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The influence of the cell inclination on the heat transport and large-scale circulation in liquid metal convection

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

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The influence of the cell inclination on the heat transport and large-scale circulation in liquid metal convection - Volume 884 - Lukas Zwirner, Ruslan Khalilov, Ilya Kolesnichenko, Andrey Mamykin, Sergei Mandrykin, Alexander Pavlinov, Alexander Shestakov, Andrei Teimurazov, Peter Frick, Olga Shishkina
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J. Fluid Mech. (2020), vol. 884, A18. c
The Author(s) 2019
This is an Open Access article, distributed under the terms of the Creative Commons Attribution
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doi:10.1017/jfm.2019.935
884 A18-1
The influence of the cell inclination on the heat
transport and large-scale circulation in liquid
metal convection
Lukas Zwirner1,, Ruslan Khalilov2, Ilya Kolesnichenko2, Andrey Mamykin2,
Sergei Mandrykin2, Alexander Pavlinov2, Alexander Shestakov2,
Andrei Teimurazov2, Peter Frick2and Olga Shishkina1
1
Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
2Institute of Continuous Media Mechanics, Korolyov 1, Perm 614013, Russia
(Received 4 April 2019; revised 25 October 2019; accepted 5 November 2019)
Inclined turbulent thermal convection in liquid sodium is studied at large Rayleigh
numbers Ra &107based on the results of both experimental measurements and
high-resolution numerical simulations. For a direct comparison, the considered system
parameters are set to be similar: Ra =1.67 ×107in the direct numerical simulations
(DNS), Ra =1.5×107in the large-eddy simulations and Ra =1.42 ×107in the
experiments, while the Prandtl number of liquid sodium is very small (Pr 0.009).
The cylindrical convection cell has an aspect ratio of one; one circular surface is
heated, while the other one is cooled. Additionally, the cylinder is inclined with
respect to gravity and the inclination angle varies from β=0, which corresponds
to Rayleigh–Bénard convection (RBC), to β=90, as in a vertical convection
(VC) set-up. Our study demonstrates quantitative agreement of the experimental
and numerical results, in particular with respect to the global heat and momentum
transport, temperature and velocity profiles, as well as the dynamics of the large-scale
circulation (LSC). The DNS reveal that the twisting and sloshing of the LSC at small
inclination angles periodically affects the instantaneous heat transport (up to ±44 %
of the mean heat transport). The twisted LSC is associated with a weak heat transport,
while the sloshing mode that brings together the hot and cold streams of the LSC is
associated with a strong heat transport. The experiments show that the heat transport
scales as Nu Ra0.22 in both limiting cases (RBC and VC) for Rayleigh numbers
around Ra 107, while any inclination of the cell, 0 < β 690, leads to an increase
of Nu.
Key words: Bénard convection, convection in cavities, turbulent convection
1. Introduction
Elucidation of the mechanisms of turbulent thermal convection in very-low-Prandtl-
number fluids is crucial for our understanding of the universe and the advancement
Email address for correspondence: lukas.zwirner@ds.mpg.de
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884 A18-2 L. Zwirner and others
of cooling technology. Turbulent thermal convection takes place, for example, on
the surfaces of stars, where the Prandtl number (Pr) varies from 108to 104
(Spiegel 1962; Hanasoge, Gizon & Sreenivasan 2016). Furthermore, turbulent thermal
convection in liquid metals (Pr 1) is relevant in engineering applications, especially
in cooling systems of tokamaks and fast breeder reactors (Zhilin et al. 2009; Belyaev
et al. 2013). Liquid sodium is of particular interest because of its very low Prandtl
number (Pr 0.009) and it is widely used as cooling agent in fast neutron reactors
(Heinzel et al. 2017).
One classical model of thermal convection is Rayleigh–Bénard convection (RBC),
where the fluid is confined between a heated lower plate and a cooled upper plate,
and the main driving force is buoyancy. The temperature inhomogeneity varies the
fluid density, which in presence of gravity leads to the convective fluid motion. For
reviews on RBC we refer to Bodenschatz, Pesch & Ahlers (2000), Ahlers, Grossmann
& Lohse (2009), Lohse & Xia (2010) and Chillà & Schumacher (2012).
Thermal convection inevitably arises in the case of a horizontal temperature gradient.
This is known as vertical convection (VC), convection in cavities or side-heated
convection. In vertical convection, the heated and cooled plates are located parallel
to the gravity vector and shear plays the key role, see Ng et al. (2015,2017) and
Shishkina (2016).
The concept of inclined convection (IC) is a generalisation of RBC and VC, i.e.
the fluid layer between the parallel plates is tilted with respect to the direction of
gravity, and both buoyancy and shear act on the flow. This type of convection was
studied previously by Daniels, Wiener & Bodenschatz (2003), Chillà et al. (2004),
Sun, Xi & Xia (2005), Ahlers, Brown & Nikolaenko (2006b), Riedinger et al. (2013),
Weiss & Ahlers (2013) and Langebach & Haberstroh (2014), and more recently by
Frick et al. (2015), Mamykin et al. (2015), Vasil’ev et al. (2015), Kolesnichenko
et al. (2015), Shishkina & Horn (2016), Teimurazov & Frick (2017), Mandrykin &
Teimurazov (2019), Khalilov et al. (2018) and Zwirner & Shishkina (2018).
In thermal convection, the global flow structures and heat and momentum transport
are determined mainly by the following system parameters: the Rayleigh number Ra,
the Prandtl number Pr and the aspect ratio of the container Γ. These are defined as
Ra αg1L3/(κν), Pr ν , Γ D/L,(1.1ac)
respectively. 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. (1.2a,b)
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; uis the velocity vector field and h·i denotes the average
in time and over the whole convection cell. Note that, even for a fixed set-up in
natural thermal convection with no additional shear imposed on the system, the
scaling relations of the mean heat and momentum transport, represented by Nu and
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Inclined liquid metal convection 884 A18-3
Re, with the input parameters Ra and Pr, are not universal and are influenced by
non-Oberbeck–Boussinesq (NOB) effects, see Kraichnan (1962), Grossmann & Lohse
(2000), Ahlers et al. (2006a,2009), Lohse & Xia (2010), Shishkina, Grossmann &
Lohse (2016a), Shishkina, Weiss & Bodenschatz (2016b) and Weiss et al. (2018).
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 (Huang
et al. 2013; Chong et al. 2015; Chong & Xia 2016). Thus, in experiments by Huang
et al. (2013) for Pr =4.38, an increase of Nu due to the cell confinement was
obtained, while in the direct numerical simulations (DNS) by Wagner & Shishkina
(2013) for Pr =0.786, the heat and mass transport gradually reduced with increasing
confinement. This virtual contradiction was recently resolved in Chong et al. (2018).
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 Γdecreases with increasing Pr, but for Pr 60.5 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 the general case of inclined thermal convection, the cell inclination angle β
(β=0in RBC and β=90in VC) is an influential input parameter of the convective
system, apart from Ra,Pr and the geometry of the container. Experimental studies
of inclined thermal liquid-sodium convection in cylinders of different aspect ratios,
showed that the convective heat transfer between the heated and cooled surfaces of
the container is most efficient neither in a standing position of the cylinder (β=0),
nor in a lying position (β=90), but at an inclined position for a certain intermediate
value of β, 0< β < 90, see Vasil’ev et al. (2015) for L=20D, Mamykin et al.
(2015) for L=5Dand Khalilov et al. (2018) for L=D. Moreover, these experiments
showed that, for Pr 1 and Ra &109, any tilt β, 0< β 690, of the cell leads to
a larger mean heat flux (Nu) than in RBC, at 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 (Shishkina & Horn 2016). For example,
for Pr 6.7 and Ra 4.4×109, a monotonic reduction of Nu with increasing βin
the interval β∈ [0,90]takes place, as it was obtained in measurements by Guo
et al. (2015).
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 of βon Nu,
reflected in a tiny reduction of Nu with increasing βclose to β=0, see Ciliberto,
Cioni & Laroche (1996), Cioni, Ciliberto & Sommeria (1997), Chillà et al. (2004),
Sun et al. (2005), Ahlers et al. (2006b), Roche et al. (2010) and Wei & Xia (2013).
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 (Weiss & Ahlers 2013). The single-roll
LSC is known to be more efficient in the heat transport than its double-roll form, as
was proved in the measurements (Xi & Xia 2008; Weiss & Ahlers 2013) and DNS
(Zwirner & Shishkina 2018).
The IC in a broad range of the inclination angle β(from 0to 90) in liquid
sodium has been studied so far by Frick et al. (2015), Mamykin et al. (2015),
Vasil’ev et al. (2015), Kolesnichenko et al. (2015) and Khalilov et al. (2018). These
sodium experiments were conducted in relatively long cylinders, in which the scaling
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884 A18-4 L. Zwirner and others
exponents are essentially increased due to the geometrical confinement. For RBC in
a cylinder with L=5D, Frick et al. (2015) reported Nu Ra0.4and for RBC in a
very long cylinder with L=20D, Mamykin et al. (2015) 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.
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.
An 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 at
different inclination angles. Experimental studies on these type of convective flows
were conducted by Zimin, Frik & Shaidurov (1982).
Although there is no scaling theory for general IC, for the limiting configurations
of 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), Grossmann & Lohse (2000,2001,2011) (hereafter 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 (u) and thermal (θ)
dissipation rates, for which analytical relations with Nu,Ra and Pr exist. Equating
uand θto their estimated either bulk or BL contributions and employing in the
BL dominated regimes the Prandtl–Blasius BL theory (Prandtl 1905; Blasius 1908;
Landau & Lifshitz 1987; Schlichting & Gersten 2000), theoretically possible limiting
scaling regimes were derived. The theory allows us to predict Nu and Re in RBC if
the pre-factors fitted with the latest experimental and numerical data are used, see
Stevens et al. (2013) and Shishkina et al. (2017).
In the other limiting case of IC, which is VC (β=90), the mean kinetic
dissipation rate ucannot be derived analytically from Ra,Pr and Nu, and this
impedes an extension of the GL theory to VC. However, for the case of laminar free
convection between two differentially 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 the
assumption that a similarity solution exists (Shishkina 2016). 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 of high Ra, the scaling exponent in the Nu versus Ra and Re versus Ra
scalings is 1/2, as in RBC (Ng et al. 2018).
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 (Bailon-Cuba
et al. 2012; Wagner & Shishkina 2013,2015; Ng et al. 2015), but the VC and RBC
flows can even be 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 (Shishkina & Horn 2016). 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 approximately 1/4 (Schmidt & Beckmann 1930; Lorenz
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Inclined liquid metal convection 884 A18-5
1934; Saunders 1939; Churchill & Chu 1975), being slightly larger for very small
Ra, where the geometrical cell confinement influences the heat transport (Versteegh
& Nieuwstadt 1999; Yu, Li & Ecke 2007; Kis & Herwig 2012; Ng et al. 2015).
The scaling exponent γis also larger for very high Ra, where, with growing Ra,
the VC flows become first transitional (Ng et al. 2017) and later on fully turbulent
(Fujii et al. 1970; George & Capp 1979). 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 Frick et al. (2015) and Mamykin
et al. (2015) 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, Frick et al. (2015) 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, Mamykin et al. (2015) found Nu Ra0.95 and
Re Gr0.63.
There exist a few measurements of the scaling relations of Nu versus Ra in liquid–
metal Rayleigh–Bénard convection (without any cell inclination). For mercury (Pr
0.024), it was reported Nu Ra0.27 for 2 ×106<Ra <8×107by Takeshita et al.
(1996), Nu Ra0.26 for 7 ×106<Ra <4.5×108and Nu Ra0.20 for 4.5×108<Ra <
2.1×109by Cioni et al. (1997) and Nu Ra0.29 for 2 ×105<Ra <7×1010 by Glazier
et al. (1999). For liquid gallium (Pr 0.025), King & Aurnou (2013) measured Nu
Ra0.25 for 2 ×106<Ra <108, and for GaInSn (Pr =0.029), Zürner et al. (2019)
measured Nu Ra0.27 for 4 ×106<Ra <6×107. For liquid sodium (Pr =0.006),
Horanyi, Krebs & Müller (1999) measured Nu Ra0.25 for 2 ×104<Ra <5×106.
Conducting accurate DNS of natural thermal convection at high Ra and very low Pr
is very challenging, since it requires very fine meshes in space and short steps in time
due to the necessity to resolve the Kolmogorov microscales in the bulk of the flows as
well in the viscous BLs (Shishkina et al. 2010). 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 at large Ra. Thus, there exist
only a few DNS of thermal convection for a combination of large Ra and very low
Prandtl numbers, Pr 60.025, which, moreover, have been conducted exclusively for
the Rayleigh–Bénard configuration (Scheel & Schumacher 2016,2017; Schumacher
et al. 2016; Horn & Schmid 2017; Vogt et al. 2018).
DNS of turbulent flows at extremely large Gr and extremely small Prandtl numbers,
for example convective flow of magnesium that develops in a titanium reduction
reactor (characterised by Gr of the order of 1012), is currently unrealisable. 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 this kind of problem (Teimurazov, Frick & Stefani 2017).
Although knowledge on 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 quite restricted due to the
known difficulties in conducting 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 probe measurements in the core part of
pipe and channel flows are possible (Heinzel et al. 2017; Onea et al. 2017a,b), since
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884 A18-6 L. Zwirner and others
λκ×106
(W mK1) (m2s1)
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 approximately
410 K.
the flow is influenced only downstream, they will unavoidably induce too strong
disturbances in the case of natural thermal convection.
Thus, IC measurements in liquid sodium at large Ra and the corresponding precise
numerical simulations are in great demand and, therefore, we devote our present
work to this topic. Combining experiments, DNS and LES, we aim to paint a
complementary picture of turbulent thermal liquid-sodium convection. Using these
three viewpoints on liquid-sodium convection helps us to overcome their individual
difficulties and simultaneously verify their observations.
The outline of this work is as follows: in §2we introduce the experiment, DNS,
LES and methods of data analysis. Section 3presents the obtained results on how the
global flow structures (mainly LSC) and their evolution in time affect the heat and
momentum transport in case of liquid-sodium convection. An analogy of the global
flow structures and global heat transport in flows with similar values of Ra Pr is also
discussed there. The final section summarises our results.
2. Methods
2.1. Experiment
All experimental data presented in this paper are obtained at the experimental facility
described in detail in Khalilov et al. (2018). 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 (Kolesnichenko et al. 2017). The entire set-up 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 conditions that are considered in numerical simulations, are
different. For example, in the simulations, the cylindrical sidewall 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
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Inclined liquid metal convection 884 A18-7
6
2
1
Œ
3
6
4
5
Î
L
D
ı
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 L
the length of the cylindrical convection cell (1), βis the cell inclination angle, Θthe
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).
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 (Kolesnichenko et al. 2015). 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 (Kolesnichenko et al. 2017).
Typical velocities of the sodium flows in the heat exchangers are approximately
1 m s1, 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 approximately 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 are measured close to the copper plates (see figure 1).
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884 A18-8 L. Zwirner and others
In any hot liquid–metal experiment, there exist unavoidable heat losses due to the
high temperature of the set-up. 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 at Θ6=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 =Qeff Rpl, 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) for different Θare calculated in a series of auxiliary
measurements for the case of β=0 and stable temperature stratification, where
the heat 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 ), 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 ) and measured Qeff as follows:
=Θpl, ∆pl =Qeff Rpl .(2.1a,b)
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, Tm139.8and 25.3 K.
The Nusselt number is then calculated as
Nu =LQeff
λNaS.(2.2)
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 that at the cooled plate, T, are
calculated as follows:
T+=Tm+∆/2,T=Tm∆/2.(2.3a,b)
The Rayleigh number in the experiment is evaluated as
Ra αg1D4/(Lκν), (2.4)
which slightly differs from the value defined in (1.1), since Dis slightly smaller
than L. Thus, for α=2.56 ×104K1,g=9.81 m s2,=25.3 K, ν=6.174 ×
107m2s1and κ=6.651 ×105m2s1, the Rayleigh number equals Ra =(1.42 ±
0.03)×107.
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Inclined liquid metal convection 884 A18-9
0.22
0.22
0.14
0.14
0.14
0.14
0.08
0.08
x
z
ı
D
L
T+
T-
g
A
B
C
DE
F
G
H
1
3
4
5
2
(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 set-up from figure (a), with shown distances
(normalised with L=D) between the neighbouring probes and between the probes and
the sidewall of the cylindrical convection cell.
For a deep analysis of the convective liquid-sodium flows, the convection cell
is equipped with 28 thermocouples, 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 2by 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 eight 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 sidewall and thus are located on five circles, which
are marked in figure 2with 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 Khalilov et al. (2018), where a single
Rayleigh number Ra =(1.42 ±0.03)×107was 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 cylindrical coordinates
(r, φ, z):
Dtu=ν2up+αg(TT0)ˆe,(2.5)
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884 A18-10 L. Zwirner and others
DtT=κ2T,(2.6)
∇ · u=0,(2.7)
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 ˆeis the unit vector,
ˆe=(sin(β) cos ), sin) sin), 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-dimensionalised by using the cylinder radius R, the free-fall
velocity Uf, the free-fall time tf,
UfgR∆)1/2,tf=RgR∆)1/2,(2.8a,b)
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 (2.5)–(2.7), 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 sidewall,
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 (Kooij et al. 2018). No turbulence model is applied
in the simulations. The utilised staggered computational grids of approximately 1.5×
108nodes, which are clustered near all rigid walls, are sufficiently fine to resolve the
Kolmogorov microscales (see table 2). Statistical averaging is usually carried out for
several hundred free-fall time units, which is sufficient not only for integral quantities
like the Nusselt number and Reynolds number to converge, but also for the flow fields.
The exceptions are the most expensive DNS for liquid sodium. Statistical averaging in
our liquid-sodium DNS is, however, the longest known for such low Prandtl numbers
and the integral quantities and also the velocity and temperature profiles are converged
to a reasonable degree.
Since simulations on such fine meshes are extremely expensive, only four 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 an almost constant
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 ×106
and Pr =1 with Ra =109. Note that the Grashof number of the auxiliary DNS
(Gr =109) is slightly different from that of the liquid-sodium DNS (Gr =1.78 ×109).
In the former additional DNS, eight different inclination angles are considered, while
in the latter additional DNS, 11 different values of βare examined (see table 2). The
computational grids, used in the auxiliary DNS for similar Ra Pr, contain only three
nodes within each viscous boundary layer, however, a convergence study, using twice
as many nodes in each direction, for three different inclination angles demonstrated
a deviation in Nu and Ra within 1 %.
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Inclined liquid metal convection 884 A18-11
Ra Pr β(deg.) tavg/tfNrNφNzNth Nvδu/Lδth/L
DNS 1.67 ×1070.0094 0 157.2 384 512 768 82 7 3.4×1035.2×102
36 89.0 384 512 768 67 7 3.2×1034.1×102
72 72.0 384 512 768 69 7 3.2×1034.2×102
90 66.8 384 512 768 77 7 3.4×1034.8×102
LES 1.5×1070.0093 0 460.5 100 160 200 32 3 3.3×1035.4×102
10 629.4 100 160 200 31 3 3.2×1035.2×102
20 806.0 100 160 200 29 3 3.1×1034.9×102
30 468.2 100 160 200 27 3 3.1×1034.4×102
40 468.2 100 160 200 27 3 3.1×1034.3×102
50 928.8 100 160 200 26 3 3.1×1034.2×102
60 560.3 100 160 200 27 3 3.1×1034.2×102
70 460.5 100 160 200 27 3 3.1×1034.3×102
80 537.3 100 160 200 28 3 3.2×1034.6×102
90 652.4 100 160 200 30 3 3.3×1035.0×102
DNS 1.67 ×1060.0940 0 4000 95 128 192 13 3 1.21 ×1026.1×102
9 1000 95 128 192 12 3 1.24 ×1025.7×102
18 2000 95 128 192 12 3 1.19 ×1025.4×102
27 1000 95 128 192 11 3 1.15 ×1025.2×102
36 2000 95 128 192 11 3 1.13 ×1025.1×102
54 2000 95 128 192 11 3 1.15 ×1025.0×102
72 1000 95 128 192 11 3 1.11 ×1025.2×102
90 1000 95 128 192 13 3 1.11 ×1025.9×102
DNS 1091 0 420 384 512 768 16 11 5.2×1037.8×103
9 385 384 512 768 15 10 5.0×1037.7×103
18 403 384 512 768 15 10 4.6×1037.6×103
27 465 384 512 768 15 9 4.3×1037.6×103
36 447 384 512 768 15 8 3.9×1037.6×103
45 356 384 512 768 15 7 3.6×1037.5×103
54 366 384 512 768 15 7 3.2×1037.4×103
63 382 384 512 768 15 6 2.9×1037.7×103
72 448 384 512 768 16 6 2.8×1038.0×103
81 182 384 512 768 15 6 2.7×1038.4×103
90 36 384 512 768 17 6 2.6×1038.7×103
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 δu/L(2.17) and the thermal boundary layer δth/L(2.16).
2.3. Large-eddy simulations
At any fixed time slice, LES generally require more computational effort 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 the possibility
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884 A18-12 L. Zwirner and others
of obtaining reliable data faster, compared to the DNS, using modest computational
resources.
In our study, the OB equations (2.5)–(2.7) of thermogravitational convection with
the LES approach for small-scale turbulence modelling are solved numerically using
the open-source package OpenFOAM 4.1 (Weller et al. 1998) 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 (Deardorff 1970) with the Smagorinsky constant Cs=0.17, which is compatible
with the value of the Kolmogorov spectrum constant for the inertial subrange. The
turbulent Prandtl number in the core part of the domain equals Prt=0.9 and smoothly
vanishes close to the rigid walls. According to different models for liquid metals, the
Prtvalue can be higher than this one (Chen et al. 2013). To estimate the effect of
the Prtvalue on the simulation results we carried out additional LES with Prtvalues
up to 4.12 and showed that the specific choice of Prtdoes not significantly affect the
results. The utilised finite-volume solver is buoyantBoussinesqPimpleFoam with the
pressure implicit with splitting of operators (PISO) algorithm by Issa (1986). Time
integration is realised with the implicit Euler scheme; the diffusive and convective
terms are also treated linearly (more precisely, using the filteredLinear scheme). The
resulting systems of linear equations are solved with the preconditioned conjugate
gradient method with the diagonal-based incomplete Cholesky preconditioner for the
pressure and preconditioned biconjugate gradient method with the diagonal-based
incomplete lower–upper (LU) preconditioner for other flow components (Fletcher
1976; Ferziger & Peri´
c2002).
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 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 Mandrykin & Teimurazov (2019).
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,(2.9)
where zis a component of the heat-flux vector along the cylinder axis:
zuzTκzT
κ∆/L,(2.10)
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.(2.11)
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Inclined liquid metal convection 884 A18-13
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,(2.12)
where h·itdenotes the averaging in time and h·iVthe averaging over the whole
convection cell. Following Teimurazov & Frick (2017), we also evaluate the Reynolds
number based on the volume-averaged velocity fluctuations, or small-scale Reynolds
number, as
Reu0(L/ν)ph(u huit)2i.(2.13)
Finally, one can calculate the Reynolds number based on the ‘wind of turbulence’ as
follows:
Rew=(L/ν) max
zUw(z), (2.14)
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,(2.15)
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 written below in §2.4.4.
2.4.2. Boundary-layer thicknesses
Close to the heated and cooled plates, thermal and viscous boundary layers develop.
The thickness of the thermal boundary layer is calculated as
δth =L/(2Nu). (2.16)
This is the standard way to define the thickness of the thermal boundary layer under
the assumption of pure conductive heat transport within this layer, cf. Ahlers et al.
(2009).
Using the slope method (cf. Zhou & Xia 2010), from the Uw(z)profile along the
cylinder axis, see (2.15), the viscous boundary-layer thickness δuis defined as follows:
δumax
z{Uw(z)}dUw
dz
z=01
.(2.17)
The same criterion was used also for vertical convection before in Ng et al. (2015),
and can also be adopted to the general case of IC.
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 2and discussed in § 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
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884 A18-14 L. Zwirner and others
Tmax
Tmin
doc
rprobes
¯T˘
t
T+
T-
ç
FIGURE 3. Sketch to illustrate the off-centre distance doc of the sloshing mode, as
obtained by the temperature extrema extraction method (cf. (2.19)), see Xi et al. (2009).
at the same locations in all three approaches. The only difference is that in the DNS
and LES there are eight virtual probes in each cross-section, while in the experiment,
there are eight probes in the cross-sections 1, 3 and 5 and only two 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 Cioni et al. (1997) is applied, which is
widely used in RBC experiments (Brown & Ahlers 2006; Xi & Xia 2007; Bai, Ji &
Brown 2016; He, Bodenschatz & Ahlers 2016; Khalilov et al. 2018) and simulations
(Mishra et al. 2011; Stevens, Clercx & Lohse 2011; Wagner, Shishkina & Wagner
2012; Ching, Dung & Shishkina 2017). 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) =Tm+δ3cosθ3)(2.18)
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 sidewall. At the warmer part of the sidewall,
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 δ1
and δ5at other heights from the heated plates, i.e. along circle 1 (closer to the heated
plate) and along circle 5 (closer to the cooled plate), respectively.
To extract the sloshing mode we use the method by Xi et al. (2009). The so-called
temperature extrema extraction (TEE) method is executed as follows: first, the
maximum and minimum of the eight probes at a certain time and horizontal level
are determined and then a second-order polynomial is fitted exactly through that
maximum/minimum using one neighbouring probe to each side (3 points in total per
fit). The off-centre distance of the sloshing mode is defined as
doc =rprobes cos), (2.19)
and it measures the shortest distance between the connecting line (of maximum and
minimum temperature) and the centreline of the cylinder (cf. figure 3).
2.4.4. Velocity estimates
While in the DNS and LES the spatial distributions of all velocity components
are available, direct measurements of the velocity in experiments on natural thermal
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Inclined liquid metal convection 884 A18-15
convection in liquid sodium remain 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). (2.20)
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 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 in a
cylindrical container with a diameter-to-height aspect ratio of 1, 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 3and 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 3and 4). This auxiliary case is interesting
for the following reasons. The ratio of the thermal diffusion time scale, tκ=R2, to
the free-fall time scale, tf=Rgis in both cases 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 approximately tenfold smaller than
that in the main liquid-sodium DNS, albeit being approximately 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 tκ140tfand tν14 900tf.
Another set of auxiliary DNS of IC is conducted for Ra =109and Pr =1 for
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=R/(αg∆) 1.3 s and is similar to
that in the main DNS and LES. Thus, the conducted DNS cover only approximately
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884 A18-16 L. Zwirner and others
Ra Pr β(deg.) Nu Ra Pr β(deg.) Nu
Exp. 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 9.59 1.67 ×1070.0094 72 11.89
1.67 ×1070.0094 36 12.24 1.67 ×1070.0094 90 10.37
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 8.21 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. The
uncertainty in the experiments is approximately 3 %, in the DNS it is approximately 4 %
and IN the LES it is below 1 %.
two minutes of the real-time experiment, which was conducted for approximately
7 h. One should note that although the DNS statistical averaging time is quite short,
collecting approximately 100tfstatistics for the case β=0consumed approximately
390 000 CPUh at the SuperMUC at the Leibniz Supercomputing Centre and required
approximately 60 days of runtime.
In the remaining part of this section we investigate integral time-averaged quantities
such as the global heat transport (Nusselt number) and the global momentum transport
(Reynolds number), provide the evidence of a quantitative agreement between the
simulations and experiment and present a complementary picture of the dynamics of
the large-scale flows in liquid-sodium IC.
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Inclined liquid metal convection 884 A18-17
Ra Pr β(deg.) Re Reu0ReU
DNS 1.67 ×1070.0094 0 1.80 ×1041.27 ×1041.27 ×104
36 1.95 ×1041.02 ×1041.66 ×104
72 1.34 ×1045.5×1031.22 ×104
90 1.01 ×1044.3×1039.1×103
LES 1.5×1070.0093 0 1.627 ×1041.136 ×1041.165 ×104
10 1.702 ×1041.145 ×1041.258 ×104
20 1.766 ×1041.113 ×1041.371 ×104
30 1.772 ×1049.70 ×1031.483 ×104
40 1.723 ×1048.24 ×1031.513 ×104
50 1.676 ×1047.11 ×1031.518 ×104
60 1.517 ×1046.02 ×1031.392 ×104
70 1.311 ×1045.32 ×1031.199 ×104
80 1.147 ×1044.60 ×1031.051 ×104
90 9.60 ×1033.85 ×1038.80 ×103
DNS 1.67 ×1060.0940 0 1.33 ×1031.24 ×1034.7×102
9 1.42 ×1031.10 ×1031.10 ×103
18 1.46 ×1038.1×1021.22 ×103
27 1.46 ×1037.0×1021.28 ×103
36 1.43 ×1036.3×1021.28 ×103
54 1.33 ×1033.7×1021.28 ×103
72 9.9×1027.7×1019.9×102
90 7.3×1020 7.3×102
DNS 1091 0 4.72 ×1034.25 ×1032.05 ×103
9 5.13 ×1033.62 ×1033.63 ×103
18 4.97 ×1032.50 ×1034.29 ×103
27 4.65 ×1032.14 ×1034.12 ×103
36 4.06 ×1031.91 ×1033.59 ×103
45 3.79 ×1031.84 ×1033.31 ×103
54 3.31 ×1031.68 ×1032.85 ×103
63 2.44 ×1031.29 ×1032.07 ×103
72 1.66 ×1037.5×1021.48 ×103
81 1.23 ×1032.9×1021.19 ×103
90 8.4×1021.2×1018.4×102
TABLE 4. Reynolds numbers, a they were obtained in the DNS and LES, see the
definitions (2.11), (2.12) and (2.13). The numerical uncertainty of the Reynolds numbers
was estimated to be approximately 3 % for DNS and below 1 % for LES.
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 4.
There is an excellent agreement of the DNS and LES data, for example, the Nusselt
number deviates less than 1 % and the Reynolds number around 5 % for the RBC
and VC cases. Also in figure 4we compare our numerical results with the DNS by
Scheel & Schumacher (2017), 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 Scheel &
Schumacher (2017), as expected. Note that our LES and DNS and the DNS by Scheel
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884 A18-18 L. Zwirner and others
11
N
u
Ra
10
9
8
7
6
5
4
107
GL Pr = 0.0094
107
106
0.19
0.18
0.17
0.16
Nu/Ra0.22
FIGURE 4. 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 (7 h) 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 inset shows the compensated
Nusselt number for RBC and VC experimental data; the DNS for Ra =1.67 ×107,Pr =
0.0094, β=0(filled diamond); the DNS for Ra =1.67 ×106,Pr =0.094, β=0(cross);
the LES for Ra =1.5×107,Pr =0.0093, β=0(open triangle). Results of the RBC
DNS by Scheel & Schumacher (2017) for Pr =0.005 (open diamonds) and for Pr =0.025
(pluses) and predictions for Pr =0.0094 of the Grossmann & Lohse (2000,2001) theory
considered with the pre-factors from Stevens et al. (2013) (dash line) are presented for
comparison. Everywhere a cylindrical convection cell of aspect ratio 1 is considered.
& Schumacher (2017) were conducted using completely different codes (the Nek5000
spectral element package in the latter case), but nevertheless lead to consistent results.
This verifies to the fact that the obtained results are independent of the numerical
method.
The experimental data exhibit generally lower Nusselt numbers (approximately 35%)
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
cannot 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 of measuring
the temperature directly at the outer surfaces of the copper plates leads to a slight
overestimation of Θand (see figure 1) and, hence, of the effective Rayleigh
numbers for the measured Nusselt numbers.
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Inclined liquid metal convection 884 A18-19
3.1.1. Scaling relations
For Pr =0.0094, the predictions by the Grossmann & Lohse (2000,2001) theory,
with the prefactors from Stevens et al. (2013), are shown in figure 4between the
obtained experimental and numerical data. The experimental data for a certain Ra-
range around Ra =107, shown in figure 4, follow a scaling relation
Nu (0.16 ±0.01)Ra0.22±0.06 (RBC, β =0). (3.1)
The fitted scaling exponent γfor the RBC case is smaller than the scaling
exponents measured in liquid gallium (γ=0.249, King & Aurnou (2013)) and
liquid mercury (γ=0.285, Takeshita et al. (1996)). The Prandtl number for both
fluids is Pr 0.025.
Horanyi et al. (1999) measured the heat transport in liquid sodium at Pr 0.006
for Ra 64.5×106. They used larger aspect ratios from 4.5 to 20, while their highest
Ra measurements were conducted in cells with the smallest aspect ratio. Nevertheless,
their Nu 5 at their highest Ra 4.5×106is similar to Nu =4.97 at our lowest
Ra =5.27 ×106. Furthermore, Horanyi et al. (1999) found that Nu scales as Ra0.25,
which agrees with our results.
The GL theory predicts a local scaling exponent ranging from 0.23 (Ra =5×106)
to 0.26 (Ra =3×107) for Pr =0.0094. This results in a mean scaling exponent of
approximately 0.25 for this interval of Rayleigh numbers.
We fit log(Nu)versus log(Ra)with a second-order polynomial to estimate the
uncertainty of the scaling exponent in our experiments. The local scaling exponent
of this fit ranges from 0.16 (Ra =5×106) to 0.27 (Ra =3×107), thus, we estimate
the uncertainty of the scaling exponent to be = ±0.06. A large variation of
the local scaling exponent can be seen in Glazier et al. (1999) for liquid mercury
in an aspect ratio one cylinder. Their local scaling exponent for small Rayleigh
numbers (106–108) is 0.25, while their average scaling exponent is 0.285. Glazier
et al. (1999) argue that the ‘strong steady bulk mean flow in the cell of aspect
ratio one’ reduces the scaling exponent. Presumably, we see a similar effect in our
measurements.
In figure 4we also present the measured scaling relations for Nu versus Ra for the
case of VC (β=90). The scaling relation in VC is found to be quite similar to that
in RBC, namely
Nu (0.18 ±0.01)Ra0.22±0.06 (VC, β =90). (3.2)
The absolute values of the Nusselt numbers in VC are, however, larger than in RBC.
3.1.2. Time-averaged heat transport in inclined convection
In figure 5(a), 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)/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. As previously discussed for the limiting cases
(RBC and VC), the deviation between DNS and LES data for the Nusselt numbers
is only a few per cent. One can see that the relative increase Nu(β)/Nu(0)of the
numerical data is up to 10 percentage points higher than that of the experimental data
for an inclination angle of approximately 50. This is accompanied by a shift of the
inclination angle with maximal heat transport from approximately 50in our numerical
results to 70in our experimental results. On the one hand, one may take into account
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884 A18-20 L. Zwirner and others
1.3
1.2
1.1
1.0
0.9
Nu(ı)/Nu(0°)Re(ı)/Re(0°)
1.0
0.5
0 183654
ı (deg.)
72 90
(a)
(b)
FIGURE 5. (a) Normalised Nusselt number Nu )/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 and LES as in (a); similar symbols are used as
in (a).
that the Rayleigh number in the experiment is slightly smaller (Ra =1.42 ×107)
compared to that in the DNS (Ra =1.67 ×107) and LES (Ra =1.5×107). On the
other hand, the heat transport in the inclined case is enhanced up to 29 % which
might reduce the heat loss Qloss. However, Qloss is measured in the isothermal case,
maintaining the mean temperature in the cylinder (cf. § 2.1). If the heat loss in the
inclined case is less than in the isothermal reference case, then also the calculated
effective power Qeff =QQloss is lower, and so is the Nusselt number.
As discussed above, we want 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 5(a) and table 3show that
the obtained Nusselt numbers for the same Ra Pr (Ra =1.67 ×106,Pr =0.094) are
in good agreement (the difference is less than 10 percentage points) with the liquid-
sodium experimental results (Ra =1.42 ×107,Pr 0.0093). Remarkable is that, not
only the relative Nusselt number, Nu)/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
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Inclined liquid metal convection 884 A18-21
increase of the Nusselt number due to the cell inclination is only approximately 6 %,
while in the liquid-sodium case it is up to 29 %.
3.1.3. Time-averaged momentum transport in inclined convection
In figure 5(b) and table 4, the results for the Reynolds number are presented for the
liquid-sodium measurements and simulations as well as for the auxiliary DNS. Again,
the agreement between the experiments, DNS and LES for liquid sodium is excellent,
usually less than 10 % between DNS and LES. The dependences of Re(β)/Re(0)
on the inclination angle, obtained in the liquid-sodium DNS and in the DNS for
the same product of Ra Pr, demonstrate perfect agreement (less than two percentage
points difference). The values of Re(β)/Re(0)first slightly increase (around 10 %)
with the inclination angle and then smoothly decrease, so that the Reynolds number
Re(90)in the VC case is significantly smaller (only approximately 50 %) 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 almost similar Grashof number, the Reynolds
numbers decrease much faster with increasing inclination angle than in the liquid-
sodium case.
3.1.4. Time-averaged flow structures in inclined convection
Since the Nusselt numbers and relative Reynolds numbers behave very similarly
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 6and 7. In these figures,
the time-averaged temperature (figure 6) and the time-averaged component of the
heat-flux vector parallel to 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 identical. In contrast to them, the corresponding
distributions for a similar Grashof number look very 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 what is presented above, one can see that at almost similar Grashof numbers
convection of a Prandtl-number-one fluid leads neither to similar integral quantities
like Nu or Re, nor to similar heat flow structures in IC. Thus, it is clearly different to
small-Prandtl-number IC flows. 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.
3.2. 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.
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884 A18-22 L. Zwirner and others
g
¯T˘t
T-T+
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
( j) (k) (l)
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 (a,d,g,j); Ra =1.67 ×107,
Pr =0.0094 (b,e,h,k) and Ra =109,Pr =1 (c,f,i,l). From (a) to (l), the inclination angle
βchanges from β=0(RBC case) through β=36and β=72to β=90(VC case).
In figure 8(a), the time-averaged temperature profiles along the lines of the probe
positions A to H are presented for the inclination angles β=36(DNS) and β=40
(LES and experiments). Figure 8(b) 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 demonstrates
quantitative 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 of the time-averaged temperatures are found to be in good agreement
(largest deviation approximately 0.1) with the numerical data, taking into account
that the Rayleigh number in the experiments is approximately 15 % smaller than in
the DNS.
At an inclination angle of approximately β=36or β=40(figure 8a), 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. In contrast to that, for the inclination angle of approximately β=70
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Inclined liquid metal convection 884 A18-23
g
01
¯Øz˘tmax
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
( j) (k) (l)
FIGURE 7. Table of vertical slices in the plane of the LSC of the time-averaged
component of the heat-flux vector parallel to the cylinder axis, hzit, normalised by
its maximal value through the entire volume, max, as obtained in the DNS for Ra =
1.67 ×106,Pr =0.094 (a,d,g,j); Ra =1.67 ×107,Pr =0.0094 (b,e,h,k) and Ra =109,
Pr =1 (c,f,i,l). From (a) to (l), the inclination angle βchanges from β=0(RBC case)
through β=36and β=72to β=90(VC case).
(figure 8b), the mean flow is stratified and the temperature profiles have non-vanishing
gradients in the z-direction.
In figure 9, the time-averaged profiles of the velocity component uzalong the line
of the probe positions (A–H) parallel to the cylinder axis are presented for the same
inclinations angles, as in figure 8. 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 agreement (largest deviation
approximately 0.27Uf) 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.
The viscous boundary-layer thickness ((2.17) and table 2) for the liquid-sodium
DNS and LES agree within a small deviation of approximately 3 %. With inclination,
the viscous boundary-layer thickness first decreases slightly (by a few per cent) and
then gradually increases. Note that the viscous boundary layers defined on the velocity
magnitude (Ching et al. 2017) and on the wall shear stress (Scheel & Schumacher
2016) are, respectively, approximately two and three times thicker than the boundary
layer defined by (2.17) and table 2.
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884 A18-24 L. Zwirner and others
0 0.2 0.4 0.6 0.8 1.0
T
+
T
m
T
-
T
+
T
m
T
-
z/L
¯T˘t(z) ¯T˘t(z)
(a)
(b)
FIGURE 8. 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. The DNS
data are for Ra =1.67 ×107,Pr =0.0094, the LES data are for Ra =1.5×107,Pr =0.0093
and the experiments are for Ra =1.42 ×107,Pr 0.0093. 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). The data are averaged
along opposite positions due to the expected symmetry and in order to reduce the number
of lines in the plot.
3.3. 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 (Brown & Ahlers 2006; Wagner et al. 2012) and
can exhibit complicated dynamics with twisting (Funfschilling & Ahlers 2004;
Funfschilling, Brown & Ahlers 2008; He et al. 2016) and sloshing (Xi, Zhou &
Xia 2006; Xi et al. 2009; Brown & Ahlers 2009; Zhou et al. 2009; Bai et al. 2016;
Zürner et al. 2019). In very-low-Prandtl-number RBC, this complicated behaviour of
the LSC was reported in experiments with mercury by Cioni et al. (1997) and in the
simulations by Schumacher et al. (2016) and Scheel & Schumacher (2016,2017).
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Inclined liquid metal convection 884 A18-25
0
0
0.2 0.4 0.6 0.8 1.0
z/L
U
f
-U
f
0
U
f
-U
f
¯uz˘t(z) ¯uz˘t(z)
(a)
(b)
FIGURE 9. 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. The DNS data are for Ra =1.67 ×107,
Pr =0.0094, the LES data are for Ra =1.5×107,Pr =0.0093 and the experiments are
for Ra =1.42 ×107,Pr 0.0093. 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). The data are averaged along opposite positions due to the expected symmetry and
in order to reduce the number of lines in the plot.
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-twisting behaviour of the
LSC is quite sharp and it is presumably caused by the increasing stratification of the
temperature at larger inclination angles (Khalilov et al. 2018).
No LSC cessations were captured in the experiments so far. LSC precession happens
only in the RBC case (Khalilov et al. 2018). Only one LSC reversal was captured in
a seven-hour-long RBC experiment in liquid sodium (Mamykin et al. 2018). Further
investigations regarding LSC cessations and reversals need to be carried out in the
future.
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884 A18-26 L. Zwirner and others
0 5 10 15 20 25 30
14
12
10
8
t/tf
z)V(t)
FIGURE 10. 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=RgR∆)1/2.
The arrows indicate the dimensionless times, which are marked in figure 12 with vertical
lines and corresponding 3-dimensional snapshots are shown in figure 13.
In figure 10, the time evolution of the volume-averaged components of the heat-flux
vector parallel to 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 to ±3 % of hziin the VC case. In figure 10 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 5(a) and
table 3provide a more detailed evidence of this fact, based on our measurements and
numerical simulations.
3.3.1. The twisting mode
In figure 11 we present the dynamics of the LSC twisting mode. There, the
temporal evolution of the phases of the LSC in circle 1 (closer to the heated plate)
and in circle 5 (closer to the cold plate) are presented for different inclinations angles
βof the convection cell filled with liquid sodium, as obtained in our DNS and
measurements.
The main evidence for the existence of the twisting mode is the visible strong anti-
correlation of the phases θ1(t)and θ5(t), which are measured via the probes at circles 1
and 5, respectively. It is present in the RBC case (figure 11a,b) and for the inclination
angles β=20(figure 11c) and β=36(figure 11d). The measurements and DNS
at the inclination angles β>40(figure 11e) show that, with increasing β, the above
anti-correlation vanishes. At large inclination angles, there is no visible anti-correlation
of the phases θ1(t)and θ5(t)and one can conclude that the twisting movement of the
LSC is not present any more (figure 11f,g,h).
The dynamics of the twisting 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
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Inclined liquid metal convection 884 A18-27
0 10 20 30 0 10 20 30
0.2
0
-0.2
0.2
0
-0.2
0.2
0
-0.2
0.2
0
-0.2
t/tft/tf
œi(t)/(2π
i(t)/(2π
i(t)/(2π
i(t)/(2π)
(a) (b)
(c) (d)
(e) (f)
(g) (h)
FIGURE 11. Temporal evolution of the phase θ1in circle 1 (solid lines) and of the phase
θ5in 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).
that the period duration Ts(β) equals Ts(0)=8.8tffor the RBC case and is equal to
Ts(36)=7.4tffor 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 is approximately proportional to the Reynolds number ω·tκRe (Cioni et al.
1997). Therefore, the slightly larger period durations in the experiments compared to
those in the DNS are consistent with slightly lower Reynolds numbers and Rayleigh
numbers there.
We also compare the twisting frequencies obtained in our DNS with results by
Schumacher et al. (2016), Xi et al. (2009), Xie, Wei & Xia (2013) and Funfschilling
& Ahlers (2004). Note that we converted all numbers to free-fall time units (tf=
RgR∆)1/2). For low Prandtl numbers, the numerical results by Schumacher et al.
(2016) give a period of 4.6tffor Pr =0.021 and Ra =107. At higher Prandtl numbers
the period is much longer. For water it is approximately 30tfat Pr =5.3 and Ra =
5×109(Xi et al. 2009). For even lager Prandtl number, Pr =19.4, Xie et al. (2013)
found Ts85tfat Ra =2×1011. The twisting frequency is lower with increasing Pr
at high Prandtl numbers (Pr >1). However, the twisting frequency is slightly lower
at Pr =0.0094 compared to Pr =0.021 for similar Rayleigh number.
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884 A18-28 L. Zwirner and others
0 5 10 15 20 25 30
14
13
12
11
10
0.1
0
-0.1
0.5
0
-0.5
T
+
T
m
T
-
t/tf
z)V(t) œi(t)/(2π)T(t)
doc/R
(a)
(b)
(c)
FIGURE 12. 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 circle 1 (solid line) and the phase θ5in circle 5 (dash-dotted line)
and (c) the volume-averaged component of the heat-flux vector along the cylinder axis,
hziV(solid line) and the off-centre distance, doc (dashed line). The three vertical lines
mark the times at which the snapshots in figure 13 are taken. The phases θ1(t)and θ5(t)
in (b) have a period of Tθ=(7.4±0.2)tf, which is determined by the Fourier analysis.
3.3.2. The sloshing mode
Besides the twisting mode there is another mode present in the flow. This is
the sloshing mode; it brings together the hot and cold streams of the LSC. Both
the twisting and the sloshing mode are observed simultaneously, and the full LSC
dynamics is determined by a combination of both modes. To quantify the sloshing
motion, we calculate the off-centre distance (2.19) and plot it over time (figure 12c)
for the example case (DNS, Ra =1.67 ×107,Pr =0.0094, β=36). The sloshing
mode has the same frequency as the twisting mode (correlation of both signals
is >0.8), but it shows a phase difference of approximately 1
4-period. From the
comparison of doc(t)and hziV(t)we learn that, first the hot and cold streams are
brought close to each other, and then, with a short delay, the heat transport increases.
The sloshing mode is coherent through the entire height of the cylinder, quantified
by a correlation of at least 0.8 between the sloshing signal (doc) at mid-height circle 3
and the lower/higher circles (1, 2, 4 and 5). The correlation of circles 1 and 5 is
approximately 0.75.
Let us investigate the IC flow in liquid sodium for the inclination angle β=36, as
in figure 11(d), where a very strong LSC twisting is observed. In figure 12 we analyse
this flow in more detail. Figure 12(a) 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.
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Inclined liquid metal convection 884 A18-29
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(d) (e) (f)
(g) (h) (i)
FIGURE 13. Three-dimensional 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=17tf, (d,e,f)t=19tf, (g,h,i)t=21tf.
The dot (green online) marks the location A3. The snapshots presented here correspond
to the times marked in figure 12 with the vertical lines.
One can see that the temperature dependencies on time at the locations B3 and F3
are synchronous. So are the temperature dependencies on time at the locations H3
and D3. The temperatures at the locations B3 and D3 are anti-correlated. 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 13). These events happen at the times t/tf=17 and t/tf=21 in figure 12,
respectively. Thus, at the times t/tf=17 and t/tf=21 a big hot and a big cold
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884 A18-30 L. Zwirner and others
plume approach each other very closely. This sloshing movement happens periodically,
alternately, near one side of the sidewall, then on the opposite side.
Figure 12(b) presents the evolution in time of the LSC phase θ1in circle 1 (close
to the heated plate) and of the phase θ5in circle 5 (close to the cold plate).
These phases are anti-correlated, and 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 sidewall.
In figure 12(c) the temporal evolution of the volume-averaged heat-flux vector
parallel to the cylinder axis, hziV, is presented. Again, a very strong relationship
with the LSC twisting and sloshing is observed. The maximal values of hziVare
obtained when the hot and cold LSC streams meet, thanks to sloshing, while the
minimum value is obtained at the time periods when the LSC is strongly twisted.
In figure 13, the above described process, namely, the azimuthal movement of the
hot and cold batches of fluid in the 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 movies to this paper, available at https://doi.org/10.1017/jfm.2019.935,
the described dynamics of the LSC can be observed in detail.
3.3.3. Statistics of the large-scale circulation
In figure 14(a) the standard deviations of the phases θiin the circles i=1, 3 and 5
are presented, while figure 14(b) 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 ) that happen between β=20and β=40, which indicate a
sharp transition between the twisting and sloshing modes of the LSC and the 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 14 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 14b) show similar trends. In the RBC case (β=0),
the LSC strength is small and grows smoothly with the inclination angle β. However,
the mean strength of the LSC in the measurements is approximately 0.02, i.e. 2 %
of the temperature difference, weaker that the one in the liquid-sodium DNS. This is
consistent with the lower Nusselt number, as well as the lower Rayleigh number of
the experiments. Surprisingly, the liquid-sodium DNS data show a similar strength of
the LSC as the data from the auxiliary DNS for the same Ra Pr.
Evaluating the LSC phase angle θiand strength δiusing the cos method (Cioni
et al. 1997) by fitting only eight probes results in an uncertainty of approximately
8.6and 12.2 %, respectively. In table 5we compare the mean deviation of the
fit with a given number of probes ncompared to the case n=512. We find that
the use of 32 probes is sufficient to reduce the uncertainty to below 1and 1 %,
respectively. However, using only eight probes in the experiment is acceptable,
especially considering that every additional probe disturbs the flow.
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Inclined liquid metal convection 884 A18-31
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¯∂i˘tSTDi/(2π))
(a)
(b)
FIGURE 14. (a) Standard deviations of the phase angles θiin radians normalised by 2π
and (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 ×10