothermal since $\mathrm{Bi}_{\mathrm{h}}$ and
$\kappa_{\textrm{f}}\mathrm{Nu}/\kappa_{\textrm{p, h}}$ are both of the order
$\mathcal{O}\left(10^{-2}\right)$. On the other hand, at the cooling plate,
both $0.88\lesssim\mathrm{Bi}_{\mathrm{c}}\lesssim 1.22$ and
$1.47\lesssim\kappa_{\textrm{f}}\mathrm{Nu}/\kappa_{\textrm{p, c}}\lesssim
2.05$ – depending on the particular Rayleigh number – indicate that the
thermal boundary conditions can neither be considered as constant temperature
nor constant heat flux, but instead only as mixed boundary condition.
Recalling the stability behavior at the onset of convection – see again Figure
3 –, these numbers also imply larger most unstable convective length scales.

$\mathrm{Ra}$ | $\mathrm{Bi}_{\mathrm{c}}$ | $\mathrm{Bi}_{\mathrm{h}}$ | $\kappa_{\mathrm{f}}\mathrm{Nu}/\kappa_{\textrm{p, c}}$ | $\kappa_{\mathrm{f}}\mathrm{Nu}/\kappa_{\mathrm{\textrm{p, h}}}$ | $\tilde{\lambda}$
---|---|---|---|---|---
$2\times 10^{5}$ | 0.88 | 0.0091 | 1.47 | 0.0092 | 9.36
$4\times 10^{5}$ | 1.16 | 0.0121 | 1.95 | 0.0121 | 10.53
$7\times 10^{5}$ | 1.22 | 0.0127 | 2.05 | 0.0128 | 11.50
Table 3: Table of the Biot number at the cooling and heating plate Bic and
Bih, the ratio of the effective thermal diffusivities of the fluid and the
plate material at the cooling and heating plate
$\kappa_{\mathrm{f}}\mathrm{Nu}/\kappa_{\mathrm{p,c}}$ and
$\kappa_{\mathrm{f}}\mathrm{Nu}/\kappa_{\mathrm{p,h}}$, and the average
characteristic size of the temperature patterns $\tilde{\lambda}$ for the
experiments at varying $\mathrm{Ra}$.

Recent numerical studies showed that the underlying stability from at and
slightly above the onset of convection is not forgotten by the dynamical
system even when the latter is driven far into the turbulent regime [20, 38].
Transferring this consideration to the present context allows us to identify
the top plate as an important factor influencing the formation of large-scale
flow structures. In other words, the primary stability behavior suggests the
formation of larger convective flow patterns even in the turbulent regime,
particularly for the top plate. This agrees, in fact, with our experimental
observations. As the Rayleigh number $\mathrm{Ra}$ is increased, so is the
effective ratio due to the larger Nusselt number, and the experimentally
observed patterns become successively larger – see again Table 3.

Clearly, the size of the large-scale flow structures manifests based on a
balance between almost perfect Dirichlet boundary conditions at the bottom
plate and conditions closer to the Neumann case at the top plate. A
significantly height-dependent flow structure size should, however, not be
expected as $\tilde{\lambda}\gg H$.

Figure 5: Plot of the temporal development of the superstructure size. The
horizontal axis shows the time in free-fall units $\tau_{\textrm{f}}$, and the
vertical axis shows the structure size. The dashed line indicates an aspect
ratio $\Gamma$=25, which is the domain size of the experiment and the
numerical simulation in the Dirichlet configuration.

As outlined in Section II, different thermal boundary conditions result
furthermore in different temporal behavior of the large-scale flow structures.
We compare this aspect in Figure 4 for numerical and experimental data. Note
that the experimental data visualize an over $33\tau_{\textrm{f}}$ slightly
time-averaged temperature field (to reduce the influence of noise on the shape
of the superstructures), whereas the numerical data shows instantaneous
snapshots throughout the gradual supergranule aggregation for the Neumann
case. The temporal behavior of both flows differs significantly. While the
structures observed in the experiment show a rather slow reorganization over
time, the numerical Neumann case shows an intense gradual aggregation process.
The fundamental difference concerning the long-term behavior or dynamics of
the large-scale flow structures suggests to term the experimentally observed
patterns are still turbulent superstructures rather than supergranules.

This is supported by an even more comprehensive comparison of the temporal
evolution of the large-scale flow structure size for all in this work
considered experimental and numerical data. We compute, therefore, the two-
dimensional power spectrum of the temperature field, average it in azimuthal
direction, determine the wave number $\hat{k}$ of the spectral peak, and
eventually calculate the corresponding characteristic wavelength via
$\tilde{\lambda}=2\pi/\hat{k}$. Note here that, as already above, the
experimental temperature fields are again averaged over $33\tau_{\textrm{f}}$
and the two-dimensional spectra zero-padded to reduce the influence of noise
and increase the spectral resolution, respectively. The results of this
analysis are summarized in Figure 5. Note here that the discrete nature of
this evolution is nothing but the consequence of the discrete wave numbers
corresponding to the Discrete Fourier Transform (DFT). Here, we want to point
out that the mentioned structure size appears larger, as reported in previous
publications. This is exclusively related to a different binning scheme of the
azimuthally averaged spectra required to capture the structure size in the
Neumann case correctly. When looking at this Figure from the bottom to the
top, the turbulent superstructures observed in the constant temperature
simulations exhibit no variation in the structure size over time. The
situation changes for the experimental runs, in which the structure size is
significantly larger and fluctuates slightly over time. This fluctuation can
be related to the rearrangement of the turbulent superstructures and the
horizontally not entirely captured flow domain. Unlike the Dirichlet
simulations, the structure size in the experiments clearly depends on the
Rayleigh number $\mathrm{Ra}$. We attribute this circumstance to the
$\mathrm{Ra}$-dependent thermal boundary conditions at the top plate, as
explained above. Although the structure size fluctuates over time in the
experiment, this is in clear contrast to the case of thermal Neumann boundary
conditions that are applied at both plates in the remaining numerical
simulation. Here, the gradual aggregation of the flow structures results in a
temporal evolution that stops only artificially once the domain size of
$\Gamma=60$ is reached.

These comparisons of the temporal evolution of the large-scale flow structures
underline the different natures of flow structures subject to different
thermal boundary conditions. In a nutshell, the experimentally observed
patterns are unambiguously more similar to the turbulent superstructures known
for Dirichlet conditions than to those found for Neumann conditions. Again,
this is in accordance with the still finite critical convective structure size
at the onset of convection for $\kappa_{\textrm{f}}/\kappa_{\textrm{p}}=2$,
recall Figure 3.

##  V The influence of the thermal boundary conditions on the heat transfer

Likewise to the difference in structure size, Moller et al. [27] reported
differences in the heat transfer between their experiments and numerical
simulations. In particular, the former showed a reduced global heat transport
as well as higher probabilities of negative local Nusselt numbers
$\mathrm{Nu}_{\textrm{loc}}$ (which will be defined in eq. (14)) compared to
the latter. First, we want to understand in more detail how the global and
local Nusselt number is altered by the experimental measurements and its
definition, respectively.

We start by defining the local Nusselt number $\mathrm{Nu}$ similar to [24]

$\mathrm{Nu}_{\textrm{loc},T}\left(\tilde{\bm{x}},\tilde{t}\right)=\sqrt{\mathrm{Ra}\mathrm{Pr}}\thickspace\tilde{u}_{z}\left(\tilde{\bm{x}},\tilde{t}\right)\tilde{T}\left(\tilde{\bm{x}},\tilde{t}\right)-\frac{\partial\tilde{T}\left(\tilde{\bm{x}},\tilde{t}\right)}{\partial\tilde{z}}$
(12)

which is proportional to the vertical component of the heat current vector.
Note that although both $\mathrm{Nu}_{\textrm{loc},T}$ and
$\mathrm{Nu}_{\textrm{loc}}$ are local measures, the latter is a subset of the
former. This is done by decomposing the temperature into the linear conduction
profile $\tilde{T}_{\mathrm{lin}}$ and its fluctuating part or deviation
$\tilde{\Theta}$ such that
$\tilde{T}\left(\tilde{\bm{x}},\tilde{t}\right)=\tilde{T}_{\mathrm{lin}}\left(\tilde{z}\right)+\tilde{\Theta}\left(\tilde{\bm{x}},\tilde{t}\right)$
with $\tilde{T}_{\mathrm{lin}}=1-\tilde{z}$. Hence, the local Nusselt number
is the combination of $4$ separate subsets such that
$\mathrm{Nu}_{\textrm{loc},T}=\sum_{i=1}^{4}\mathrm{Nu}_{\textrm{loc,}i}$ with

$\displaystyle\mathrm{Nu}_{\textrm{loc,}1}$
$\displaystyle:=\sqrt{\mathrm{Ra}\mathrm{Pr}}\thickspace\tilde{u}_{z}\left(\tilde{\bm{x}},\tilde{t}\right)\tilde{T}_{\textrm{lin}},$
(13) $\displaystyle\mathrm{Nu}_{\textrm{loc,}2}$
$\displaystyle:=\sqrt{\mathrm{Ra}\mathrm{Pr}}\thickspace\tilde{u}_{z}\left(\tilde{\bm{x}},\tilde{t}\right)\tilde{\Theta}\left(\tilde{\bm{x}},\tilde{t}\right)\equiv\mathrm{Nu}_{\textrm{loc}},$
(14) $\displaystyle\mathrm{Nu}_{\textrm{loc,}3}$
$\displaystyle:=-\frac{\partial\tilde{T}_{\textrm{lin}}}{\partial\tilde{z}}=1,$
(15) $\displaystyle\mathrm{Nu}_{\textrm{loc,}4}$
$\displaystyle:=-\frac{\partial\tilde{\Theta}\left(\tilde{\bm{x}},\tilde{t}\right)}{\partial\tilde{z}}.$
(16)

Here we dropped the designation of dependence of the local Nusselt numbers on
$\left(\tilde{\bm{x}},\tilde{t}\right)$ for the sake of conciseness. To
justify a restriction on $\mathrm{Nu}_{\textrm{loc}}$, one may consider (i)
horizontal averages across the entire cross-section and (ii) an analysis at
midplane $z=0.5$. In this case, $\langle\tilde{u}_{z}\rangle_{\tilde{A}}=0$
due to continuity and
$\langle\partial\tilde{\Theta}/\partial\tilde{z}\left(\tilde{z}=0.5\right)\rangle_{\tilde{A}}=1$,
so $\langle\mathrm{Nu}_{\textrm{loc,}1}\rangle_{\tilde{A}}=0$ and
$\langle\mathrm{Nu}_{\textrm{loc,}3}\rangle_{\tilde{A}}+\langle\mathrm{Nu}_{\textrm{loc},4}\rangle_{\tilde{A}}=0$
follows at midplane, respectively. Given these considerations, one ends up
with

$\langle\mathrm{Nu}_{\textrm{loc},T}\left(\tilde{z}=0.5\right)\rangle_{\tilde{A}}=\langle\mathrm{Nu}_{\textrm{loc}}\left(\tilde{z}=0.5\right)\rangle_{\tilde{A}}.$
(17)

Note that these Nusselt numbers yield the global values of the heat transfer,
so
$\mathrm{Nu}=\langle\mathrm{Nu}_{\textrm{loc}}\left(\tilde{z}=0.5\right)\rangle_{\tilde{A}}$.

Clearly, the experimental measurements do not meet the two assumptions or
considerations from above perfectly due to the limited field of view and the
light sheet thickness, respectively. Besides the circumstance of not being
able to measure the vertical gradient of the temperature field, the statistic
of $\mathrm{Nu}_{\textrm{loc}}$ in the experiment is further altered by a
coarser measurement resolution. In more detail, the experimental measurements
cannot resolve the Kolmogorov scale, which in turn affects the statistics.
Thus, there are effects of the experimental measurement on both the global and
local Nusselt number.

Figure 6: Local Nusselt number analysis at or around the horizontal midplane
for the numerical data at $\mathrm{Ra}_{\textrm{D}}$ = 1$\times 10^{5}$.

In order to investigate the impact of these experimental effects on the
statistics, we compute the probability density functions (PDFs) of descendants
of $\mathrm{Nu}_{\textrm{loc},T}$ for the well-resolved numerical data at
$\mathrm{Ra}=1\times 10^{5}$ with different pre-processing. For a more concise
representation, we omit here the effect of $\mathrm{Nu}_{\textrm{loc,}1}$ as
its statistic is symmetric around zero if being evaluated across the entire
cross-section. We start with $\mathrm{Nu}_{\textrm{loc, fine}}$, which
represents the PDF of
$\mathrm{Nu}_{\textrm{loc},T}-\mathrm{Nu}_{\textrm{loc,}1}$ based on high-
resolution data at $z=0.5$. Starting from this quantity, we additionally
neglect the effect of heat conduction via $\mathrm{Nu}_{\textrm{loc,}3}$ and
$\mathrm{Nu}_{\textrm{loc,}4}$, so only $\mathrm{Nu}_{\textrm{loc,}2}$ remains
on the high-resolution grid.

In the next step, we add the effect of the light sheet thickness by vertically
averaging $\tilde{u}_{z}$ and $\tilde{\Theta}$ over the light sheet thickness
$\approx 0.14H$ centered around the midplane. We designate the resulting local
Nusselt number as $\mathrm{Nu}_{\textrm{loc}}^{\textrm{lst}}$. As a final step
and in addition to the averaging over the light sheet thickness, we perform an
in-plane binning from the high resolution down to a spatial resolution that
matches experimental conditions – we denote this final quantity by
$\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$. Figure 6 summarises the PDFs for
these four introduced quantities.

We find that the PDFs collapse quite well as long as no horizontal binning is
performed. On the one hand, this underlines that heat conduction is not
important in the bulk region, while, on the other hand, it judges the
influence of the light sheet thickness as negligible. In contrast, in-plane
binning shows the largest influence on the statistics. In essence, this
averaging leads to an underestimation of the probability of large-magnitude
events – thus, this effect concerns the positive $\mathrm{Nu}_{\textrm{loc}}$
range more than its negative counterpart. As the effect is not overwhelmingly
strong, this underlines that the measurements are well suited to determine
$\mathrm{Nu}_{\textrm{loc}}\equiv\mathrm{Nu}_{\textrm{loc,}2}\approx\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$
experimentally.

Figure 7: Instantaneous $\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ field at
the horizontal midplane. The data corresponds to
$\mathrm{Ra}_{\textrm{D}}=1\times 10^{5}$.

Yet, this sparser in-plane resolution must be taken into account when
comparing experimental and numerical data. An exemplary snapshot of the
resulting $\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ is shown in Figure 7 for
numerically generated data. As can be seen, the superstructures can – together
with their increased heat transfer at the ridges, see also Fonda et al. [57] –
easily be recognized despite the spatial binning. Note that deviations caused
by this binning mostly affect the large-magnitude ridges but can practically
not be recognized in such a plot. Hence, the binning does not distort the
general picture of the superstructures and may also be applied in our
subsequent data analysis.

Figure 8: PDF of the $\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ for the
experiments and the Dirichlet DNS.

We contrast $\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ statistics from
experimental and numerical data in Figure 8, indicating two interesting
aspects. First, we observe a qualitatively reasonable trend with increasing
$\mathrm{Ra}$ up to $\mathrm{Ra}=7\times 10^{5}$ for the positive tails.
Afterward, the trend seems to be broken, and the data contradicting. However,
in our view, the data is still plausible since experimental measurements are
subject to measurement uncertainties – their PDFs spread unavoidably across
broader ranges, leading to relatively higher probabilities of far-tail events.
Thus, we conclude the trend to be quite consistent across the entire range of
investigated $\mathrm{Ra}$. This first aspect is in clear contrast to the
second one, the latter of which represents a clear mismatch for the negative
tails. More precisely, we observe a striking trend towards more frequent and
extreme negative $\mathrm{Nu}_{\textrm{loc}}$ events with increasing
$\mathrm{Ra}$. This overshooting cannot be explained by the weak effect of
measurement uncertainties from above, asking clearly for a more detailed
analysis in the following.

Figure 9: PDFs of the decomposed $\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$.
The black and red lines correspond to contribution to
$\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ caused by $\tilde{\Theta}<0$ and
$\tilde{\Theta}_{z}>0$, respectively.

Recalling eq. (14) (to which $\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ is
directly related), negative $\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ can
either be caused by up-welling cold fluid or by down-welling hot fluid – thus,
we disentangle $\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ subsequently based
on the sign of the temperature deviation field. We contrast the resulting
conditioned PDFs of
$\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}|_{\tilde{\Theta}<0}$ (black) and
$\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}|_{\tilde{\Theta}>0}$ (red) for the
entire range of $\mathrm{Ra}$ in Figure 9. First, we notice a perfect match of
both PDFs for the numerically obtained data due to the top-down symmetry as a
consequence of the underlying Oberbeck-Boussinesq approximation. In contrast,
we find an increasing probability of (negative and positive) far-tail events
in the case of hot fluid for experimental data and successively larger
$\mathrm{Ra}$. This observation confirms an incapability of the cooling plate
to generate comparable localized thermal variance – that leads to high-
magnitude $\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ events – due to the
smaller thermal conductivity of the glass plate. In other words, the cooling
plate cannot transfer the heat out of up-welling thermal plumes fast enough.
As a consequence, hot fluid that rose upwards is not sufficiently cooled in a
region close to the top plate and is subsequently pulled down again by down-
welling thermal plumes due to viscosity. Figure 10 visualizes this concept in
a sketch.

Figure 10: Sketch of a negative $\mathrm{Nu}_{\textrm{loc}}$ event. A hot
thermal plume rises towards the cooling plate. Due to the lower thermal
conductivity of the cooling plate, the heat of the hot thermal plumes is only
slowly transferred through the plate. Meanwhile, detaching cold thermal plumes
drag warm fluid with them, leading to negative $\mathrm{Nu}_{\textrm{loc}}$
events.

In order to validate this view, we turn our attention to the entire
$\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ fields. Here, we decompose them
into the four different possible combinations of $\tilde{u}_{z}$ and
$\tilde{\Theta}$, and visualize exemplary fields averaged over
33$t_{\mathrm{f}}$ for $\mathrm{Ra}=2\times 10^{5}$ and $\mathrm{Ra}=7\times
10^{5}$ in Figure 11 (a) and (b), respectively. Note the indication of these
different combinations via different colors, whereas their corresponding
intensity encodes the magnitude of
$\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$. As down-welling hot fluid (green)
occurs mostly between hot up-welling (red) and cold down-welling (blue)
regions, we find our concept of dragged-down hot fluid due to the limited
thermal conductivity of the glass plate supported. Since we visualize time-
averaged fields, we find these regions of reversed heat transfer to be quite
persistent rather than spontaneous.

Figure 11: Decomposition of $\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ for
experimental data at (a) $\mathrm{Ra}=2\times 10^{5}$ and (b)
$\mathrm{Ra}=7\times 10^{5}$ depending on the signs of $\tilde{u}_{z}$ and
$\tilde{\Theta}$. Both red and blue regions imply
$\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}>0$, whereas green and purple
regions correspond to $\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}<0$. The
intensity indicates the magnitude – note here the different scalings for
positive and negative $\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$. The fields
are averaged over 33 $t_{\mathrm{f}}$, which indicates a certain persistence
of the inverted heat transfer events.

Having a closer look at these two panels, we exclusively find ridges of hot
up-welling fluid which are mostly connected with each other together with
regions of cold down-welling fluid that are surrounded by the former. As
outlined by Fonda et al. [57], such an arrangement also impacts the heat
transfer. The authors categorized the convection pattern based on a deep-
learning approach into ridges, trisectors (connection points of ridges) and
wedges (endpoints of ridges). Interestingly, the individual structure classes
affect also heat transfer – for instance, they attributed twice as much heat
transfer to trisectors compared to wedges at
$\mathrm{Ra}_{\textrm{D}}=10^{5}$. Adopting their terminology to our present
configuration, we find several trisectors for hot up-welling fluid but none
for cold down-welling fluid – instead, we only observe wedges for the latter.
One might thus expect extended regions of the latter to account for the
increased or localized heat transfer of the former. As this is confirmed by
our observations, it simultaneously explains why reversed heat transfer due to
dragged-down regions appears closer to the hot up-welling regions.

Our conceptual sketch consistently explains the far negative tails only –
however, returning to Figure 9, one might wonder why the experimental positive
tails agree only up to $\mathrm{Ra}=4\times 10^{5}$ well but do not match
anymore for $\mathrm{Ra}=7\times 10^{5}$. Figure 9 (d) shows that these
positive far-tail events are dominantly caused by hot (up-welling) fluid. In
turn, this implies that cold (down-welling) fluid cannot account for a similar
inverse transfer of heat. In fact, this underlines once more the limits that
the cooling plate sets on the heat transfer of the entire system and the
resulting broader extent of down-welling fluid regions, as elaborated above.

Run |  | $\langle\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}\rangle_{\tilde{A},\tilde{t}}$
---|---|---
DNS | $\mathrm{Ra}_{\textrm{D}}=1\times 10^{5}$ | $4.04$
Exp. | $\mathrm{Ra}\thickspace\thickspace=2\times 10^{5}$ | $4.00$
Exp. | $\mathrm{Ra}\thickspace\thickspace=4\times 10^{5}$ | $5.31$
Exp. | $\mathrm{Ra}\thickspace\thickspace=7\times 10^{5}$ | $5.58$
DNS | $\mathrm{Ra}_{\textrm{D}}=1\times 10^{6}$ | $7.55$
Table 4: Global Nusselt numbers as inferred from their local measurements. The
time average is taken across the entire corresponding evolution times provided
in Figure 5. In contrast to the simulations, the latter of which analyze the
entire horizontal cross-section, the average across the horizontal plane
$\tilde{A}$ is restricted in the experiment to the limited field of view.

Lastly, we turn our focus from the local to the global heat transfer by
computing
$\langle\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}\rangle_{\tilde{A},\tilde{t}}$
as summarised in Table 4 for both the simulations and experiments across the
entire $\mathrm{Ra}$-range. Note here that an average across the horizontal
plane $\tilde{A}$ is restricted in the experiment to the limited field of
view. Nevertheless, the experiment shows a rather good agreement at the lowest
Rayleigh number. However, it falls increasingly behind the trend – that is set
by the numerically obtained data for Dirichlet boundary conditions – as higher
$\mathrm{Ra}$ result in successively stronger non-Dirichlet effects at the top
plate as explained in Section IV. Hence, these effects explain the mismatch of
the global Nusselt number when comparing the experiments with numerically
obtained results.

##  VI Discussion and perspectives

In this work, we first investigated the effect of mixed and asymmetric
boundary conditions on the size of long-living large-scale flow structures in
thermal convection. A comparison of the experimentally observed patterns with
numerically obtained patterns for Dirichlet conditions showed an increasing
deviation of the former’s size from the latter for increasing $\mathrm{Ra}$.
We related this trend to a gradual change in the thermal boundary condition at
the cooling plate. Despite this difference in the structure size compared to
perfect Dirichlet conditions, we confirmed that the patterns in the case of
complementary Neumann conditions offered fundamentally different long-term
dynamics. In this case, the large-scale flow structures grow gradually until
domain size, being in contrast to the fluctuation around a time-independent
mean value as observed in the experiments. This confirmed the experimentally
obtained structures to be categorized as turbulent superstructures. We
underlined our observations based on considerations concerning the linear
stability at the onset of convection.

As the thermal boundary conditions vary between the top and bottom plate – and
change in the case of the former even with $\mathrm{Ra}$ –, the primary
instability is similarly affected. However, the final flow establishes a
balance between both conditions. To investigate the effect of potential slight
variations of the patterns over the layer height, further experiments are
necessary. These would need to incorporate a simultaneous and fully volumetric
temperature and velocity measurement across the entire height since the
temperature gradients must not be neglected within the thermal boundary
layers. Motivated by the infinite growth of critical patterns at the onset of
convection in the Neumann case, it would further be of interest at which
critical ratio of thermal diffusivities such behavior sets in.

The second part of our work addresses the influence of the experimentally
asymmetric boundary conditions on local and global heat transfer. From a
systematic derivation based on high-resolution numerical data, we found that
the coarse horizontal resolution of the experimental measurement affects the
statistics of the local Nusselt number most strongly. Contrasting the
probability density functions of the local Nusselt number
$\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ from experimental and numerical
data signalized higher probabilities of inverted heat transfer for the
experiment. A subsequent decomposition of
$\mathrm{Nu}_{\textrm{loc}}^{\textrm{exp}}$ depending on the sign of the
temperature deviation $\Theta$ unveiled an asymmetry due to the limited
thermal conductivity of the cooling plate. In other words, the cooling plate
falls short of generating large magnitude $\mathrm{Nu}_{\textrm{loc}}$ events
or pronounced cold plumes. We confirm this interpretation by detailed
horizontal cross-sections of the local Nusselt number, which exhibits extended
regions of hot down-welling fluid located between regions of hot up-welling
and cold down-welling fluid.

Our study highlights the importance of systematic investigations of convection
with mixed or asymmetric boundary conditions beyond the linear stability.
Hence, engineers and scientists are asked to develop advanced measurement
techniques for combined volumetric temperature and velocity measurements in
laboratory experiments – this would eventually allow for long-term
investigations of the slow dynamics of turbulent superstructures as well as
other long-living large-scale flow structures and the influence of asymmetric
thermal boundary conditions that go beyond the present possibilities of the
present studies.

## Acknowledgement

The authors gratefully acknowledge Sebastian Moller and Ambrish Pandey for
providing the experimental measurement and numerical data, respectively. This
work was supported by the DFG Priority Programme SPP 1881 on “Turbulent
Superstructures”. The work of T.K. was partly funded by the Carl Zeiss
Foundation within project no. P2018-02-001 “Deep Turb – Deep Learning in and
of Turbulence”.

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