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Towards DNS of the Ultimate Regime of Rayleigh--B\'enard Convection

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Towards DNS of the Ultimate Regime of
enard Convection
Richard J.A.M. Stevens, Detlef Lohse, and Roberto Verzicco
1 Introduction
Heat transfer mediated by a fluid is omnipresent in Nature as well as in technical ap-
plications and it is always among the fundamental mechanisms of the phenomena.
The performance of modern computer processors has reached a plateau owing to the
inadequacy of the fluid based cooling systems to get rid of the heat flux which in-
creases with the operating frequency [1]. On much larger spatial scales, circulations
in the atmosphere and oceans are driven by temperature differences whose strength
is key for the evolution of the weather and the stability of regional and global climate
The core of the problem, which is referred to as natural convection, is relatively
simple since it reduces to determining the strength of the heat flux crossing the
system for given flow conditions. Unfortunately, the governing equations (Navier–
Stokes) are complex and non–linear, thus preventing the possibility to obtain analyti-
cal solutions. On the other hand laboratory experiments, aimed at tackling these phe-
nomena, have to cope with the issue of how to make a setup of size hm=O(cm–m)
dynamically similar to a system of h=O(m–Km) (figure 1). Indeed, this upscaling
problem is common to many experiments in hydrodynamics; in thermal convection
however it is exacerbated since, as we will see in the next section, the most relevant
governing parameter, the Rayleigh number (Ra), depends on the third power of the
leading spatial scale. This implies that in real applications Ra easily attains huge
values while it hardly hits its low–end in laboratory experiments.
Numerical simulations are subjected to similar limitations because of the spa-
tial and temporal resolution requirements that become more severe as the Rayleigh
R.J.A.M. Stevens ·D. Lohse ·R. Verzicco
PoF, UTwente, e-mail: {r.j.a.m.stevens,d.lohse, r.verzicco}
R. Verzicco
DII, Uniroma2 e-mail:
arXiv:2005.06874v1 [physics.flu-dyn] 14 May 2020
2 Richard J.A.M. Stevens, Detlef Lohse, and Roberto Verzicco
Fig. 1 Cartoon of the scaling
problem for a Rayleigh–
enard flow: a model system
of size hmhas to be operated
in dynamic similarity with a
real system of size h.
number increases [3]. Only recently the former have become a viable alternative to
experiments thanks to the continuously growing power of supercomputers.
Indeed, if the dynamics of the system could be expressed by power laws of the
form ARaβ, experiments and numerical simulations, performed at moderate val-
ues of the driving parameters, could be scaled up to determine the response of the
real systems at extreme driving values. Unfortunately, this strategy works only as-
suming that the coefficients of the power law (Aand β) remain constant for every Ra
and this could not be the case for thermal convection [4]. More in detail, Malkus [5]
and Priestley [6], conjectured that all the mean temperature profile variations occur
within the thermal boundary layers at the heated plates while the mean temperature
in the bulk of the flow is essentially constant. Assuming also that the thermal bound-
ary layers are far enough to evolve independently, one immediately obtains β=1/3.
A few years later, however, Kraichnan [7] noticed that as Ra increases, also the flow
strengthens and the viscous boundary layers eventually must become turbulent. In
this case, referred to as ultimate regime, velocity profiles are logarithmic with the
wall normal distance, and it results β=1/2 (times logarithmic corrections) which
yields huge differences with respect to the previous theory.
This last observation and the fact that most of the practical applications evolve in
the range of very high Rayleigh numbers motivate the effort to study turbulent ther-
mal convection in the ultimate regime even if it requires the solution of formidable
difficulties that we will detail in the next section.
2 A trap problem
One of the most appealing features of thermal convection is that the essence of the
phenomenon can be reduced to a very simple model problem in which a fluid layer
of thickness h, kinematic viscosity νand thermal diffusivity κis heated from below
Towards DNS of the Ultimate Regime of Rayleigh–B´
enard Convection 3
ν κ
Fig. 2 Cartoon of the ideal Rayleigh–
enard flow.
Fig. 3 Simplified sketch of a real ex-
perimental set–up. Outside of the cell
only an insulating layer of foam has
been reported while all the details
about heating and cooling systems and
thermal shields have been neglected.
and cooled from above with a temperature difference ThTc=. The temperature
field, in a constant gravity field gproduces a flow motion via the thermal expansion,
parameterized by a constant coefficient α: this is the Rayleigh–B´
enard flow. The
heat flux ˙
Qbetween the plates will be a function of the form ˙
which involves N=7 different quantities whose minimum number of independent
dimensions is K=4. The Buckingham Π–theorem assures that, in non–dimensional
form, the above relation is equivalent to one with only NK=3 parameters that
can be written as Nu =F(Ra,Pr)being Ra =gαh3/(νκ),Pr =ν/κand Nu =
Qdi f f with Qdi f f the diffusive heat flux through the fluid in absence of motion.
The above relation looks very attractive since it depends only on two indepen-
dent parameters Ra and Pr, the latter being determined solely by the fluid properties.
Many efforts have been made to study the function F(·)through laboratory experi-
ments and, in the last two decades, also by numerical simulations as a viable alter-
native. Unfortunately, the practical realization of the RB convection is substantially
different from the theoretical problem and many additional details come into play.
The first point is that while in the ideal flow the fluid layer is laterally unbounded,
in real experiments, by necessity, it must be somehow confined thus introducing a
second length dor equivalently an additional parameter Γ=d/h, the aspect–ratio.
The parameters become more than one if the tank does not have a cylindrical or
square cross section.
Several additional variables are introduced by the physical realization of the ther-
mal boundary conditions; in fact, in the ideal problem, all the heat entering the
fluid through the lower isothermal hot plate leaves the fluid only crossing the up-
per cold plate, without heat leakage across the side boundaries. In the real flow,
the isothermal surfaces are obtained by thick metal plates (a=O(1–5)cm) of
high thermal diffusivity κpl (copper or aluminum) that provide stable temperature
values at the fluid interface for every flow condition. The sidewall, in contrast,
4 Richard J.A.M. Stevens, Detlef Lohse, and Roberto Verzicco
should minimize the heat transfer and it is therefore made of low thermal diffu-
sivity κsw materials (steel or Plexiglas) and with reduced thickness (e=O(1–5)
mm). To further prevent parasite heat currents, insulation foam layers and even
active thermal shields are installed outside of the cell, thus further increasing the
number of input parameters. Within this scenario, the heat flux function looks
like ˙
Q=f0(h,ν,κ,g,α,,d,e,κsw,a,κpl ,b,κf oam ,...)whose variable counting is
N>14 while it results always K=4 thus implying that the non–dimensional coun-
terpart (Nu =F0(Ra,Pr,...)) involves more than NK>10 independent parame-
Fig. 4 Compilation of exper-
imental and numerical data
for the compensated Nusselt
versus Rayleigh numbers:
the solid black line is the
theory by [4], blue bullets
are the experiments by [8],
purple squares [9], dark red
diamonds [10], red right trian-
gles [11], orange left triangles
[12], yellow up triangles [13],
yellow down triangles [14],
yellow stars [15], black bul-
lets, numerical simulations by
[16], black squares, prelim-
inary numerical simulations
by Stevens (2019), (Personal
The hope is that when the function F0(·)is explored, experimentally or numer-
ically, most of the parameters introduced by the experimental technicalities do not
affect significantly the phenomena and the relevant variables reduce to a tractable
number. Unfortunately, some of the recent, and not so recent, experiments have
shown that it is not always the case since dynamically equivalent flows do not yield
identical results. In figure 4 we report a collection of Nusselt numbers taken from
different sources showing some disagreement both at the low– and high–end of
Ra. While the former differences have been attributed mainly to the heat leakage
through the sidewall [17, 18, 19, 20] and some of the latter to the non–perfect ther-
mal sources [21], most of the discrepancies are still unexplained and they are the
subject of intense investigation by many research groups worldwide. The situation
is even more complex if one does not restrict the analysis only to the Nusselt number
since higher order statistics show higher sensitivity to external perturbations.
In this context, numerical experiments can be particularly helpful since they can
be used as ideal tests to isolate the different perturbations of the basic problem and
assess their effect. However, performing direct numerical simulation of Rayleigh-
enard convection implies several non obvious choices and requires huge compu-
tational resources that, for the parameter range of the ultimate regime, are not fully
Towards DNS of the Ultimate Regime of Rayleigh–B´
enard Convection 5
available yet. In the following we will present estimates of the computational costs
and discuss some of the possible open choices.
The first relevant point is whether the simulation should be aimed at the ideal
RB flow or rather has to mimic a laboratory experiment. In the first case the com-
putational domain should be laterally unbounded which can be approximated by
periodic boundary conditions applied to a rectangular domain of horizontal size
d; this configuration results in a horizontally homogeneous flow that benefits from
easy and efficient uniform spatial discretizations and fast converging statistics. On
the other hand, in a real set–up, the boundedness of the fluid layer generally results
in a smaller fluid volume although the kinematic boundary layer at the sidewall has
to be resolved by additional gridpoints.
Fig. 5 Nusselt number versus
domain aspect–ratio Γ=d/h
at Ra =108and Pr =1. The
dashed line for the ‘unifying
theory’ [4] is computed from
experimental data at Γ=1.
Which of the two configurations is more advantageous is not obvious since the
computational efficiency of the fully bounded flow solvers is smaller than those with
homogeneous directions but the former setup involves smaller flow volumes.
Besides computational considerations there are also physical issues since the
presence of lateral boundaries, and even the shape of the container affect the flow
dynamics [22]. In figure 5 we report the dependence of the Nusselt number on the
domain aspect–ratio Γfor rectangular and cylindrical geometries. It is immediately
evident that, for slender domains Nu depends on Γin the opposite way and only for
Γ1 the two geometries yield similar values while for Γ4 the Nusselt number
converges to the asymptotic value for unbounded domains. All the laboratory ex-
periments reported in figure 4 have been performed in low aspect ratio cylindrical
cells and accordingly also the numerical simulations have been run in a cylinder at
Γ=0.5. In contrast, if a horizontally homogeneous flow has to be simulated, in or-
der to get rid of the numerical confinement effect, it must be computed on domains
at least of Γ=4, even if spectra and higher order statistics indicate that Γ=8 or
Γ=16 is needed to eliminate confinement effects [23].
6 Richard J.A.M. Stevens, Detlef Lohse, and Roberto Verzicco
In the following we give an estimate of the computational resources needed for
direct numerical simulation of turbulent RB convection in rectangular and cylindri-
cal geometries by evaluating the number of nodes contained in the relative mesh.
The basic assumption is that the flow can be divided into bulk and boundary layer
regions, the former discretized by a mesh of the same size as the smallest between
the Kolmogorov and Batchelor scales and the latter with the resolution criteria sug-
gested by [3]. We further assume that the rectangular box has a size d×d×hdis-
cretized in Cartesian coordinates while the cylinder has a diameter dand a height h
discretized in polar coordinates.
For the ease of discussion we will restrict to Pr =1 keeping in mind that as the
Prandtl number deviates substantially from unity the simulation becomes even more
demanding either because the velocity field develops finer scales than the tempera-
ture (Pr 1) or vice versa (Pr 1).
For the mean Kolmogorov scale ηwe can easily write η/h(RaNu)1/4that
with a fit Nu =ARaβ(A'0.05 and β=1/3 from the high end of Ra in figure
4) yields a number of nodes per unit length in the bulk Nbu =0.473Ra1/3. For the
resolution of each boundary layer we rely on the correlation derived by [3] which
suggest a number of nodes Nbl 0.35Ra0.15. Within these figures, the total number
of nodes for the rectangular domain reads NCar =Γ2(0.105Ra +0.156Ra0.816).
We proceed along the same lines for the cylindrical domain keeping in mind
that there is an extra boundary layer at the sidewall and that the polar coordinates
have azimuthal isolines that diverge radially. Therefore the resolution requirements
in this direction are dictated by the location farthest from the symmetry axis. Using
the same correlations as above we obtain NCyl =0.5πΓ 2(0.105Ra +0.156Ra0.816 )+
πΓ (0.223Ra0.816 +0.116Ra0.633 ).
It is worth mentioning that these expressions have been obtained by simplifying
assumptions therefore their results should be taken as coarse estimates and not as
precise measures. For example, Aand βhave been assumed constant and equal to
the high–end Ra values of figure 4 and we have used h2δbl h(with δbl the
boundary layer thickness): all these positions concur to an overestimate of the num-
ber of nodes. On the other hand, the correlation Nbl 0.35Ra0.15 of [3] was obtained
for a Prandtl–Blasius laminar boundary layer that is expected to underestimates the
resolution when the ultimate regime sets in and the boundary layers transition to tur-
bulence. At the transitional Rayleigh number, the above factors might compensate
each other and the estimates could give reasonable numbers.
A comparison of the two expressions immediately shows that the leading or-
der term increases at the same rate with Ra and Γalthough the cylindrical mesh
has asymptotically 60% more nodes than the Cartesian counterpart. This is true al-
though, for the same aspect–ratio, the latter has a volume (Γ2h3) which is more than
20% bigger than the former (πΓ 2h3/4).
If now we focus on the onset of the ultimate regime we have to determine the
critical Rayleigh number at which the boundary layer undertakes the transition to
the turbulent state. This is triggered by the large scales of convection that sweep
the plates by the induced winds; according to Reference [24] the boundary layer
Towards DNS of the Ultimate Regime of Rayleigh–B´
enard Convection 7
Fig. 6 Number of nodes
N(in a cylindrical cell of
Γ=0.5) and achievable flow
Rayleigh number Ra versus
the years for direct numerical
simulations of Rayleigh–
enard convection. Black
squares for various data from
the literature, blue bullets for
simulations from our research
group, big red bullet final
goal for the ultimate regime
transition occurs for a shear Reynolds number of ReS420 that Grossmann &
Lohse [4] have estimated to happen around RaC1014.
In a rectangular domain with Γ=4 this Rayleigh number implies a mesh with
NCar >1014 nodes that is clearly infeasible in the mid–term future. In a cylindrical
cell of Γ=0.5, however, it results NCyl 1012 nodes that could be achieved within
the next five years (see figure 6). Indeed, we are already running simulations at
Ra =1013 at Γ=0.5 and even Ra =1014 at Γ=0.25 with meshes of the order of
1011 nodes (R. Stevens, Personal Communication) although we expect to tackle the
ultimate regime only by the ‘next generation’ simulations.
We wish to point out that if we compare the numbers coming from the present
formulas with those currently used for the highest Rayleigh number simulations we
find that the former produce a systematic overestimate of the required resolution. For
example, for a cylindrical mesh of aspect–ratio Γ=0.5 at Ra =1013 our prediction
yields a number of nodes NCyl 4×1011 while a simulation on a mesh 4608 ×
1400 ×4480 (NCyl 2.9×1010) yielded the same Nusselt number as another run
on the finer grid 6144 ×1536 ×6144 with NCyl 5.8×1010 nodes (R. Stevens,
Personal Communication).
A possible explanation for this difference is that in our model we have assumed
that in the bulk the mesh has to be as fine as the mean Kolmogorov scale η. How-
ever, looking at the dissipation spectra of turbulence [25] one finds its peak around
10ηthus implying that also a mesh of size 1.5–2ηalready resolves most of the dis-
sipation. In three–dimensions this difference yields a factor 6.25–8 less in the node
counting that is about the mismatch between our prediction and the actual meshes.
For a while we have been working at improving the simulation code [26, 27, 16]
by more efficient implementations of the solution algorithms and of the paralleliza-
8 Richard J.A.M. Stevens, Detlef Lohse, and Roberto Verzicco
tion strategies in order to reduce the time–to–solution. In addition we are also figur-
ing out alternatives to achieve the ultimate regime in more affordable problems.
One possible way is to exploit the analogy between Rayleigh–B´
enard and Taylor–
Couette (TC) flow [28]. The latter is the flow developing in the gap between two
coaxial cylinders rotating at different angular velocities and whose angular momen-
tum flux across the cylinders behaves as the heat flux between the plates in a RB flow
[29]. It turns out, however, that the mechanical forcing of the TC flow is more effi-
cient in producing turbulent boundary layers than the thermal forcing of RB flows
and the ultimate regime can be achieved for smaller values of the driving parame-
ters that are affordable by numerical simulation [30]. In Reference [31], thanks to
the presence of baffled cylinders, that disrupted the logarithmic part of the turbulent
boundary layer profiles, it has been possible to get rid of the logarithmic correction
and obtain a pure 1/2 power law in the analogous of the Nu versus Ra relationship.
Another possibility is to simulate a two–dimensional RB flow that allows, already
now, to tackle Rayleigh numbers >1014 ; indeed in Reference [32] (and successive
developments) simulations have been run up to Ra '5×1014 with the appearance
of a transition already for Ra 1013.
3 Closing remarks
In this contribution we have briefly introduced the problem of turbulent thermal
convection with a particular look at its transition to the ultimate regime and the
resolution requirements needed for the direct numerical simulation of this flow.
Leaving aside all the complications related to the spurious heat currents through
the sidewall and the imperfect character of the thermal sources, already addressed
in some of the referred papers, it appears that a preliminary fundamental question is
whether the simulation should be aimed at replicating an experimental set–up with
a lateral confinement or to mimic the truly Rayleigh–B´
enard flow that is virtually
infinite in the horizontal directions.
We have shown that in the latter case a domain with aspect–ratio no smaller
than Γ=4 is required and this implies, at the estimated critical Rayleigh number
RaC1014, a computational mesh with more than 1014 nodes that is not likely to
be tractable within the next decade. On the other hand, although for a given Ra and
Γcylindrical, laterally confined geometries contain about 60% more nodes than the
rectangular ‘unbounded’ domains, when restricted to the existing, slender cylinders
of the laboratory experiments the number of nodes becomes more feasible. In partic-
ular, for Γ=0.5 and Ra =1014 the present estimate gives a mesh slightly larger that
a trillion of nodes. Even if this number might look impressive, it is ‘only’ one order
of magnitude bigger than the current state–of–the–art simulations and, according to
figure 6, such meshes will become affordable within the next five years or so. It is
also worthwhile mentioning that the present estimates assume a mesh in the bulk of
the flow that is everywhere as fine as the mean Kolmogorov scale ηwhile actual
grid refinement checks performed on Rayleigh–B´
enard turbulence have shown con-
Towards DNS of the Ultimate Regime of Rayleigh–B´
enard Convection 9
verged results already for meshes of size 2η. This implies that in three–dimensional
flows the actual mesh sizes can be about one order of magnitude smaller and this is
fully confirmed by our ongoing simulations.
Needless to say, once the ultimate regime will have been hit by numerical simula-
tions also in three–dimensions, a terra incognita will be entered. Turbulent boundary
layers have more severe resolution requirements than the laminar counterparts and
once the ballistic plumes of Kraichnan [7], which can be thought of as pieces of de-
tached thermal boundary layer, are shot into the bulk also the resolution of that flow
region is likely to become more demanding. Clearly, attempting resolution estimates
beyond the onset of the transition would be even more speculative than those of the
present paper and only with the data of those simulations at hand, further reasonable
projections can be made.
While waiting for adequate computational resources to tackle thermal convec-
tion in the ultimate regime we can nevertheless compute turbulent flows that exhibit
similar dynamics or that can be reduced to a tractable size by simplifying assump-
tions. These include the Taylor–Couette flow, that can be rigorously shown to be
analogous to Rayleigh-B´
enard convection, or two–dimensional thermal convection
that already now can be simulated well beyond Ra =1014 and has indeed shown
evidence of transition to the ultimate state.
Acknowledgements This work is supported by the Twente Max-Planck Cen-
ter and the ERC (European Research Council) Starting Grant no. 804283 Ulti-
mateRB. The authors gratefully acknowledge the Gauss Centre for Supercomputing
e.V. ( for funding this project by providing computing
time on the GCS Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre
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Full-text available
We report the observation of superstructures, i.e., very large-scale and long living coherent structures in highly turbulent Rayleigh-Bénard convection up to Rayleigh Ra=109. We perform direct numerical simulations in horizontally periodic domains with aspect ratios up to Γ=128. In the considered Ra number regime the thermal superstructures have a horizontal extend of six to seven times the height of the domain and their size is independent of Ra. Many laboratory experiments and numerical simulations have focused on small aspect ratio cells in order to achieve the highest possible Ra. However, here we show that for very high Ra integral quantities such as the Nusselt number and volume averaged Reynolds number only converge to the large aspect ratio limit around Γ≈4, while horizontally averaged statistics such as standard deviation and kurtosis converge around Γ≈8, the integral scale converges around Γ≈32, and the peak position of the temperature variance and turbulent kinetic energy spectra only converge around Γ≈64.
Full-text available
The possible transition to the so-called ultimate regime, wherein both the bulk and the boundary layers are turbulent, has been an outstanding issue in thermal convection, since the seminal work by Kraichnan [Phys. Fluids 5, 1374 (1962)]. Yet, when this transition takes place and how the local flow induces it is not fully understood. Here, by performing two-dimensional simulations of Rayleigh-Bénard turbulence covering six decades in Rayleigh number Ra up to 1014 for Prandtl number Pr=1, for the first time in numerical simulations we find the transition to the ultimate regime, namely, at Ra*=1013. We reveal how the emission of thermal plumes enhances the global heat transport, leading to a steeper increase of the Nusselt number than the classical Malkus scaling Nu∼Ra1/3 [Proc. R. Soc. A 225, 196 (1954)]. Beyond the transition, the mean velocity profiles are logarithmic throughout, indicating turbulent boundary layers. In contrast, the temperature profiles are only locally logarithmic, namely, within the regions where plumes are emitted, and where the local Nusselt number has an effective scaling Nu∼Ra0.38, corresponding to the effective scaling in the ultimate regime.
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Turbulence governs the transport of heat, mass and momentum on multiple scales. In real-world applications, wall-bounded turbulence typically involves surfaces that are rough; however, characterizing and understanding the effects of wall roughness on turbulence remains a challenge. Here, by combining extensive experiments and numerical simulations, we examine the paradigmatic Taylor–Couette system, which describes the closed flow between two independently rotating coaxial cylinders. We show how wall roughness greatly enhances the overall transport properties and the corresponding scaling exponents associated with wall-bounded turbulence. We reveal that if only one of the walls is rough, the bulk velocity is slaved to the rough side, due to the much stronger coupling to that wall by the detaching flow structures. If both walls are rough, the viscosity dependence is eliminated, giving rise to asymptotic ultimate turbulence—the upper limit of transport—the existence of which was predicted more than 50 years ago. In this limit, the scaling laws can be extrapolated to arbitrarily large Reynolds numbers.
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Direct numerical simulations of Taylor-Couette flow (TC). Shear Reynolds numbers of up to $3\cdot10^5$, corresponding to Taylor numbers of $Ta=4.6\cdot10^{10}$, were reached. Effective scaling laws for the torque are presented. The transition to the ultimate regime, in which asymptotic scaling laws (with logarithmic corrections) for the torque are expected to hold up to arbitrarily high driving, is analysed for different radius ratios $\eta$, different aspect ratios $\Gamma$ and different rotation ratios $Ro$. It is shown that the transition is approximately independent of $Ro$ and $\Gamma$, but depends significantly on $\eta$. We furthermore calculate the local angular velocity profiles and visualize different flow regimes that depend both on the shearing of the flow, and the Coriolis force originating from the outer cylinder rotation. Two main regimes are distinguished, based on the magnitude of the Coriolis force, namely the co-rotating and weakly counter-rotating regime dominated by Rayleigh-unstable regions, and the strongly counter-rotating regime where a mixture of stable and unstable regions exist. Furthermore, an analogy between $\eta$ and outer-cylinder rotation is revealed, namely that smaller gaps behave like a wider gap with co-rotating cylinders, and that wider gaps behave like smaller gaps with weakly counter-rotating cylinders. Finally, the effect of $\Gamma$ on the effective torque versus $Ta$ scaling is analysed and it is shown that different branches of the torque-versus-$Ta$ relationships associated to different aspect ratios are found to cross within $15%$ of the $Re$ associated to the transition to the ultimate regime. The paper culminates in phase diagram in the inner vs outer $Re$ number parameter space and in the $Ta$ vs $Ro$ parameter space, which can be seen as the extension of the Andereck \emph{et al.} phase diagram towards the ultimate regime.
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We investigate the influence of the temperature boundary conditions at the sidewall on the heat transport in Rayleigh–Bénard (RB) convection using direct numerical simulations. For relatively low Rayleigh numbers Ra the heat transport is higher when the sidewall is isothermal, kept at a temperature Tc+Δ/2 (where Δ is the temperature difference between the horizontal plates and Tc the temperature of the cold plate), than when the sidewall is adiabatic. The reason is that in the former case part of the heat current avoids the thermal resistance of the fluid layer by escaping through the sidewall that acts as a short-circuit. For higher Ra the bulk becomes more isothermal and this reduces the heat current through the sidewall. Therefore the heat flux in a cell with an isothermal sidewall converges to the value obtained with an adiabatic sidewall for high enough Ra (≃1010). However, when the sidewall temperature deviates from Tc+Δ/2 the heat transport at the bottom and top plates is different from the value obtained using an adiabatic sidewall. In this case the difference does not decrease with increasing Ra thus indicating that the ambient temperature of the experimental apparatus can influence the heat transfer. A similar behaviour is observed when only a very small sidewall region close to the horizontal plates is kept isothermal, while the rest of the sidewall is adiabatic. The reason is that in the region closest to the horizontal plates the temperature difference between the fluid and the sidewall is highest. This suggests that one should be careful with the placement of thermal shields outside the fluid sample to minimize spurious heat currents. When the physical sidewall properties (thickness, thermal conductivity and heat capacity) are considered the problem becomes one of conjugate heat transfer and different behaviours are possible depending on the sidewall properties and the temperature boundary condition on the ‘dry’ side. The problem becomes even more complicated when the sidewall is shielded with additional insulation or temperature-controlled surfaces; some particular examples are illustrated and discussed. It has been observed that the sidewall temperature dynamics not only affects the heat transfer but can also trigger a different mean flow state or change the temperature fluctuations in the flow and this could explain some of the observed differences between similar but not fully identical experiments.
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We report on the experimental results for heat-transport measurements, in the form of the Nusselt number Nu, by turbulent Rayleigh–Bénard convection (RBC) in a cylindrical sample of aspect ratio Γ ≡ D/L = 0.50 (D = 1.12 m is the diameter and L = 2.24 m the height). The measurements were made using sulfur hexafluoride at pressures up to 19 bar as the fluid. They are for the Rayleigh-number range 3 × 1012 Ra 1015 and for Prandtl numbers Pr between 0.79 and 0.86. For Ra < Ra*1 1.4 × 1013 we find Nu = N0 Raγeff with γeff = 0.312 ± 0.002, which is consistent with classical turbulent RBC in a system with laminar boundary layers below the top and above the bottom plate. For Ra*1 < Ra < Ra*2 (with Ra*2 5 × 1014) γeff gradually increases up to 0.37 ± 0.01. We argue that above Ra*2 the system is in the ultimate state of convection where the boundary layers, both thermal and kinetic, are also turbulent. Several previous measurements for Γ = 0.50 are re-examined and compared with our results. Some of them show a transition to a state with γeff in the range from 0.37 to 0.40, albeit at values of Ra in the range from 9 × 1010 to 7 × 1011 which is much lower than the present Ra*1 or Ra*2. The nature of the transition found by them is relatively sharp and does not reveal the wide transition range observed in this work. In addition to the results for the genuine Rayleigh–Bénard system, we present measurements for a sample which was not completely sealed; the small openings permitted external currents, imposed by density differences and gravity, to pass through the sample. That system should no longer be regarded as genuine RBC because the externally imposed currents modified the heat transport in a major way. It showed a sudden decrease of γeff from 0.308 for Ra < Rat 4 × 1013 to 0.25 for larger Ra. A number of possible experimental effects are examined in a sequence of appendices; none of these effects is found to have a significant influence on the measurements.
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A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh-Benard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large-scale convection roll ('wind of turbulence') and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number Ra versus Prandtl number Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the kinetic boundary layer is thicker. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra ≤ 1011) the leading terms are Nu ~ Ra(1/4) Pr(1/8), Re ~ Ra(1/2) Pr(-3/4) for Pr ≤ 1 and Nu ~ Ra(1/4) Pr(-1/12), Re ~ Ra(1/2) Pr(-5/6) for Pr ≥ 1. In most measurements these laws are modified by additive corrections from the neighbouring regimes so that the impression of a slightly larger (effective) Nu vs. Ra scaling exponent can arise. The most important of the neighbouring regimes towards large Ra are a regime with scaling Nu ~ Ra(1/2) Pr(1/2), Re ~ Ra(1/2) Pr(-1/2) for medium Pr ('Kraichnan regime'), a regime with scaling Nu ~ Ra(1/5) Pr(1/5), Re ~ Ra(2/5) Pr(-3/5) for small Pr, a regime with Nu ~ Ra(1/3), Re ~ Ra(4/9) Pr(-2/3) for larger Pr, and a regime with scaling Nu ~ Ra(3/7) Pr(-1/7), Re ~ Ra(4/7) Pr(-6/7) for even larger Pr. In particular, a linear combination of the 1/4 and the (1/3) power laws for Nu with Ra, Nu = 0.27Ra(1/4) + 0.038Ra(1/3) (the prefactors follow from experiment), mimics a (2/7) power-law exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through the non-normal-nonlinear transition to turbulence and another regime emerges. The theory presented is best summarized in the phase diagram figure 2 and in table 2.
As climate change unfolds, weather systems in the United States have been shifting in patterns that vary across regions and seasons. Climate science research typically assesses these changes by examining individual weather indicators, such as temperature or precipitation, in isolation, and averaging their values across the spatial surface. As a result, little is known about population exposure to changes in weather and how people experience and evaluate these changes considered together. Here we show that in the United States from 1974 to 2013, the weather conditions experienced by the vast majority of the population improved. Using previous research on how weather affects local population growth to develop an index of people's weather preferences, we find that 80% of Americans live in counties that are experiencing more pleasant weather than they did four decades ago. Virtually all Americans are now experiencing the much milder winters that they typically prefer, and these mild winters have not been offset by markedly more uncomfortable summers or other negative changes. Climate change models predict that this trend is temporary, however, because US summers will eventually warm more than winters. Under a scenario in which greenhouse gas emissions proceed at an unabated rate (Representative Concentration Pathway 8.5), we estimate that 88% of the US public will experience weather at the end of the century that is less preferable than weather in the recent past. Our results have implications for the public's understanding of the climate change problem, which is shaped in part by experiences with local weather. Whereas weather patterns in recent decades have served as a poor source of motivation for Americans to demand a policy response to climate change, public concern may rise once people's everyday experiences of climate change effects start to become less pleasant.
The mixing-length theory of turbulent thermal convection in a gravitationally unstable fluid is extended to yield the dependence of Nusselt number H∕H0 on both Prandtl number σ and Rayleigh number Ra. The analysis assumes a layer of Boussinesq fluid contained between infinite, horizontal, perfectly conducting, rigid plates. Also obtained is the dependence of mean temperature deviation T¯(z), rms temperature fluctuation &psgr;˜(z), and rms velocity upon height z above the bottom plate. The theory gives H∕H0 ∝ Ra1∕3 (high σ), H∕H0 ∝ (σ Ra)1∕3 (low σ), and H∕H0 ∼ 1 (very low σ). The boundaries of the several σ ranges are determined. At one intermediate Prandtl number only, the behavior of T¯(z) and &psgr;˜(z) reduces to that previously found by Priestly. At high σ, there is a range of z, outside the molecular conduction region, where T¯(z) ∝ z−1, &psgr;˜(z) ∝ z−1. The results at very low σ reduce to those of Ledoux, Schwarzschild, and Spiegel. The dynamics are found to be importantly modified at extremely large Ra because of the stirring action of small-scale turbulence generated in shear boundary layers attached to the eddies of largest scale. The consequent corrected asymptotic law of heat transport at fixed σ is H∕H0 ∝ [Ra∕(In Ra)3]1∕2.