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Inclined turbulent thermal convection in liquid sodium

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  • Institute of Continuous Media Mechanics of the Ural Branch of Russian Academy of Science
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Abstract and Figures

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 $Pr\approx0.0093$ 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, $Ra=1.67\times10^7$ in the DNS, $Ra=1.5\times10^7$ in the LES and $Ra=1.42\times10^7$ in 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 $\beta=0^\circ$, which corresponds to a Rayleigh-B\'enard configuration (RBC), to $\beta=90^\circ$, 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 $Ra=10^7$ and $Pr\approx0.009$. 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 $Ra=1.67\times10^6$, $Pr=0.094$ and $Ra=10^9$, $Pr=1$.
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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 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 108to
104[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 <102). 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 αgL3/(κν),(1)
Pr ν/κ, ΓD/L.
Here αdenotes the isobaric thermal expansion coefficient, νthe kinematic viscosity, κthe thermal diffusivity
of the fluid, gthe acceleration due to gravity, ∆ T+Tthe 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= 0in the RBC configuration and β= 90in 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 0and 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= 0and
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= 45was 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 0to 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= 139.8C, 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 =QQloss 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 =QeRpl.
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, Tm139.8
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 αgD4/(Lκν),(4)
which slightly differs from the value defined in (1), since Dis slightly smaller than L. Thus, for α=
2.56 ×104K1,g= 9.81 m/s2, ∆ = 25.3 K, ν= 6.174 ×107m2/s and κ= 6.651 ×105m2/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 45and 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=ν2u− ∇p+αg(TT0)ˆ
e,(5)
DtT=κ2T, (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 (Tor 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= 72and 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= 0and 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 0106.9 384 512 768 82 7 3.4×103
3653.1 384 512 768 67 7 3.2×103
7250.4 384 512 768 69 7 3.2×103
9051.4 384 512 768 77 7 3.4×103
LES 1.5×1070.0093 0460.5 100 160 200 32 3 3.3×103
10629.4 100 160 200 31 3 3.2×103
20806.0 100 160 200 29 3 3.1×103
30468.2 100 160 200 27 3 3.1×103
40468.2 100 160 200 27 3 3.1×103
50928.8 100 160 200 26 3 3.1×103
60560.3 100 160 200 27 3 3.1×103
70460.5 100 160 200 27 3 3.1×103
80537.3 100 160 200 28 3 3.2×103
90652.4 100 160 200 30 3 3.3×103
DNS 1.67 ×1060.0940 04000 95 128 192 13 3 1.21 ×102
91000 95 128 192 12 3 1.24 ×102
182000 95 128 192 12 3 1.19 ×102
271000 95 128 192 11 3 1.15 ×102
362000 95 128 192 11 3 1.13 ×102
542000 95 128 192 11 3 1.15 ×102
721000 95 128 192 11 3 1.11 ×102
901000 95 128 192 13 3 1.11 ×102
DNS 1091 0420 384 512 768 16 11 5.2×103
9385 384 512 768 15 10 5.0×103
18403 384 512 768 15 10 4.6×103
27465 384 512 768 15 9 4.3×103
36447 384 512 768 15 8 3.9×103
45356 384 512 768 15 7 3.6×103
54366 384 512 768 15 7 3.2×103
63382 384 512 768 15 6 2.9×103
72448 384 512 768 16 6 2.8×103
81182 384 512 768 15 6 2.7×103
9036 384 512 768 17 6 2.6×103
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 =hziz,(8)
where Ωzis a component of the heat flux vector along the cylinder axis,
zuzTκ∂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=01
.(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(θθ3),(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 04.97 1.42 ×1070.0093 206.51
6.43 ×1060.0097 05.18 1.42 ×1070.0093 307.02
9.32 ×1060.0096 05.53 1.42 ×1070.0093 407.07
1.12 ×1070.0095 05.75 1.42 ×1070.0093 507.15
1.18 ×1070.0093 05.84 1.42 ×1070.0093 607.23
1.28 ×1070.0094 05.91 1.42 ×1070.0093 707.30
1.42 ×1070.0093 06.04 1.42 ×1070.0093 807.13
1.43 ×1070.0091 06.09 6.52 ×1060.0095 905.87
1.55 ×1070.0093 06.18 8.91 ×1060.0094 906.19
1.80 ×1070.0091 06.39 1.11 ×1070.0093 906.53
2.06 ×1070.0088 06.79 1.32 ×1070.0091 906.77
2.18 ×1070.0088 06.55 1.42 ×1070.0093 906.84
2.37 ×1070.0086 06.92 1.60 ×1070.0090 907.07
1.42 ×1070.0093 106.17 1.88 ×1070.0086 907.47
DNS 1.67 ×1070.0094 09.66 1.67 ×1070.0094 7211.91
1.67 ×1070.0094 3612.24 1.67 ×1070.0094 9010.38
LES 1.5×1070.0093 09.27 1.5×1070.0093 5011.95
1.5×1070.0093 109.65 1.5×1070.0093 6011.80
1.5×1070.0093 2010.28 1.5×1070.0093 7011.61
1.5×1070.0093 3011.41 1.5×1070.0093 8010.97
1.5×1070.0093 4011.71 1.5×1070.0093 9010.06
DNS 1.67 ×1060.0940 08.16 1.67 ×1060.0940 369.79
1.67 ×1060.0940 98.78 1.67 ×1060.0940 549.95
1.67 ×1060.0940 189.23 1.67 ×1060.0940 729.55
1.67 ×1060.0940 279.64 1.67 ×1060.0940 908.55
DNS 1091 063.74 1091 5467.24
1091 964.58 1091 6364.78
1091 1865.81 1091 7262.53
1091 2765.58 1091 5459.60
1091 3666.05 1091 9057.52
1091 4567.15
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, to the
12
Ra Pr βRe ReuReU
DNS 1.67 ×1070.0094 017927 12828 12523
3619430 10081 16609
7213372 5527 12176
9010110 4406 9100
LES 1.5×1070.0093 016271 11363 11646
1017015 11454 12582
2017658 11128 13711
3017722 9699 14832
4017230 8241 15131
5016760 7106 15178
6015167 6022 13920
7013114 5319 11987
8011472 4598 10511
909602 3845 8798
DNS 1.67 ×1060.0940 01326 1241 467
91421 1103 1103
181460 809 1216
271457 702 1277
361430 634 1281
541328 366 1277
72991 77 988
90725 0 725
DNS 1091 04721 4254 2047
95126 3624 3625
184968 2504 4291
274648 2144 4124
364063 1905 3589
453785 1836 3310
543307 1676 2851
632439 1288 2070
721661 752 1481
811226 292 1191
90839 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/tfRa Pr . In contrast, in this
auxiliary DNS case, the ratio tν/tfof the viscous diffusion time scale tν=R2to 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 tκ140 tfand
tν14900 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= 0consumed 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
hziv(t)
Figure 4: Time dependences of the volume-averaged component of the heat flux vector along the cylinder
axis, hziV, 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)
01836547290
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, hziVare 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 hzi. The strength
of the fluctuations gradually decreases with growing inclination angle βand amount only ±3 % of hziin
the VC case. In figure 4 one can see that for the inclination angles β= 36and 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 hzit(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= 36there
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= 36and β= 72to 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
hzit/max
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, hzit, 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= 36and β= 72to 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
hziv(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, hziV. 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 β40(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·tκRe [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 90azimuthally below B3. Analogously, when the fluid is hot at the location H3, its
lowest temperature is obtained near the location F3, which is 90below 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= 36at 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))
01836547290
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, hziV, is presented. Again, a very strong relationship with the LSC phases and LSC sloshing
is observed. The maximal values of hziVare 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= 20and 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= 36or 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= 36and the LES and experiments for β= 40and in (b) the DNS for β= 72and 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= 36and the LES and
experiments for β= 40and in (b) the DNS for β= 72and 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 LD.
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/tfRa 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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