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Towards DNS of the Ultimate Regime of

Rayleigh–B´

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

[2].

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}@utwente.nl

R. Verzicco

DII, Uniroma2 e-mail: verzicco@uniroma2.it

1

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–

B´

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

.

h

T

Th

cQ

Q

ν κ

c

∼Q

∼

∼

Q

h

νκ

a

cb

.

d

Th

T

Fig. 2 Cartoon of the ideal Rayleigh–

B´

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 Th−Tc=∆. 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 ˙

Q=f(h,ν,κ,g,α,∆)

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 N−K=3 parameters that

can be written as Nu =F(Ra,Pr)being Ra =gα∆h3/(νκ),Pr =ν/κand Nu =

˙

Q/˙

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 N−K>10 independent parame-

ters.

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

Communication).

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-

B´

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

Γ≈1 the two geometries yield similar values while for Γ≥4 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 h−2δbl ≈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–

B´

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

simulation.

transition occurs for a shear Reynolds number of ReS≈420 that Grossmann &

Lohse [4] have estimated to happen around RaC≈1014.

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

RaC≈1014, 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. (www.gauss-centre.eu) for funding this project by providing computing

time on the GCS Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre

(www.lrz.de).

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