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arXiv:1904.02110v1 [physics.flu-dyn] 3 Apr 2019

Inclined turbulent thermal convection in liquid sodium

Lukas Zwirner1, Ruslan Khalilov2, Ilya Kolesnichenko2,

Andrey Mamykin2, Sergei Mandrykin2, Alexander Pavlinov2, Alexander Shestakov2, Andrei Teimurazov2,

Peter Frick2& Olga Shishkina1

1Max Planck Institute for Dynamics and Self-Organization,

Am Fassberg 17, 37077 G¨ottingen, Germany

2Institute of Continuous Media Mechanics,

Korolyov 1, Perm, 614013, Russia

Abstract

Inclined turbulent thermal convection by large Rayleigh numbers in extremely small-Prandtl-number

ﬂuids is studied based on results of both, measurements and high-resolution numerical simulations. The

Prandtl number Pr ≈0.0093 considered in the experiments and the Large-Eddy Simulations (LES)

and Pr = 0.0094 considered in the Direct Numerical Simulations (DNS) correspond to liquid sodium,

which is used in the experiments. Also similar are the studied Rayleigh numbers, which are, respectively,

Ra = 1.67 ×107in the DNS, Ra = 1.5×107in the LES and Ra = 1.42 ×107in the measurements. The

working convection cell is a cylinder with equal height and diameter, where one circular surface is heated

and another one is cooled. The cylinder axis is inclined with respect to the vertical and the inclination

angle varies from β= 0◦, which corresponds to a Rayleigh–B´enard conﬁguration (RBC), to β= 90◦, as in

a vertical convection (VC) setup. The turbulent heat and momentum transport as well as time-averaged

and instantaneous ﬂow structures and their evolution in time are studied in detail, for diﬀerent inclination

angles, and are illustrated also by supplementary videos, obtained from the DNS and experimental data.

To investigate the scaling relations of the mean heat and momentum transport in the limiting cases of

RBC and VC conﬁgurations, additional measurements are conducted for about one decade of the Rayleigh

numbers around Ra = 107and Pr ≈0.009. With respect to the turbulent heat transport in inclined

thermal convection by low Pr, a similarity of the global ﬂow characteristics for the same value of RaPr is

proposed and analysed, based on the above simulations and measurements and on complementary DNS

for Ra = 1.67 ×106,Pr = 0.094 and Ra = 109,Pr = 1.

Keywords: Rayleigh–B´enard convection, vertical convection, inclined convection, liquid metals, con-

vection in cavities, turbulent heat transport, Direct Numerical Simulations (DNS), Large-Eddy Simulations

(LES), measurements, liquid sodium

1 Introduction

Elucidation of the mechanisms of turbulent thermal convection in very-low-Prandtl ﬂuids that takes place,

for example, on surfaces of stars, including the Sun, where the Prandtl number (Pr) varies from 10−8to

10−4[73, 33], is crucial for understanding of the universe. One of the possible ways to get one step closer

to this goal is to investigate laboratory convective ﬂows, which can be classiﬁed as turbulent and which are

characterised by very small Prandtl numbers (Pr <10−2). In the present experimental and numerical study

we focus on the investigation of turbulent natural thermal convection in liquid sodium (Pr ≈0.0093), where

the imposed temperature gradient, like in nature and in many engineering applications, is not necessarily

parallel to the gravity vector.

Generally, a turbulent ﬂuid motion, which is driven by an imposed temperature gradient, is a very common

phenomenon in nature and is important in many industrial applications. In one of the classical models of

thermal convection, which is Rayleigh–B´enard convection (RBC), the ﬂuid is conﬁned between a heated lower

horizontal plate and an upper cooled plate, and buoyancy is the main driving force there: The temperature

inhomogeneity leads to the ﬂuid density variation, which in presence of gravity leads to a convective ﬂuid

motion. For reviews on RBC we refer to [8, 3, 48, 12].

In the case of convection under a horizontal temperature gradient, which is known as vertical convection

(VC) or convection in cavities, the heated and cooled plates are parallel to each other, as in RBC, but are

1

located parallel to the gravity vector. Therefore, in this case shear plays the key role, see [53, 54, 67]. The

concept of inclined convection (IC) is a generalisation of RBC and VC. There, the ﬂuid layer, heated on one

surface and cooled from the opposite surface, is tilted with respect to the gravity direction, so that both,

buoyancy and shear drive the ﬂow in this case. This type of convection was studied previously, in particular,

by [20, 11, 76, 2, 59, 87, 47] and more recently by [24, 50, 80, 43, 70, 78, 51, 40, 98].

In thermal convection, the global ﬂow structures and heat and momentum transport are determined

mainly by the following system parameters: the Rayleigh number Ra, Prandtl number Pr and the aspect

ratio of the container Γ:

Ra ≡αg∆L3/(κν),(1)

Pr ≡ν/κ, Γ≡D/L.

Here αdenotes the isobaric thermal expansion coeﬃcient, νthe kinematic viscosity, κthe thermal diﬀusivity

of the ﬂuid, gthe acceleration due to gravity, ∆ ≡T+−T−the diﬀerence between the temperatures at the

heated plate (T+) and at the cooled plate (T−), Lthe distance between the plates and Dthe diameter of the

plates.

The main response characteristics of a natural convective system are the mean total heat ﬂux across the

heated/cooled plates, q, normalised by the conductive part of the total heat ﬂux, ˆq, i.e. the Nusselt number

Nu, and the Reynolds number Re,

Nu ≡q/ˆq, Re ≡LU/ν. (2)

Here Uis the reference velocity, which is usually determined by either the maximum of the time-averaged

velocity along the plates or by hu·ui1/2, i.e. it is based on the mean kinetic energy, with ubeing the

velocity vector-ﬁeld and h·i denotes the average in time and over the whole convection cell. Note, that even

for a ﬁxed setup in natural thermal convection, where no additional shear is imposed into the system, the

scaling relations of the mean heat and momentum transport, represented by Nu and Re, with the input

parameters Ra and Pr, are not universal and are inﬂuenced by non-Oberbeck–Boussinesq (NOB) eﬀects, see

[45, 29, 1, 3, 48, 69, 72, 88].

Here one should note that apart from Pr and Ra, the geometrical conﬁnement of the convection cell also

determines the strength of the heat transport [38, 14, 16]. Thus, in experiments by [38] for Pr = 4.38, an

increase of Nu due to the cell conﬁnement was obtained, while in the Direct Numerical Simulations (DNS)

by [83] for Pr = 0.786, the heat and mass transport gradually reduced with increasing conﬁnement. This

virtual contradiction was recently resolved in [15]. It was found that Pr determines whether the optimal Γ,

at which the maximal heat transport takes place, exists or not. For Pr >0.5 (Ra = 108) an enhancement of

Nu was observed, where the optimal Γ decreases with increasing Pr , but for Pr ≤0.5 a gradual reduction of

the heat transport with increasing conﬁnement was obtained. For all Pr, the conﬁnement induced friction

causes a reduction of Re.

In a general case of inclined thermal convection, apart from Ra,Pr and the geometry of the container,

also the cell inclination angle β(β= 0◦in the RBC conﬁguration and β= 90◦in VC) is the inﬂuential input

parameter of the convective system. Experimental studies of turbulent thermal liquid sodium convection in

cylinders of diﬀerent aspect ratios, showed that the convective heat transfer between the heated and cooled

parallel surfaces of the container is most eﬃcient neither in a standing position of the cylinder (as in RBC,

with a cell inclination angle β= 0◦), nor in a lying position (as in VC, β= 90◦), but in an inclined position

for a certain intermediate value of β, 0◦< β < 90◦, see [80] for L= 20D, [50] for L= 5Dand [40] for L=D.

Moreover, these experiments showed that for Pr ≪1 and Ra &109, any tilt β, 0◦< β ≤90◦, of the cell

leads to a larger mean heat ﬂux (Nu ) than in RBC, by similar values of Ra and Pr. Note that the eﬀect

of the cell tilting on the convective heat transport in low-Pr ﬂuids is very diﬀerent from that in the case of

large Pr [70]. For example, for Pr ≈6.7 and Ra ≈4.4×109, a monotone reduction of Nu with increasing β

in the interval β∈[0◦,90◦] takes place, as it was obtained in measurements by [32].

One should mention that there are only a few experimental and numerical studies of IC in a broad range

of β, whereas most of the investigations of the cell-tilt eﬀects on the mean heat transport were conducted in

a narrow region of βclose to 0◦and mainly for large-Pr ﬂuids. These studies showed generally a small eﬀect

2

of βon Nu, reﬂected in a tiny reduction of Nu with increasing βclose to β= 0◦, see [18, 19, 11, 76, 2, 60, 86].

A tiny local increase of Nu with a small inclination of the RBC cell ﬁlled with a ﬂuid of Pr >1 is possible

only when a two-roll form of the global Large Scale Circulation (LSC) is present in RBC, which usually

almost immediately transforms into a single-roll form of the LSC with any inclination [87]. The single-roll

LSC is known to be more eﬃcient in the heat transport than its double-roll form, as it was proved in the

measurements [91, 87] and DNS [98]. Thus, all available experimental and numerical results on IC show

that the Nu(β)/Nu(0) dependence is a complex function of Ra,Pr and Γ, which cannot be represented as a

simple combination of their power functions.

A strong analogy can be seen between the IC ﬂows and convective ﬂows, which occur from the imposed

temperature diﬀerences at both, the horizontal and vertical surfaces of a cubical container. With a diﬀerent

balance between the imposed horizontal and vertical temperature gradients, where the resulting eﬀective

temperature gradient has non-vanishing horizontal and vertical components, one can mimic the IC ﬂows by

diﬀerent inclination angles. Experimental studies on these type of convective ﬂows were conducted by [97].

Although there is no scaling theory for general IC, for the limiting conﬁgurations in IC (β= 0◦and

β= 90◦) and suﬃciently wide heated/cooled plates, there are theoretical studies of the scaling relations

of Nu and Re with Pr and Ra. For RBC (β= 0), [29, 30, 31] (GL) developed a scaling theory which is

based on a decomposition into boundary-layer (BL) and bulk contributions of the time- and volume-averaged

kinetic (ǫu) and thermal (ǫθ) dissipation rates, for which there exist analytical relations with Nu,Ra and Pr .

Equating ǫuand ǫθto their estimated either bulk or BL contributions and employing in the BL dominated

regimes the Prandtl–Blasius BL theory [58, 7, 46, 64], theoretically possible limiting scaling regimes were

derived. The theory allows to predict Nu and Re in RBC if the pre-factors ﬁtted with the latest experimental

and numerical data are used, see [75] and [68].

In contrast to RBC, in the other limiting case of IC, which is VC (β= 90◦), the mean kinetic dissipation

rate ǫucannot be derived analytically from Ra,Pr and Nu , as in RBC, and this impedes an extension of

the GL theory to VC. However, for the case of laminar free convection between two diﬀerently heated plates

(i.e., VC), it is possible to derive the dependences of Re and Nu on Ra and Pr from the BL equations, under

an assumption that a similarity solution exists [67]. Although this problem is solved for the laminar case, to

our knowledge, there is no theoretical model to predict Nu and Re in turbulent VC. It is expected, however

that in the asymptotic regime by high Ra, the scaling exponents in the Nu vs. Ra and Re vs. Ra scalings is

1/2, as in RBC [55].

Generally, the dependences of Nu and Re on Ra and Pr in VC have been less investigated than those

in RBC. For similar cell geometry and ranges of Ra and Pr , not only the heat transport in VC diﬀers

quantitatively from that in RBC [5, 83, 84, 53], but the VC and RBC ﬂows can be even in diﬀerent states.

For example, for Pr = 1, Ra = 108and a cylindrical container of Γ = 1, the VC ﬂow is steady, while the

RBC ﬂow is already turbulent [70]. Previous experimental and numerical studies of free thermal convection

under an imposed horizontal temperature gradient (i.e., VC) reported the scaling exponent γin the power

law Nu ∼Raγ, varying from 1/4 to 1/3. In laminar VC, it is about 1/4 [65, 49, 61, 17], being slightly

larger for very small Ra, where the geometrical cell conﬁnement inﬂuences the heat transport [81, 94, 41, 53].

The scaling exponent γis also larger for very high Ra, where, with growing Ra, the VC ﬂows become ﬁrst

transitional [54] and later on fully turbulent [28, 25]. Note that all the mentioned experiments and simulations

of VC were conducted for ﬂuids of Pr about or larger than 1.

For the case of small Pr, in the experiments by [24] and [50] on turbulent VC in liquid sodium (Pr ≈

0.01) in elongated cylinders, signiﬁcantly larger scaling exponents were observed, due to the geometrical

conﬁnement. Thus, for a cylinder with L= 5Dand the Rayleigh numbers, based on the cylinder diameter,

up to 107, [24] obtained Nu ∼Ra0.43 and Re ∼Gr0.44, where Gr is the Grashof number, Gr ≡Ra/Pr . For

an extremely strong geometrical conﬁnement, namely, for a cylindrical convection cell with L= 20D, and a

similar Rayleigh number range, [50] found Nu ∼Ra0.95 and Re ∼Gr0.63.

As already mentioned, natural thermal convection for Pr ≪1 is signiﬁcantly less studied than that for

Pr ∼1, whereas the knowledge of convection in very-low-Prandtl-number ﬂuids is desired for understanding

of many astrophysical phenomena and needed in engineering and technology. Thus, liquid metals, being very

eﬃcient heat transfer ﬂuids, are used in numerous applications. Of particular interest is liquid sodium, which

is one of the liquid metals that has been mostly wide used in liquid metal-cooled fast neutron reactors being

3

developed during the last several decades [35].

Although the knowledge on natural thermal convection in liquid metals is required for the development of

safe and eﬃcient liquid metal heat exchangers, the experimental database of the corresponding measurements

remains to be quite restricted due to the known diﬃculties in conduction of thermal measurements in liquid

metals. Apart from general problems, occurring from the high temperature and aggressivity of liquid sodium,

natural convective ﬂows are known to be relatively slow and are very sensitive to the imposed disturbances.

While the probe measurements in the core part of the ﬂows are possible in pipe and channel ﬂows [35, 56, 57],

since the interference inﬂuences the ﬂows only downstreams in those cases, they unavoidably induce too strong

disturbances in the case of natural thermal convection.

There exist a few measurements of the scaling relations of Nu vs. Ra in liquid-metal Rayleigh–B´enard

convection (without any cell inclination). Thus, for mercury (Pr = 0.025), it was reported Nu ∼Ra0.27

for 2 ×106<Ra <8×107by [77], Nu ∼Ra0.26 for 7 ×106<Ra <4.5×108and Nu ∼Ra0.20 for

4.5×108<Ra <2.1×109by [19]. For liquid sodium (Pr = 0.006), [36] measured Nu ∼Ra0.25 for

2×104<Ra <5×106.

IC in liquid sodium has been studied so far by [24, 50, 80, 43, 40]. These sodium experiments were

conducted in relatively long cylinders, in which the scaling exponents are essentially increased due to the

geometrical conﬁnement. For RBC in a cylinder with L= 5D, [24] reported Nu ∼Ra0.4and for RBC in a

very long cylinder with L= 20D, [50] obtained Nu ∼Ra0.77. In both studies, also IC in liquid sodium for

β= 45◦was investigated, and the following scaling laws were obtained for this inclination angle: Nu ∼Ra0.54

for L= 5Dand Nu ∼Ra0.7for L= 20D. Note that in both IC measurements, much higher mean heat

ﬂuxes were obtained, compared to those in VC or RBC conﬁgurations. Liquid sodium IC in a broad range

of the inclination angle β(from 0◦to 90◦) has been studied so far by [80] for a long cylinder with L= 20D

and by [40] for a cylinder of the aspect ratio one (L=D).

Conduction of accurate DNS of natural thermal convection by high Ra and very low Pr is also very

challenging, since it requires very ﬁne meshes in space and in time, due to the necessity to resolve the

Kolmogorov microscales in the bulk of the ﬂows as well in the viscous BLs [71]. When Pr ≪1, the thermal

diﬀusion, represented by κ, is much larger than the momentum diﬀusion, represented by ν, and therefore, the

viscous BLs become extremely thin by large Ra. Thus, there exist only a few DNS of thermal convection for

a combination of large Ra and very low Prandtl numbers, Pr ≤0.025, which, moreover, have been conducted

exclusively for the Rayleigh–B´enard conﬁguration, i.e., only for β= 0 [66, 62, 63, 37, 82].

In engineering applications, however, of particular interest is natural thermal convection with diﬀerent

orientation in the gravity ﬁeld of the imposed temperature gradient. Turbulent thermal convection of liquid

metals in such ﬂow conﬁgurations is especially relevant in cooling systems in tokamaks and fast breeder

reactors, and therefore investigation of such ﬂows are of special importance [95, 6]. The above nonferrous-

metal devices are known to be strongly aﬀected by turbulent thermal convection. For example, convective

ﬂows of magnesium that develop in the titanium reduction reactors, are characterised by the Grashof number

of order of 1012 . DNS of such ﬂows by that large Gr and extremely small Prandtl numbers, would be extremely

expensive and currently are unrealizable and, therefore, further development of reduced mathematical models

is still required. Computational codes for Large-Eddy Simulations (LES) of turbulent thermal convection in

liquid metals, veriﬁed against the corresponding experiments and DNS, can be useful in solving of such kind

of problems [79].

Thus, IC measurements in liquid sodium by large Ra and the corresponding precise numerical simulations

are in great demand and, therefore, we devote our present work to this topic. By combining experiments,

DNS and LES, we are aimed to paint a complementary picture of this kind of convection. In chapter 2 of

this paper, the experiment, DNS, LES and methods of data analysis are described. Chapter 3 presents the

obtained results with respect to the heat and momentum transport in IC in liquid sodium and the global ﬂow

structures and their evolution in time. An analogy of the global ﬂow structures and global heat transport in

ﬂows of similar values of Ra Pr is also discussed there. Conclusion chapter summarises the obtained results

and gives an outlook on future research.

4

6

2

1

3

6

4

5

Θ

L

D

∆

β

Figure 1: Sketch of the experimental facility, which consists of: (1) a cylindrical convection cell, (2) a hot

heat exchanger chamber, (3) a cold heat exchanger chamber, (4) a heated copper plate, (5) a cooled copper

plate and (6) inductor coils. The convection cell (1) and heat exchangers (2, 3) are ﬁlled with liquid sodium.

Dis the diameter and Lthe length of the cylindrical convection cell (1), βis the cell inclination angle, Θ

the temperature drop between the hot and cold heat exchanger chambers, ∆ the resulting temperature drop

between the inner surfaces of the heated and cooled plates of the convection cell (1).

λ κ ×106

(W/mK) (m2/s)

Stainless steel 17 4.53

Na 84.6 66.5

Cu 391.5 111

Table 1: The thermal conductivity λand the thermal diﬀusivity κof stainless steel, liquid sodium (Na) and

copper (Cu) at the mean temperature of the experiment of about 410 K.

2 Methods

2.1 Experiment

All experimental data presented in this paper are obtained at the experimental facility, described in detail in

[40]. The convection cell is made of a stainless steel pipe with a 3.5 mm thick wall. The inner length of the

convection cell is L= 216 mm and the inner diameter D= 212 mm. Both end faces of the convection cell

are separated from the heat exchanger chambers by 1 mm thick copper discs, see a sketch in ﬁgure 1. The

convection cell is ﬁlled with liquid sodium. The heat exchanger chambers are ﬁlled also with liquid sodium

and the temperature there is kept constant. The thin end-face copper plates are intensively washed with

liquid sodium from the chamber sides, ensuring homogeneous temperature distributions at their surfaces [42].

The entire setup is placed on a swing frame, so that the convection cell can be tilted from a vertical position

to a horizontal one. Inclination of the convection cell is then characterised by the angle βbetween the vertical

and the cylinder axis, see ﬁgure 1.

Obviously, the boundary conditions in a real liquid-metal experiment and the idealised boundary con-

ditions that are considered in numerical simulations, are diﬀerent. For example, in the simulations, the

5

cylindrical side wall is assumed to be adiabatic, while in the real experiment it is made from 3.5 mm thick

stainless steel and is additionally covered by a 30 mm thick layer of mineral wool. In table 1, the thermal

characteristics of stainless steel, liquid sodium and copper are presented. One can see that the thermal

diﬀusivity of the stainless steel, although being smaller compared to that of liquid sodium, is not negligible.

Copper, which is known to be the best material for the heat exchangers in the experiments with moderate or

high Pr ﬂuids, has the thermal diﬀusivity of the same order as the thermal diﬀusivity of sodium. Therefore,

massive copper plates would not provide a uniform temperature at the surfaces of the plates [43]. To avoid

this undesirable inhomogeneity, in our experiment, instead of thick copper plates, we use rather thin ones,

which are intensively washed from the outside by liquid sodium of prescribed temperature (see ﬁgure 1). The

latter process takes place in two heat exchanger chambers, a hot one and a cold one, which are equipped with

induction coils. A sodium ﬂow in each heat exchanger chamber is provided by a travelling magnetic ﬁeld.

The induction coils are attached near the outer end faces of the corresponding heat exchanger, thus ensuring

that the electromagnetic inﬂuence of the inductors on the liquid metal in the convection cell is negligible [42].

Typical velocities of the sodium ﬂows in the heat exchangers are about 1 m/s, being an order of magnitude

higher than the convective velocity inside the convection cell.

In all conducted experiments, the mean temperature of liquid sodium inside the convection cell is about

Tm= 139.8◦C, for which the Prandtl number equals Pr ≈0.0093. Each experiment is performed for a

prescribed and known applied temperature diﬀerence Θ = Thot −Tcold , where Thot and Tcold are the time-

averaged temperatures of sodium in, respectively, the hot and cold heat exchanger chambers, which measured

close to the copper plates (see ﬁgure 1).

In any hot liquid-metal experiment, there exist unavoidable heat losses due to the high temperature of

the setup. To estimate the corresponding power losses Qloss, one needs to measure additionally the power,

which is required to maintain the same mean temperature Tmof liquid sodium inside the convection cell,

but under the condition of equal temperatures in both heat exchanger chambers, that is, for Θ = 0. From

Qloss and the total power consumption Qin the convective experiment by Θ 6= 0, one calculates the eﬀective

power Qeﬀ =Q−Qloss in any particular experiment.

Although the thermal resistance of the two thin copper plates themselves is negligible, the sodium-copper

interfaces provide additional eﬀective thermal resistance of the plates mainly due to inevitable oxide ﬁlms.

The temperature drop ∆pl through both copper plates covered by the oxide ﬁlms could be calculated then

from ∆pl =QeﬀRpl , as soon as the eﬀective thermal resistance of the plates, Rpl, is known. Note that for a

ﬁxed mean temperature Tm, the value of Rpl depends on Θ, since the eﬀective thermal conductivities of the

two plates are diﬀerent due to their diﬀerent temperatures.

The values of Rpl(Θ) for diﬀerent Θ are calculated in series of auxiliary measurements for the case of

β= 0 and stable temperature stratiﬁcation, where the heating is applied from above, to suppress convection.

In this purely conductive case, the eﬀective thermal resistance of the plates, Rpl, can be calculated from Θ

and the measured eﬀective power ¯

Qeﬀ from the relation Θ = ¯

Qeﬀ(RNa +Rpl ), where the thermal resistance

of the liquid sodium equals RNa =L/(λNaS) with λNa being the liquid sodium thermal conductivity and

S=πR2with the cylinder radius R.

Using the above measured eﬀective thermal resistances of the plates, Rpl (Θ), in any convection experiment

for a given Ra and βand the mean temperature Tm, the previously unknown temperature drop ∆ inside the

convection cell can be calculated from Θ, Rpl (Θ) and measured Qeﬀ as follows:

∆ = Θ −∆pl,∆pl =QeﬀRpl.

Note that in all our measurements in liquid sodium, for all considered inclination angles of the convection cell,

the obtained mean temperature and the temperature drop within the cell equal, respectively, Tm≈139.8◦

and ∆ ≈25.3 K.

The Nusselt number is then calculated as

Nu =L Qeﬀ

λNa S∆.(3)

For a comparison of the experimental results with the DNS and LES results, where the temperatures at the

plates inside the convection cell are known a priori, the experimental temperature at the hot plate, T+, and

6

0.22

0.14

0.14

0.08

0.22

0.14

0.14

0.08

T+

T−

D

L

β

g

z

x

1

23

4

5

A

B

C

DE

F

G

H

(a) (b)

Figure 2: (a) Sketch of an inclined convection cell. Dis the diameter and Lthe height of the cylindrical

sample, βis the inclination angle, T+(T−) the temperature of the heated (cooled) surfaces. Positioning and

naming of the 40 probes inside the cylinder, as considered in the DNS (all combinations of the azimuthal

locations A, B, C, D, E, F, G, H and circles 1,...,5) and 28 probes in the experiments (all combinations of

the eight locations A, ..., H for the circles 1, 3 and 5 plus four additional probes: A2, A4, E2, E4). Note that

the azimuthal locations are shown only for the circle 3, not to overload the sketch. For any inclination angle

β > 0◦, the upper azimuthal location is A. (b) Sketch of a central vertical cross-section of the setup from

ﬁgure (a), with shown distances (normalised with L=D) between the neighbouring probes and between the

probes and the side wall of the cylindrical convection cell.

that at the cooled plate, T−, are calculated as follows:

T+=Tm+ ∆/2, T−=Tm−∆/2.

The Rayleigh number in the experiment is evaluated as

Ra ≡αg∆D4/(Lκν),(4)

which slightly diﬀers from the value deﬁned in (1), since Dis slightly smaller than L. Thus, for α=

2.56 ×10−4K−1,g= 9.81 m/s2, ∆ = 25.3 K, ν= 6.174 ×10−7m2/s and κ= 6.651 ×10−5m2/s, the Rayleigh

number equals Ra = 1.42 ×107.

For a deep analysis of the convective liquid sodium ﬂows, the convection cell is equipped with 28 thermo-

couples, each with an isolated junction of 1 mm. The thermocouples are located on 8 lines aligned parallel to

the cylinder axis, see ﬁgure 2. The azimuthal locations of these lines are distributed with an equal azimuthal

step of 45◦and are marked in ﬁgure 2 by capital letters A to H (counterclockwise, if looking from the cold

end face). The line A has the upper position if β > 0◦. On each of the 8 lines (A to H), 3 or 5 thermocouples

are placed. Thus, all thermocouples are located in ﬁve cross-sections of the convection cell, which are parallel

to the end faces. The thermocouples are installed inside the convection cell at the same distance of 17 mm

from the inner cylinder side wall and thus are located on 5 circles, which are marked in ﬁgure 2 with numbers

1,...,5. The circles 1, 3, and 5 include eight thermocouples, and the circles 2 and 4 only two thermocouples

(A and E).

In this paper, besides the measurements by [40], where a single Rayleigh number Ra = (1.42 ±0.03) ×107

was considered, we present and analyse also new experimental data, which are obtained for a certain range

of the Rayleigh number, based on diﬀerent imposed temperature gradients.

2.2 Direct numerical simulations

The problem of inclined thermal convection within the Oberbeck–Boussinesq (OB) approximation, which

is studied in the DNS, is deﬁned by the following Navier–Stokes, temperature and continuity equations in

7

cylindrical coordinates (r, φ, z):

Dtu=ν∇2u− ∇p+αg(T−T0)ˆ

e,(5)

DtT=κ∇2T, (6)

∇ · u= 0,(7)

where Dtdenotes the substantial derivative, u= (ur, uφ, uz) the velocity vector ﬁeld, with the component

uzin the direction z, which is orthogonal to the plates, pis the reduced kinetic pressure, Tthe temperature,

T0= (T++T−)/2 and ˆ

eis the unit vector, ˆ

e= (−sin(β) cos(φ),sin(β) sin(φ),cos(β)). Within the considered

OB approximation, it is assumed that the ﬂuid properties are independent of the temperature and pressure,

apart from the buoyancy term in the Navier–Stokes equation, where the density is taken linearly dependent

on the temperature.

These equations are non-dimensionalized by using the cylinder radius R, the free-fall velocity Uf, the

free-fall time tf,

Uf≡(αgR∆)1/2, tf=R(αgR∆)−1/2,

and the temperature drop between the heated plate and the cooled plate, ∆, as units of length, velocity, time

and temperature, respectively.

To close the system (5)–(7), the following boundary conditions are considered: no-slip for the velocity

at all boundaries, u= 0, constant temperatures (T−or T+) at the face ends of the cylinder and adiabatic

boundary condition at the side wall, ∂T/∂r = 0.

The resulting dimensionless equations are solved numerically with the ﬁnite-volume computational code

goldfish, which uses high-order interpolation schemes in space and a direct solver for the pressure [44]. No

turbulence model is applied in the simulations. The utilised staggered computational grids of about 1.5×108

nodes, which are clustered near all rigid walls, are suﬃciently ﬁne to resolve the Kolmogorov microscales, see

table 2.

Since the simulations on such ﬁne meshes are extremely expensive, only 4 inclination angles are considered

for the main case of Pr = 0.0094 and Ra = 1.67 ×107, which are β= 0◦,β= 36◦,β= 72◦and β= 90◦. To

study similarities of the ﬂows with respect to the global heat transport and global ﬂow structures for a ﬁxed

Grashof number, Gr ≡Ra/Pr , and for a ﬁxed value of Ra Pr, some additional DNS of IC were conducted for

the combinations of Pr = 0.094 with Ra = 1.67 ×106and Pr = 1 with Ra = 109. In the former additional

DNS, 8 diﬀerent inclination angles are considered, while in the latter additional DNS, 11 diﬀerent values of

βare examined, see table 2.

2.3 Large Eddy Simulations

At any ﬁxed time slice, LES generally require more computational eﬀorts per computational node, than the

DNS. However, since the LES are relieved from the requirement to resolve the spatial Kolmogorov microscales,

one can use signiﬁcantly coarser meshes in the LES compared to those in the DNS, as soon as the LES are

veriﬁed against the measurements from the physical point of view and against the DNS from the numerical

point of view. Thus, the veriﬁed LES open a possibility to obtain reliable data faster, compared to the DNS,

using modest computational resources.

In our study, the OB equations (5)–(7) of thermogravitational convection with the LES approach for

small-scale turbulence modelling are solved numerically using the open-source package OpenFOAM 4.1 [89]

for Pr = 0.0093 and Ra = 1.5×107and 10 diﬀerent inclination angles, equidistantly distributed between

β= 0◦and β= 90◦.

The package is conﬁgured as follows. The used LES model is that by Smagorinsky-Lilly [21] with the

Smagorinsky constant Cs= 0.17. The turbulent Prandtl number in the core part of the domain equals

Prt= 0.9 and smoothly vanishes close to the rigid walls. The utilised ﬁnite-volume solver is buoyantBoussi-

nesqPimpleFoam with PISO algorithm by [39]. Time integration is realised with the implicit Euler scheme;

linearly are treated also the diﬀusive and convective terms (more precisely, using the ﬁlteredLinear scheme).

8

Ra Pr β tavg/tfNrNφNzNth Nvδv/L

DNS 1.67 ×1070.0094 0◦106.9 384 512 768 82 7 3.4×10−3

36◦53.1 384 512 768 67 7 3.2×10−3

72◦50.4 384 512 768 69 7 3.2×10−3

90◦51.4 384 512 768 77 7 3.4×10−3

LES 1.5×1070.0093 0◦460.5 100 160 200 32 3 3.3×10−3

10◦629.4 100 160 200 31 3 3.2×10−3

20◦806.0 100 160 200 29 3 3.1×10−3

30◦468.2 100 160 200 27 3 3.1×10−3

40◦468.2 100 160 200 27 3 3.1×10−3

50◦928.8 100 160 200 26 3 3.1×10−3

60◦560.3 100 160 200 27 3 3.1×10−3

70◦460.5 100 160 200 27 3 3.1×10−3

80◦537.3 100 160 200 28 3 3.2×10−3

90◦652.4 100 160 200 30 3 3.3×10−3

DNS 1.67 ×1060.0940 0◦4000 95 128 192 13 3 1.21 ×10−2

9◦1000 95 128 192 12 3 1.24 ×10−2

18◦2000 95 128 192 12 3 1.19 ×10−2

27◦1000 95 128 192 11 3 1.15 ×10−2

36◦2000 95 128 192 11 3 1.13 ×10−2

54◦2000 95 128 192 11 3 1.15 ×10−2

72◦1000 95 128 192 11 3 1.11 ×10−2

90◦1000 95 128 192 13 3 1.11 ×10−2

DNS 1091 0◦420 384 512 768 16 11 5.2×10−3

9◦385 384 512 768 15 10 5.0×10−3

18◦403 384 512 768 15 10 4.6×10−3

27◦465 384 512 768 15 9 4.3×10−3

36◦447 384 512 768 15 8 3.9×10−3

45◦356 384 512 768 15 7 3.6×10−3

54◦366 384 512 768 15 7 3.2×10−3

63◦382 384 512 768 15 6 2.9×10−3

72◦448 384 512 768 16 6 2.8×10−3

81◦182 384 512 768 15 6 2.7×10−3

90◦36 384 512 768 17 6 2.6×10−3

Table 2: Details on the conducted DNS and LES, including the time of statistical averaging, tavg, normalised

with the free-fall time tf; number of nodes Nr,Nφ,Nzin the directions r,φand z, respectively; the number

of the nodes within the thermal boundary layer, Nth, and within the viscous boundary layer, Nv, and the

relative thickness of the viscous boundary layer, δv/L.

9

The resulting systems of linear equations are solved with the Preconditioned Conjugate Gradient (PCG)

method with the Diagonal Incomplete-Cholesky (DIC) preconditioner for the pressure and Preconditioned

Biconjugate Gradient (PBiCG) method with the Diagonal Incomplete-LU (DILU) preconditioner for other

ﬂow components [23, 22].

All simulations are carried out on a collocated non-equidistant computational grid consisting of 2.9 million

nodes, see table 2. The grid has a higher density of the nodes near the boundaries, in order to resolve the

boundary layers. Further details on the numerical method, construction of the computational grid and model

veriﬁcation can be found in [51].

2.4 Methods of data analysis

2.4.1 Nusselt number and Reynolds number

The main response characteristics of the convective system are the global heat and momentum transport

represented by the dimensionless Nusselt number Nu and Reynolds number Re, respectively. Within the OB

approximation, the Nusselt number equals

Nu =hΩziz,(8)

where Ωzis a component of the heat ﬂux vector along the cylinder axis,

Ωz≡uzT−κ∂zT

κ∆/L ,(9)

and h·izdenotes the average in time and over a cross-section at any distance zfrom the heated plate.

The Reynolds number can be deﬁned in diﬀerent ways and one of the common deﬁnitions is based on the

total kinetic energy of the system:

Re ≡(L/ν)phu·ui.(10)

Here h·i denotes the averaging in time and over the entire volume. We consider also the large-scale Reynolds

number

ReU≡(L/ν)qhhui2

tiV,(11)

where h·itdenotes the averaging in time and h·iVthe averaging over the whole convection cell. Following

[78], we also evaluate the Reynolds number based on the volume-averaged velocity ﬂuctuations, or small-scale

Reynolds number, as

Reu′≡(L/ν)ph(u− huit)2i.(12)

Finally, one can calculate the Reynolds number based on the ”wind of turbulence” as follows:

Rew= (L/ν) max

zUw(z),(13)

where the velocity of the wind, which is parallel to the heated or cooled plates, can be estimated by

Uw(z) = qhu2

φ+u2

riz,(14)

where uφand urare the azimuthal and radial components of the velocity, respectively.

In the experiments, the Reynolds number is evaluated based on the average of the estimated axial velocities

between the probes along the positions A and E. The velocities are estimated as it is written below, in section

2.4.4.

10

2.4.2 Boundary-layer thicknesses

Close to the heated and cooled plates, the thermal and viscous boundary layers develop. The thickness of

the thermal boundary layer is calculated as

δθ=L/(2Nu).(15)

Using the slope method, from the Uw(z) proﬁle along the cylinder axis, see equation (14), the viscous

boundary-layer thickness δuis calculated as follows:

δu= max

z{Uw(z)}dUw

dz

z=0−1

.(16)

2.4.3 Properties of the large-scale circulation

In the DNS and LES, the information on all ﬂow components is available at every small ﬁnite volume

associated with any grid node. In the experiments, all the information about the ﬂow structures is obtained

from the 28 temperature probes located as shown in ﬁgure 2 and discussed in section 2.1. The probes are

placed in ﬁve diﬀerent horizontal cross-sections of the cylindrical sample, which are parallel to the heated

or cooled surfaces. To make a comparison between the DNS, LES and the experiment possible, we measure

the temperature and monitor its temporal evolution at the same locations in all three approaches. The

only diﬀerence is that in the DNS and LES there are 8 virtual probes in each cross-section, while in the

experiment, there are 8 probes in the cross-sections 1, 3 and 5 and only 2 probes in the cross-sections 2 and

4. The azimuthal locations A to H in the experiment, DNS and LES are exactly the same.

From the temperature measurements at the above discussed locations, one can evaluate the phase and

the strength of the so-called wind of turbulence, or large scale circulation (LSC). To do so, the method by

[19] is applied, which is widely used in the RBC experiments [9, 90, 34, 4, 40] and simulations [52, 74, 85, 13].

Thus, the temperature measured at 8 locations in the central cross-section, from A3 to H3 along the central

circle 3, is ﬁtted by the cosine function

T(θ) = Tm+δ3cos(θ−θ3),(17)

to obtain the orientation of the LSC, represented by the phase θ3, and the strength of the ﬁrst temperature

mode, i.e., the amplitude δ3, which indicates the temperature drop between the opposite sides of the cylinder

side wall. At the warmer part of the side wall, the LSC carries the warm plumes from the heated plate

towards the cold plate and on the opposite colder part, it carries the cold plumes in the opposite direction.

In a similar way one can evaluate the LSC phases θ1and θ5and the strengths δ1and δ5at other heights

from the heated plates, i.e., along the circle 1 (closer to the heated plate) and along the circle 5 (closer to

the cooled plate), respectively.

2.4.4 Velocity estimates

While in the DNS and LES the spatial distributions of all velocity components are available, the direct

measurements of the velocity in the experiments on natural thermal convection in liquid sodium remain to be

impossible so far. In order to estimate the velocities from the temperature measurements in the experiment,

the cross-correlations for all combinations of any two neighbouring probes along the azimuthal locations from

A to H are used.

For example, the normalised cross-correlation function C|A1, A2(τ) for the temporal dependences of the

temperatures T|A1and T|A2measured by the probes at the locations A1 and A2 is calculated as follows:

C|A1, A2(τ)∝X

j

(T|A1(tj)− hT|A1it)·(T|A2(tj+τ)− hT|A2it).(18)

The ﬁrst maximum of the function C|A1, A2(τ) at τ=τcprovides the correlation time τc. From the known

distance between the probes A1 and A2 and the estimated time, τc, which is needed for the ﬂow to bring

11

Ra Pr βNu Ra Pr βNu

Experiment 5.27 ×1060.0094 0◦4.97 1.42 ×1070.0093 20◦6.51

6.43 ×1060.0097 0◦5.18 1.42 ×1070.0093 30◦7.02

9.32 ×1060.0096 0◦5.53 1.42 ×1070.0093 40◦7.07

1.12 ×1070.0095 0◦5.75 1.42 ×1070.0093 50◦7.15

1.18 ×1070.0093 0◦5.84 1.42 ×1070.0093 60◦7.23

1.28 ×1070.0094 0◦5.91 1.42 ×1070.0093 70◦7.30

1.42 ×1070.0093 0◦6.04 1.42 ×1070.0093 80◦7.13

1.43 ×1070.0091 0◦6.09 6.52 ×1060.0095 90◦5.87

1.55 ×1070.0093 0◦6.18 8.91 ×1060.0094 90◦6.19

1.80 ×1070.0091 0◦6.39 1.11 ×1070.0093 90◦6.53

2.06 ×1070.0088 0◦6.79 1.32 ×1070.0091 90◦6.77

2.18 ×1070.0088 0◦6.55 1.42 ×1070.0093 90◦6.84

2.37 ×1070.0086 0◦6.92 1.60 ×1070.0090 90◦7.07

1.42 ×1070.0093 10◦6.17 1.88 ×1070.0086 90◦7.47

DNS 1.67 ×1070.0094 0◦9.66 1.67 ×1070.0094 72◦11.91

1.67 ×1070.0094 36◦12.24 1.67 ×1070.0094 90◦10.38

LES 1.5×1070.0093 0◦9.27 1.5×1070.0093 50◦11.95

1.5×1070.0093 10◦9.65 1.5×1070.0093 60◦11.80

1.5×1070.0093 20◦10.28 1.5×1070.0093 70◦11.61

1.5×1070.0093 30◦11.41 1.5×1070.0093 80◦10.97

1.5×1070.0093 40◦11.71 1.5×1070.0093 90◦10.06

DNS 1.67 ×1060.0940 0◦8.16 1.67 ×1060.0940 36◦9.79

1.67 ×1060.0940 9◦8.78 1.67 ×1060.0940 54◦9.95

1.67 ×1060.0940 18◦9.23 1.67 ×1060.0940 72◦9.55

1.67 ×1060.0940 27◦9.64 1.67 ×1060.0940 90◦8.55

DNS 1091 0◦63.74 1091 54◦67.24

1091 9◦64.58 1091 63◦64.78

1091 18◦65.81 1091 72◦62.53

1091 27◦65.58 1091 54◦59.60

1091 36◦66.05 1091 90◦57.52

1091 45◦67.15

Table 3: Nusselt numbers, as they were obtained in the experiments, DNS and LES.

a thermal plume from the location A1 to A2, one can estimate the mean velocity of the ﬂow between the

locations A1 and A2.

In a similar way one estimates the mean velocities between the probes A2 and A3, etc., along the azimuthal

location A. The mean velocities along the other azimuthal locations, from B to H, are calculated analogously.

3 Results and discussion

In this section, we directly compare the results for inclined convection (IC) in a cylindrical sample of the

diameter-to-height aspect ratio one, as they were obtained in the liquid-sodium DNS for Ra = 1.67 ×107and

Pr = 0.0094, LES for Ra = 1.5×107and Pr = 0.0093 and liquid-sodium experiments for Ra = 1.42 ×107

(Pr ≈0.0093), see tables 3 and 4.

Further liquid-sodium experiments were conducted to measure the scaling relations of the Nusselt number

versus the Rayleigh number in the RBC (the inclination angle β= 0◦) and VC conﬁgurations (β= 90◦), for

the Ra-range around Ra = 107.

Additionally, we make a comparison with the auxiliary DNS results for Ra = 1.67 ×106and Pr = 0.094,

where the product of the Rayleigh number and Prandtl number, Ra Pr ≈1.57 ×105, is the same as in the

main DNS for liquid sodium with Ra = 1.67 ×107and Pr = 0.0094, see tables 3 and 4. This auxiliary

case is interesting by the following reasons. The ratio of the thermal diﬀusion time scale, tκ=R2/κ, to the

12

Ra Pr βRe Reu′ReU

DNS 1.67 ×1070.0094 0◦17927 12828 12523

36◦19430 10081 16609

72◦13372 5527 12176

90◦10110 4406 9100

LES 1.5×1070.0093 0◦16271 11363 11646

10◦17015 11454 12582

20◦17658 11128 13711

30◦17722 9699 14832

40◦17230 8241 15131

50◦16760 7106 15178

60◦15167 6022 13920

70◦13114 5319 11987

80◦11472 4598 10511

90◦9602 3845 8798

DNS 1.67 ×1060.0940 0◦1326 1241 467

9◦1421 1103 1103

18◦1460 809 1216

27◦1457 702 1277

36◦1430 634 1281

54◦1328 366 1277

72◦991 77 988

90◦725 0 725

DNS 1091 0◦4721 4254 2047

9◦5126 3624 3625

18◦4968 2504 4291

27◦4648 2144 4124

36◦4063 1905 3589

45◦3785 1836 3310

54◦3307 1676 2851

63◦2439 1288 2070

72◦1661 752 1481

81◦1226 292 1191

90◦839 12 839

Table 4: Reynolds numbers, as they were obtained in the DNS and LES, see the deﬁnitions (10), (11) and

(12).

13

free-fall time scale, tf=pR/αg∆, in both cases is the same, since tκ/tf∼√Ra Pr . In contrast, in this

auxiliary DNS case, the ratio tν/tfof the viscous diﬀusion time scale tν=R2/ν to the free-fall time scale

tfis about tenfold smaller than that in the main liquid-sodium DNS, albeit being about tenfold larger than

tκ/tf. Hence, thermal diﬀusion dominates over viscous diﬀusion in both considered sets of parameters and

for similar Ra Pr one might expect similar global temperature distributions and quantitatively similar heat

and momentum transport in IC. Note that in the liquid-sodium DNS, the diﬀusion times are tκ≈140 tfand

tν≈14900 tf.

Another set of auxiliary DNS of IC is conducted for Ra = 109and Pr = 1 for a comparison. In this case,

the Grashof number, Gr ≡Ra/Pr = 109, is similar to that in the main liquid-sodium case (Gr ≈1.8×109),

but for this Prandtl-number-one case and the liquid-sodium case we generally do not expect a close similarity

of the global ﬂow characteristics.

A summary of the conducted simulations and experiments can be found in table 2. The free-fall time

in the experiments equals tf=pR/(αg∆) ≈1.3 s and is similar to that in the main DNS and LES. Thus,

the conducted DNS cover about two minutes of the real-time experiment only, which was conducted for

about 7 hours. One should note that although the DNS statistical averaging time is quite short, collecting of

about 100 tfstatistics for the case β= 0◦consumed about 390 000 CPUh at the SuperMUC at the Leibniz

Supercomputing Center and required about 60 days of runtime.

In the remaining part of this section we investigate the integral time-averaged quantities like the global heat

transport (Nusselt number) and the global momentum transport (Reynolds number), provide the evidence

of a very good agreement between the simulations and experiment and present a complementary picture of

the dynamics of the large-scale ﬂows in liquid-sodium IC.

3.1 Time-averaged heat and momentum transport

First we examine the classical case of RBC without inclination (β= 0◦). The time-averaged mean heat

ﬂuxes, represented by the Nusselt numbers, are presented in ﬁgure 3.

Numerical data, i.e., the DNS and LES for liquid sodium, demonstrate an excellent agreement. Also

in ﬁgure 3 we compare our numerical results with the DNS by [63], for Pr = 0.005 and Pr = 0.025. Our

numerical results for Pr = 0.0094 and Pr = 0.0093 take place between the cited results by [63], as expected.

Note that our LES and DNS and the DNS by [63] were conducted using completely diﬀerent codes (Nek5000

spectral element package in the latter case), which nevertheless lead to very similar results. This veriﬁes the

independence of the obtained results from the used numerical method.

For Pr = 0.0094, the predictions by [29, 30] theory, with the prefactors from [75], take place in ﬁgure 3

between the obtained experimental and numerical data. The experimental data for a certain Ra-range around

Ra = 107, shown in ﬁgure 3, follow a scaling relation

Nu ≈0.177Ra0.215 (RBC, β= 0◦).(19)

The experimental data exhibit generally lower Nusselt numbers compared to the numerical data and this

can be explained by the following two reasons. First, the ideal boundary conditions of constant temperatures

at the plates can not be provided in the experiments, since each emission of a suﬃciently strong thermal

plume aﬀects, at least for a short time, the local temperature at the plate, which results in a reduction of

the averaged heat ﬂux compared to that in the simulations with the ideal boundary conditions. Second, the

impossibility to measure the temperature directly at the outer surfaces of the copper plates leads to a slight

overestimation of Θ and ∆ (see ﬁgure 2) and, hence, of the eﬀective Rayleigh numbers for the measured

Nusselt numbers. Anyway, the numerical, theoretical and experimental data for the Nu versus Ra scaling

are found to follow similar scaling laws.

In ﬁgure 3 we also present the measured scaling relations for Nu versus Ra for the case of VC, where the

inclination angle equals β= 90◦. The scaling relation in VC is found to be quite similar to that in RBC,

namely

Nu ≈0.178Ra0.222 (VC, β= 90◦).(20)

14

106107

4

5

6

7

8

9

10

11

GL Pr = 0.0094

Ra

Nu

Figure 3: Nusselt number versus Rayleigh number, as obtained in the RBC experiments for diﬀerent Rayleigh

numbers, Pr ≈0.009, β= 0◦(open circles) with the eﬀective scaling Nu ≈0.177Ra0.215 (solid line); the

RBC experiment for Ra = 1.42 ×107,Pr = 0.0093, β= 0◦(ﬁlled circle; this run is the longest one (7h) in

the series of measurements. For the same Ra and Pr, the Nusselt numbers were also measured for diﬀerent

β, see table 3); the VC experiments for diﬀerent Rayleigh numbers, Pr ≈0.009, β= 90◦(open squares) with

the eﬀective scaling Nu ≈0.178Ra0.222 (dash-dotted line); the DNS for Ra = 1.67 ×107,Pr = 0.0094, β= 0◦

(ﬁlled diamond); the LES for Ra = 1.5×107,Pr = 0.0093, β= 0◦(open triangle). Results of the RBC

DNS by [63] for Pr = 0.005 (open diamonds) and for Pr = 0.025 (pluses) and predictions for Pr = 0.0094

of the [29, 30] theory considered with the pre-factors from [75] (dash line) are presented for comparison.

Everywhere a cylindrical convection cell of the aspect ratio 1 is considered.

0 5 10 15 20 25 30

8

10

12

14

t/tf

hΩziv(t)

Figure 4: Time dependences of the volume-averaged component of the heat ﬂux vector along the cylinder

axis, hΩziV, as obtained in the DNS for Ra = 1.67 ×107,Pr = 0.0094 and four diﬀerent inclination angles

β= 0◦(solid line), β= 36◦(dash line), β= 72◦(dotted line) and β= 90◦(dash-dotted line). Time is

normalised with tf=R(αgR∆)−1/2. The arrows indicate the dimensionless times, which are marked in

ﬁgure 9 with vertical lines and to the snapshots in ﬁgure 10.

15

0.9

1

1.1

1.2

1.31.3

Nu(β)/Nu(0◦)

0◦18◦36◦54◦72◦90◦

0

0.5

1

β

Re(β)/Re(0◦)

(b)

(a)

Figure 5: (a) Normalised Nusselt number Nu (β)/Nu(0◦) versus the inclination angle β, as obtained in the

DNS for Ra = 1.67 ×107,Pr = 0.0094 (ﬁlled diamonds), the LES for Ra = 1.5×107,Pr = 0.0093 (open

triangles), the DNS for Ra = 1.67 ×106,Pr = 0.094 (crosses), the experiments for Ra = 1.42 ×107,

Pr ≈0.0093 (open circles) and the DNS for Ra = 109,Pr = 1 (squares). (b) Normalised Reynolds number

versus the inclination angle β, as obtained in the same DNS, LES and experiments as in (a); similar symbols

are used as in (a).

16

The absolute values of the Nusselt numbers in VC are, however, larger than in RBC.

In ﬁgure 4, the time evolution of the volume-averaged components of the heat ﬂux vector along the

cylinder axis, hΩziVare presented, as they are obtained in the DNS for Ra = 1.67 ×107,Pr = 0.0094 and

four diﬀerent inclination angles between β= 0◦(RBC) and β= 90◦(VC). Obviously, in the RBC case, the

ﬂuctuations of the heat ﬂux around its mean value are extreme and reach up to ±44 % of hΩzi. The strength

of the ﬂuctuations gradually decreases with growing inclination angle βand amount only ±3 % of hΩziin

the VC case. In ﬁgure 4 one can see that for the inclination angles β= 36◦and β= 72◦, the mean heat

transport is stronger than in the RBC or VC cases. This supports a general tendency that in small-Pr ﬂuids

the heat transport becomes more eﬃcient, when the convection cell is tilted. Figure 5a and table 3 provide

a more detailed evidence of this fact, based on our measurements and numerical simulations.

In ﬁgure 5a, the Nusselt numbers in IC are presented, which are normalised by Nu of the RBC case, for the

same Ra and Pr, i.e. the dependence of Nu(β)/Nu(0◦) on the inclination angle β. Very remarkable is that

for similar Ra and Pr, the DNS and LES deliver very similar values of Nu. One can see that the numerical

data are in good agreement with the experimental data, taking into account that the Rayleigh number in

the experiment is slightly smaller (Ra = 1.42 ×107) compared to that in the DNS (Ra = 1.67 ×107).

As discussed above, we want also to compare our results for liquid sodium with the DNS data for similar

Ra Pr and with the DNS data for similar Ra/Pr, as in our liquid-sodium measurements and numerical

simulations. Figure 5a and table 3 show that the obtained Nusselt numbers for the same Ra Pr (Ra = 1.67 ×

106,Pr = 0.094) are in very good agreement with the liquid-sodium experimental results (Ra = 1.42 ×107,

Pr ≈0.0093). Remarkable is that not only the relative Nusselt number, Nu(β)/Nu(0◦), but also the absolute

values of Nu are very similar in the liquid-sodium case and in the case of diﬀerent Pr <1 but the same

Ra Pr. In contrast to that, the Nu -dependence on the inclination angle in the case of the same Grashof

number is diﬀerent, as expected. In that case, the maximal relative increase of the Nusselt number due to

the cell inclination is only about 6%, while in the liquid-sodium case it is up to 29%.

In ﬁgure 5b and table 4, the results on the Reynolds number are presented for the liquid-sodium measure-

ments and simulations as well as for the auxiliary DNS. Again, the agreement between the experiments, DNS

and LES for liquid sodium is excellent. The dependences of Re(β)/Re(0◦) on the inclination angle, obtained

in the liquid-sodium experiments and in the DNS for the same Ra Pr, demonstrate perfect agreement. The

values of Re(β)/Re(0◦) ﬁrst slightly increase with the inclination angle and then smoothly decrease, so that

the Reynolds number Re(90◦) in the VC case is signiﬁcantly smaller than the Reynolds number Re(0◦) in the

RBC case. Here one should notice that the absolute values of Re in the liquid-sodium case are signiﬁcantly

larger than in the IC ﬂows for a similar Ra Pr . In the case of the same Grashof number, the Reynolds

numbers decrease much faster with growing inclination angle than in the liquid-sodium case.

Since the Nusselt numbers and relative Reynolds numbers behave very similar in the liquid-sodium IC

experiments and in the DNS for similar Ra Pr, we compare the time-averaged ﬂow structures for these cases in

ﬁgures 6 and 7. In these ﬁgures, the time-averaged temperature (ﬁgure 6) and the time-averaged component of

the heat ﬂux vector along the cylinder axis hΩzit(ﬁgure 7) are presented in the plane of the LSC, for diﬀerent

inclination angles. One can see that both, the temperature distributions and the heat ﬂux distributions, in

the liquid-sodium case and in the case of a similar Ra Pr look almost identically. In contrast to them,

the corresponding distributions for a similar Grashof number look considerably diﬀerent. The diﬀerence is

especially pronounced for the inclination angle β= 36◦. While in the liquid-sodium ﬂow for β= 36◦there

persist two intertwined plumes, a hot one and a cold one, the temperature in the Prandtl-number-one case is

better mixed (ﬁgure 6) and the heat-ﬂux distribution appears in a form of two triangular-shaped separated

spots (ﬁgure 7).

From the above presented one can conclude that similar Grashof numbers lead neither to similar integral

quantities like Nu or Re, nor to similar heat ﬂow structures in IC. In contrast to that, the small-Prandtl-

number IC ﬂows of similar Ra Pr have similar Nusselt numbers Nu, similar relative Reynolds numbers

Re(β)/Re(0◦) and similar mean temperature and heat ﬂux distributions.

17

Ra = 1.67 ×106

Pr = 0.094

Ra = 1.67 ×107

Pr = 0.0094

Ra = 109

Pr = 1

β= 0◦(RBC)

β= 36◦

β= 72◦

β= 90◦(VC)

T−

hTit

T+

Figure 6: Table of vertical slices through the time-averaged temperature in the plane of the LSC, as obtained

in the DNS for Ra = 1.67 ×106,Pr = 0.094 (left column); Ra = 1.67 ×107,Pr = 0.0094 (central column)

and Ra = 109,Pr = 1 (right column). From top to bottom, the inclination angle βchanges from β= 0◦

(RBC case) through β= 36◦and β= 72◦to β= 90◦(VC case). The black dash-dotted lines indicate the

cylinder axis and the arrows on the left show the direction of the gravity vector in the corresponding row of

the temperature slices.

18

Ra = 1.67 ×106

Pr = 0.094

Ra = 1.67 ×107

Pr = 0.0094

Ra = 109

Pr = 1

β= 0◦(RBC)

β= 36◦

β= 72◦

β= 90◦(VC)

0

hΩzit/Ωmax

1

Figure 7: Table of vertical slices in the plane of the LSC of the time-averaged component of the heat ﬂux

vector along the cylinder axis, hΩzit, normalised by its maximal value through the entire volume, Ωmax, as

obtained in the DNS for Ra = 1.67 ×106,Pr = 0.094 (left column); Ra = 1.67 ×107,Pr = 0.0094 (central

column) and Ra = 109,Pr = 1 (right column). From top to bottom, the inclination angle βchanges from

β= 0◦(RBC case) through β= 36◦and β= 72◦to β= 90◦(VC case). The black dash-dotted lines indicate

the cylinder axis and the arrows on the left show the direction of the gravity vector in the corresponding row

of the slices.

19

Experiments DNS

−0.2

0

0.2

β= 0◦

(a)

θi(t)/(2π)

−0.2

0

0.2

β= 20◦

(c)

θi(t)/(2π)

−0.2

0

0.2

β= 40◦

(e)

θi(t)/(2π)

0 10 20 30

−0.2

0

0.2

β= 90◦

(g)

t/tf

θi(t)/(2π)

β= 0◦

(b)

β= 36◦

(d)

β= 72◦

(f)

0 10 20 30

β= 90◦

(h)

t/tf

Figure 8: Temporal evolution of the phase θ1in the circle 1 (solid lines) and of the phase θ5in the circle 5

(dash-dotted line) of the convection cell (see locations of the circles in ﬁgure 2), as obtained in the experiments

for Ra = 1.42 ×107,Pr ≈0.0093 (a, c, e, g) and in the DNS for Ra = 1.67 ×107,Pr = 0.0094 (b, d, f, h)

for the cell inclination angles β= 0◦(a, b), β= 20◦(c), β= 36◦(d), β= 40◦(e), β= 72◦(f) and β= 90◦

(g, h).

3.2 Dynamics of the large scale ﬂow

In this section, we focus on the reconstruction of the rich structural dynamics of the large scale IC ﬂows in

liquid sodium.

It is well known from the previous RBC studies that the LSC in RBC can show diﬀerent azimuthal

orientations [9, 85] and can exhibit complicated dynamics with twisting [26, 27, 34] and sloshing [92, 93,

96, 10, 4]. In very-low-Prandtl-number RBC, this complicated behaviour of the LSC was reported in the

experiments in mercury by [19] and in the simulations by [66, 62, 63].

In our simulations and experiments in liquid sodium, we observe the twisting and sloshing dynamics of

the LSC in the RBC conﬁguration of the ﬂow, i.e. without any cell inclination, as well as for small inclination

angles βuntil a certain critical β=βs. The experimental data suggest that a transition to the non-sloshing

behaviour of the LSC is quite sharp and it is presumably caused by the increasing stratiﬁcation of the

temperature at larger inclination angles [40].

In ﬁgure 8 we present the dynamics of the LSC twisting and sloshing mode. There, the temporal evolution

of the phases of the LSC in the circle 1 (closer to the heated plate) and in the circle 5 (closer to the cold

plate) are presented for diﬀerent inclinations angles βof the convection cell ﬁlled with liquid sodium, as it is

obtained in our DNS and measurements.

The main evidence for the existence of the sloshing mode is the visible strong anticorrelation of the phases

θ1(t) and θ5(t), which are measured via the probes at the circles 1 and 5, respectively. It is present in the

RBC case (ﬁgures 8 a, b) and for the inclination angles β= 20◦(ﬁgure 8 c) and β= 36◦(ﬁgure 8 d).

20

(a)

(b)

(c)

T−

Tm

T+

T(t)

−0.1

0

0.1

θi(t)/(2π)

0 5 10 15 20 25 30

10

11

12

13

14

t/tf

hΩziv(t)

Figure 9: Temporal evolution of diﬀerent quantities obtained in the DNS for Ra = 1.67 ×107,Pr = 0.0094

and the cell inclination angle β= 36◦: (a) the temperature Tat the probes B3 (solid line), D3 (dash-dotted

line), F3 (dotted line) and H3 (dash line); (b) the phase θ1in the circle 1 (solid line) and the phase θ5in the

circle 5 (dash-dotted line) and (c) the volume-averaged component of the heat ﬂux vector along the cylinder

axis, hΩziV. The three vertical lines mark the times at which the snapshots in ﬁgure 10 are taken. The

phases θ1(t) and θ5(t) in (b) have a period of Tθ= 7.6tf, which is determined by the Fourier analysis.

The measurements and DNS at the inclination angles β≥40◦(ﬁgure 8 e) show that with the increasing

βthe above anticorrelation vanish. At large inclination angles, there is no visible anticorrelation of the

phases θ1(t) and θ5(t) and one can conclude that the sloshing movement of the LSC is not present anymore

(ﬁgures 8 f, g, h).

The dynamics of the twisting and sloshing mode of the LSC can be further studied with the Fourier

analysis. Thus, from the DNS data (Ra = 1.67 ×107,Pr = 0.0094) we obtain that the period duration Ts(β)

equals Ts(0◦) = 6.9tffor the RBC case and is equal to Ts(36◦) = 7.6tffor the inclination angle β= 36◦. The

experimental data give Ts(0◦)≈9.2tffor RBC and Ts(20◦)≈8.7tffor β= 20◦. Note that the frequency

ωof the LSC twisting and sloshing is approximately proportional to the Reynolds number ω·tκ∼Re [19].

Therefore the slightly larger period duration in the experiments compared to those in the DNS are consistent

with slightly lower Reynolds numbers and Rayleigh numbers there.

Let us investigate the IC ﬂow in liquid sodium for the inclination angle β= 36◦, as in ﬁgure 8d, where a

very strong LSC sloshing is observed. In ﬁgure 9 we analyse this ﬂow in more detail. Figure 9a presents the

evolution of the temperature in time, which is measured by the probes B3, D3, F3 and H3 that are placed

in the central circle 3. One can see that the temperature dependences on time at the locations B3 and F3

are perfectly synchronous. So are the temperature dependences on time at the locations H3 and D3. The

temperatures at the locations B3 and D3 are anticorrelated. So are the temperatures at the locations H3 and

F3. Thus, when at the location B3 the ﬂuid is extremely hot, the lowest temperature is obtained near D3,

which is located only 90◦azimuthally below B3. Analogously, when the ﬂuid is hot at the location H3, its

lowest temperature is obtained near the location F3, which is 90◦below H3 (see also ﬁgure 10). These events

happen at the times t/tf= 17 and t/tf= 21 in ﬁgure 9, respectively. Thus, at the times t/tf= 17 and

t/tf= 21 a big hot and big cold plume approach each other very closely. This sloshing movement happens

periodically, alternately, near one side of the sidewall, then on the opposite side.

Figure 9b presents the evolution in time of the LSC phase θ1in the circle 1 (close to the heated plate)

and of the phase θ5in the circle 5 (close to the cold plate). These phases are anticorrelated and at the times

21

•

•••

•

•

••

• •

GC

A

E

FD

B H

t= 17 tf

(a) (b) (c)

••

•

•

••

• •

GC

A

E

FD

B H

•

•

t= 19 tf

(d) (e) (f)

•

•••

•

•

••

• •

GC

A

E

FD

B H

t= 21 tf

(g) (h) (i)

Figure 10: 3D side views (a, d, g) and views orthogonal to the side views and to the cylinder axis (b, e, h) of

the temperature isosurfaces and the corresponding horizontal slices at the mid-height of the instantaneous

temperature ﬁelds as seen from the cold plate (c, f, i), which are obtained in the DNS for Ra = 1.67 ×107,

Pr = 0.0094 and the cell inclination angle β= 36◦at the times (a, b, c)t= 17 tf, (d, e, f)t= 19 tf, (g, h, i)

t= 21 tf. The dot (green online) marks the location A3. The here presented snapshots correspond to the

times, marked in the ﬁgure 9 with the vertical lines.

22

0

0.04

0.08

0.12

STD(θi/(2π))

0◦18◦36◦54◦72◦90◦

0

0.1

0.2

0.3

0.4

β

hδiit/∆

(b)

(a)

Figure 11: (a) Standard deviations of the phases θiand (b) the time-averaged strengths of the large scale

circulation, hδiit, as obtained at the probe circle 1 (pink colour symbols and lines), circle 3 (grey colour) and

circle 5 (blue colour) in the experiments (dash-dotted lines), the DNS for Ra = 1.67 ×107,Pr = 0.0094 (solid

lines) and DNS for Ra = 106,Pr = 0.094 (dash lines).

when the sloshing brings together the hot and the cold streams of the LSC near the sidewall, as described

above (at t/tf= 17 and t/tf= 21), these phases are equal. This means that at t/tf= 17 and t/tf= 21,

there is no twisting of the LSC near the plates. The twisting of the LSC near the plates is maximal (at

t/tf= 19) when the hot and the cold streams of the LSC in the central cross-section are located near the

opposite sides of the cylinder side wall.

The thermal plumes are emitted from the heated and cooled plates when the phase diﬀerence between

θ1and θ5is maximal. They travel towards the mid-plane and approach each other very closely when the

phase diﬀerence vanishes. At each plate, the emission of the thermal plumes takes place, roughly speaking,

from two diﬀerent spots at the plate, and again, this happens periodically and alternately. Thus, the thermal

plumes, while being emitted from the one spot, leave near the other spot suﬃcient space for new plumes to

grow and detach from the thermal boundary layer.

In ﬁgure 9c the temporal evolution of the volume-averaged component of the heat ﬂux vector along the

cylinder axis, hΩziV, is presented. Again, a very strong relationship with the LSC phases and LSC sloshing

is observed. The maximal values of hΩziVare obtained when the hot and cold LSC streams meet, thanks

sloshing, while the minimum value is obtained at the time periods when the LSC is strongly twisted.

In ﬁgure 10, the above described process, namely, the azimuthal movement of the hot and cold batches of

ﬂuid in a form of an oscillatory motion against each other, is illustrated with three-dimensional side views in

two perpendicular directions. Additionally, the corresponding horizontal cross-sections of the instantaneous

temperature ﬁelds at the mid-height of the cylinder are presented there. In the supplementary videos to this

paper, the described dynamics of the LSC can be observed in detail.

23

In ﬁgure 11a the standard deviations of the phases θiin the circles i= 1,3 and 5 are presented, while

ﬁgure 11b shows the corresponding time-averaged strengths of the LSC, hδiit, as they are obtained in the

liquid-sodium measurements and DNS. The measurements show that the standard deviations of the phases

θ1(near the heated plate) and θ5(near the cooled plate) are relatively large for small inclination angles, while

being small for large inclination angles. There exist almost immediate drops of θ1(β) and θ5(β) that happen

between β= 20◦and β= 40◦, which indicate a sharp transition between the twisting and sloshing mode

of the LSC and usual mode of the LSC, when it is not twisted and located basically in the central vertical

cross-section along the axis of the cylindrical sample. The standard deviations of θ1,θ3and θ5, obtained in

the DNS, show generally a similar behaviour as those measured in the experiments, but due to only a few

considered inclination angles in the DNS, it is impossible to resolve the sudden drop which is observed in the

measurements. Also one should notice that the data in ﬁgure 11 are very sensitive to the time of statistical

averaging, which is extremely short in the DNS compared to the experiment.

The results for the time-averaged strengths of the LSC, hδiit, obtained in the measurements and DNS

(ﬁgure 11b) show good agreement. In the RBC case (β= 0◦), the LSC strength is small and grows smoothly

with the inclination angle β. Surprisingly good here is the agreement between the liquid-sodium DNS data

and the data from the auxiliary DNS for the same Ra Pr.

3.3 Temperature and velocity proﬁles

In this section we analyse the temperature and velocity proﬁles. The focus thereby is on the following

two aspects. First, we compare the experimentally and numerically obtained proﬁles through the probes

(positions A to H) along the lines aligned parallel to the cylinder axes. Second, we compare the velocity

proﬁles, obtained in the DNS and LES, with the velocities evaluated from the correlation times between two

neighbouring probes in the experiment, in order to validate the method used in the experiment to estimate

the Reynolds number.

In ﬁgure 12a, the time-averaged temperature proﬁles along the cylinder axis at the positions A to H are

presented for the inclination angles β= 36◦(DNS) and β= 40◦(LES and experiments). Figure ﬁgure 12b

shows analogous proﬁles for the inclination angles β= 72◦(DNS) and β= 70◦(LES and experiments). In

both ﬁgures, the proﬁles at the positions A and E are presented, as well as the average of the proﬁles at the

positions B and H, the average of the proﬁles at the positions D and F and the average of the C-proﬁle and

G-proﬁle. One can see that for the same locations, the LES and DNS proﬁles are almost indistinguishable,

which again demonstrate excellent agreement between the DNS and LES. The experimental data are available

pointwise there, according to the 5 or 3 probes along each location, from A to H. The measurements are

found to be also in good agreement with the numerical data, taking into account that the Rayleigh number

in the experiments is about 15% smaller than in the DNS.

By the inclination angle about β= 36◦or β= 40◦(ﬁgure 12a), the mean temperature gradient with

respect to the direction zacross the plates is close to zero in the core part of the domain. This means that

the turbulent mixing in this case is very eﬃcient, which is also reﬂected in the increased Nusselt numbers

that we studied before. This eﬃcient mixing is provided by the sloshing dynamics of the LSC. In contrast

to that, for the inclination angle about β= 70◦(ﬁgure 12b), the mean ﬂow is stratiﬁed and the temperature

proﬁles have a non-vanishing gradients in the z-direction.

In ﬁgure 13, the time-averaged proﬁles along the cylinder axis of the velocity component uzare presented

for the same inclinations angles, as in ﬁgure 12. Again, a very good agreement between the DNS, LES and

experiments is obtained. The velocity estimates at the locations between the neighbouring thermocouples,

which are derived from the correlation times obtained in the temperature measurements, are found to be in a

very good agreement with the DNS and LES data. Thus, this method to estimate the LSC velocity from the

temperature measurement is proved to be a very reliable instrument in the IC liquid-sodium experiments.

24

T−

Tm

T+

hTit(z)

0 0.2 0.4 0.6 0.8 1

T−

Tm

T+

z/L

hTit(z)

(b)

(a)

Figure 12: Time-averaged temperature proﬁles at the positions A to H of the probes, as obtained in (a) the

DNS for β= 36◦and the LES and experiments for β= 40◦and in (b) the DNS for β= 72◦and the LES

and experiments for β= 70◦. Thick lines are the DNS data, thin lines are the LES data and symbols are

the experimental data. Data at the position A (pink solid lines, squares) and the position E (blue solid lines,

circles); the average of the data at the positions B and H (pink dash-dotted lines, pentagons), the average

of the data at the positions D and F (blue dash-dotted lines, triangles) and the average of the data at the

positions C and G (black dotted and grey dash lines, diamonds).

25

−Uf

0

Uf

huzit(z)

0 0.2 0.4 0.6 0.8 1

−Uf

0

Uf

z/L

huzit(z)

(b)

(a)

Figure 13: Time-averaged proﬁles of the velocity component uz, which is parallel to the cylinder axis,

considered at the positions A to H of the probes, as obtained in (a) the DNS for β= 36◦and the LES and

experiments for β= 40◦and in (b) the DNS for β= 72◦and the LES and experiments for β= 70◦. Thick

lines are the DNS data, thin lines are the LES data and symbols are the experimental data. Data at the

position A (pink solid lines, squares) and the position E (blue solid lines, circles); the average of the data

at the positions B and H (pink dash-dotted lines), the average of the data at the positions D and F (blue

dash-dotted lines) and the average of the data at the positions C and G (black dotted and grey dash lines).

26

4 Conclusions

In our complementary and cross-validating experimental and numerical studies, we have investigated inclined

turbulent thermal convection in liquid sodium (Pr ≈0.009) in a cylindrical container of the aspect ratio one.

The conducted measurements, DNS and LES demonstrated generally a very good agreement. It was proved,

in particular, that the usage of the cross-correlation time of the neighbouring temperature probes is a reliable

tool to evaluate velocities during the temperature measurements in liquid sodium.

For the limiting cases of inclined convection, which are Rayleigh–B´enard convection (with the cell incli-

nation angle β= 0◦) and vertical convection (β= 90◦), we have also studied experimentally the scaling

relations of the mean heat ﬂux (Nusselt number) with the Rayleigh number, for Ra around 107. The scaling

exponents were found to be about 0.22 in both cases, but the absolute values of Nu are found to be larger in

VC, compared to those in RBC. At the considered Rayleigh number about 1.5×107, any inclination of the

RBC cell generally leads to an increase of the mean heat ﬂux. The maximal Nu is obtained, however, for a

certain intermediate value of β.

For small inclination angles, the large-scale circulation exhibits a complex dynamics, with twisting and

sloshing. When the LSC is twisted, the volume-average vertical heat ﬂux is minimal, and it is maximal,

when the LSC sloshing brings together the hot and cold streams of the LSC. Figures 9 and 10 and additional

videos illustrate the studied LSC dynamics. Additional investigations will be needed to study the even more

complex behaviour of the LSC in IC of low-Pr ﬂuids in elongated containers with L≫D.

Furthermore we have found that for small Prandtl numbers there exist a similarity of the IC ﬂows of

the same Ra Pr, due to the similar ratio between the thermal diﬀusion time scale, tκ, and the free-fall time

scale, tf, for which holds tκ/tf∼√Ra Pr. Since in the small-Pr convective ﬂows, the viscous diﬀusion time

scale, tν, is much larger than tκ, the value of Ra Pr determines basically the mean temperature and heat

ﬂux distributions. The Nusselt numbers and the relative Reynolds numbers by inclination of the convection

cell with respect to the gravity vector, are also similar by similar values of Ra Pr . This property can be

very useful for the investigation of, e.g., the scaling relations of Nu and Re with Ra and Pr or of the mean

temperature or heat ﬂux distributions, by extremely high Ra and/or extremely small Pr .

Acknowledgements

This work is supported by the Priority Programme SPP 1881 Turbulent Superstructures of the Deutsche

Forschungsgemeinschaft (DFG) under the grant Sh405/7. O.S. also thanks the DFG for the support under

the grant Sh405/4 – Heisenberg fellowship. The authors acknowledge the Leibniz Supercomputing Centre

(LRZ) for providing computing time and the Institute of Continuous Media Mechanics (ICMM UB RAS) for

providing resources of the Triton supercomputer.

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