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Using complementary experiments and direct numerical simulations, we study turbulent thermal convection of a liquid metal (Prandtl number $\textit {Pr}\approx 0.03$ ) in a box-shaped container, where two opposite square sidewalls are heated/cooled. The global response characteristics like the Nusselt number ${\textit {Nu}}$ and the Reynolds number $\textit {Re}$ collapse if the side height $L$ is used as the length scale rather than the distance $H$ between heated and cooled vertical plates. These results are obtained for various Rayleigh numbers $5\times 10^3\leq {\textit {Ra}}_H\leq 10^8$ (based on $H$ ) and the aspect ratios $L/H=1, 2, 3$ and $5$ . Furthermore, we present a novel method to extract the wind-based Reynolds number, which works particularly well with the experimental Doppler-velocimetry measurements along vertical lines, regardless of their horizontal positions. The extraction method is based on the two-dimensional autocorrelation of the time–space data of the vertical velocity.
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J. Fluid Mech. (2022), vol.932, A9, doi:10.1017/jfm.2021.977
Dynamics and length scales in vertical
convection of liquid metals
Lukas Zwirner1,, Mohammad S. Emran1, Felix Schindler2,SanjaySingh
2,
Sven Eckert2,TobiasVogt
2and Olga Shishkina1
1Max Planck Institute for Dynamics and Self-Organisation, Am Fassberg 17, 37077 Göttingen
2Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr. 400, 01328 Dresden
(Received 16 April 2021; revised 7 October 2021; accepted 1 November 2021)
Using complementary experiments and direct numerical simulations, we study turbulent
thermal convection of a liquid metal (Prandtl number Pr 0.03) in a box-shaped
container, where two opposite square sidewalls are heated/cooled. The global response
characteristics like the Nusselt number Nu and the Reynolds number Re collapse if the side
height Lis used as the length scale rather than the distance Hbetween heated and cooled
vertical plates. These results are obtained for various Rayleigh numbers 5 ×103RaH
108(based on H) and the aspect ratios L/H=1,2,3 and 5. Furthermore, we present a
novel method to extract the wind-based Reynolds number, which works particularly well
with the experimental Doppler-velocimetry measurements along vertical lines, regardless
of their horizontal positions. The extraction method is based on the two-dimensional
autocorrelation of the time–space data of the vertical velocity.
Key words: convection in cavities
1. Introduction
The understanding of turbulent thermal convection is of great importance for astrophysics,
geophysics, climate research and engineering purposes alike. Although convection has
been under investigation for centuries, even today researchers struggle to unveil its
complex nature. Usually, to study turbulent thermal convection, a model system is
considered, where the fluid is confined between horizontal plates heated from below
and cooled from above, commonly known as Rayleigh–Bénard convection (RBC)
(Bodenschatz, Pesch & Ahlers 2000; Ahlers, Grossmann & Lohse 2009; Chillà &
Schumacher 2012). In this work we study a different model system, where the fluid
Email address for correspondence: lukas.zwirner@ds.mpg.de
© The Author(s), 2021. Published by Cambridge University Press. This is an Open Access article,
distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.
org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium,
provided the original work is properly cited. 932 A9-1
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L. Zwirner and others
is confined between vertical heated/cooled plates, also known as convection inside a
differentially heated enclosure, side-heated convection or vertical convection. Throughout
this work, we will refer to this system as vertical convection (VC).
RBC and VC are two limiting cases of the more general type of inclined convection
(Daniels, Wiener & Bodenschatz 2003; Chillà et al. 2004; Sun, Xi & Xia 2005; Ahlers,
Brown & Nikolaenko 2006; Weiss & Ahlers 2013; Shishkina & Horn 2016; Zwirner &
Shishkina 2018; Zwirner et al. 2020a) and therefore are of particular importance. Note,
that VC, unlike RBC, is inherently unstable even for the smallest temperature difference
between the plates.
Batchelor (1954) was one of the first to investigate VC for very small temperature
differences, and, in the beginning, the main interest was in studying the heat transport
through double layer windows. Therefore, the studied geometry was of very large aspect
ratio Lz/H, where Lzis the height of the convection cell and His the distance between
the heated and cooled vertical plates. Fujii et al. (1970) performed a detailed experimental
study on the evolution of boundary layers (BLs) and local heat transport in VC, using
two concentric cylinders, where the inner one was heated and the outer one cooled, and
using oil and water as working fluids. Belmonte, Tilgner & Libchaber (1995) used a cubic
cell and different gases of Prandtl number Pr 0.7 for Rayleigh numbers RaH1011
and found by shadowgraph visualization a stably stratified bulk, while the temperature
fluctuations were mainly observed close to the boundaries. Additionally, they measured
a scaling of the thermal BL thickness Ra0.29. Koster, Seidel & Derebail (1997)used
X-ray radiography to measure and visualize the density distribution inside a narrow VC
cell filled with liquid gallium. This was a milestone in the visualization of liquid metal
flows, however, it is only applicable to narrow cavities and also no conclusions about
turbulent convection could be drawn. Braunsfurth et al. (1997) conducted experiments
with liquid gallium inside a long VC cells of aspect ratios L/H=1/3and1/4, for small
Grashof numbers (Ra/Pr <5×104) and compared the results with two-dimensional
numerical simulations. Here, we investigate flow at much larger Ra and large aspect ratios.
As in RBC (Shishkina 2021), in VC, the aspect ratio of the container influences the
flow in a finite fluid layer confined between two differently heated plates (Batchelor 1954;
Bejan 1980,1985,2013; Paolucci 1994). However, the effects of the container’s size and
the distance between its walls have not been investigated separately. Understanding this
phenomenon is of special significance in convection, especially in a small-Pr fluid, as both
these length scales influence the flow field. In particular, this sort of flow configuration is
typically encountered in liquid metal batteries (Kelley & Weier 2018) – a technology with
potential in grid-scale storage. In this paper, we attempt to disentangle the aspect ratio
dependency, and shift the focus onto the two predominant length scales using numerical
and experimental data across several aspect ratios. Additionally, we also present a robust
new technique which is capable of computing the Reynolds number based on the wind
velocity.
There are only a few examples of ongoing investigations of VC and in addition to
Churchill & Chu (1975); Graebel (1981); Tsuji & Nagano (1988); Chen & Pearlstein
(1989); Paolucci (1990); Versteegh & Nieuwstadt (1999); Pallares et al. (2010); Kis &
Herwig (2012)and Wang et al. (2021); there are even more studies, but most of them
use air, oil or water with relatively large Prandtl numbers, Pr 0.7. However, recently,
liquid metals with very small Prandtl numbers, Pr 1, have become a particular focus
of investigations in the thermal convection community (King & Aurnou 2013; Scheel
& Schumacher 2016; Schumacher et al. 2016; Aurnou et al. 2018; Vogt et al. 2018a;
Zwirner & Shishkina 2018; Zürner et al. 2019; Zwirner et al. 2020a; Zwirner, Tilgner &
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Vertical convection of liquid metals
Shishkina 2020b). The aim of the present work is to shed more light on VC of
low-Prandtl-number fluids.
One important step that led to a better understanding of RBC was the development
of the scaling theory by Grossmann & Lohse (2000,2002) (GLT). This theory is based
on the exact relationships between the heat transport, represented by the dimensionless
Nusselt number Nu, and the kinetic and thermal dissipation rates. The GLT assumes
various scaling regimes (Nu Raγ) depending on which part of the flow determines the
scalings of the dissipation rates: the BLs or the bulk. This theory also concludes that there
is no simple scaling law applicable for the entire range of Rayleigh numbers, but that the
exponent γchanges smoothly within these regimes. A similar approach has been applied
by Ng et al. (2015)toVC, however, the difficulty in this case is a non-closed term in the
relationship between Nu and the kinetic energy dissipation rate. Nevertheless, Ng et al.
(2015) concluded that a similar approach to the GLT is applicable to VC, when a suitable
closure model for that specific term is found.
Although one might deem the question about which scales determine the flow settled,
there is still active research on the most suitable scales. For example,Wei(2020) found
the proper scale for the Reynolds shear stress in a differentially heated vertical channel is
a mixed scale of the friction and the maximum mean velocity.
Scaling relations for heat transport and BL thicknesses in vertical convection have been
investigated in the past (Batchelor 1954; Gill 1966; Saville & Churchill 1969;Ostrach
1972). A recent approach agrees upon the same exponents for vertical convection in the
laminar regime, there the exponents were derived from BL theory and confirmed by direct
numerical simulations (DNS) in a theoretical work (Shishkina 2016).
It was found that for Pr 1 the Nusselt number and the wind-based Reynolds number
scale with respect to the Rayleigh and Prandtl numbers as
Nu Pr1/4Ra1/4and Rewind Pr1/2Ra1/2,(1.1a,b)
respectively. Note that the scaling relations are valid regardless of the length scale chosen
and therefore the indices Hand Lare omitted. However, the pre-factor depends on this
choice and in the following we show that the data collapse for different aspect ratios when
the scale is chosen appropriately.
The remaining article is structured as follows: in §2we introduce the experimental
and numerical set-up, in § 3we present a detailed comparison of the experimental and
numerical results and discuss the relevant length scale, heat and momentum transport and
finally we conclude with § 4.
2. Experimental and numerical methods
Here, we present our numerical and experimental methods. A sketch of the basic set-up
used in the experiments and the three-dimensional DNS is shown in figure 1.Weusea
rectangular cell with variable aspect ratio, L/H, where the heated and cooled plates are
squares of length L=Ly=Lz. The distance between the heated and cooled plates is H.
2.1. Numerical set-up
The incompressible Navier–Stokes equations in Oberbeck–Boussinesq approximation
Dtu=ν2up+αg(TT0)ˆz,(2.1)
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L. Zwirner and others
z
x
y
AB
H
Ly
L
z
T+T
g
Figure 1. Sketch of the VC cell with a heated wall at the left side (temperature T+) and cooled wall at the right
side (temperature T). The velocity is measured along two lines parallel to the z-direction, using the Doppler
probes Aand B. The direction of gravity is indicated by g. This sketch represents the set-up for the DNS as well
as the experiments.
DtT=κ2T,(2.2)
∇·u=0 (2.3)
are solved using the high-order finite volume code GOLDFISH (Kooij et al. 2018)in
Cartesian coordinates. Here, Dtdenotes the substantial derivative, u=(ux,uy,uz)the
velocity vector field, pis the reduced kinetic pressure, Tthe temperature, T+the
temperature of the hot vertical plate, Tthe temperature of the cold vertical plate, T0
is the mean temperature (T++T)/2, νis the kinematic viscosity and κthe thermal
diffusivity. The equations are transformed into their non-dimensional representation using
the distance Hbetween the hot and cold plates, the time tfHgHΔ)1/2and the
temperature difference ΔT+T,gis the gravitational acceleration. Thus, our VC
system depends on the following three non-dimensional input parameters:
RaHαgH3/(κν), Pr ν/κ, Γ =L/H,(2.4ac)
which are the Rayleigh number, Prandtl number and aspect ratio, respectively. All DNS
are conducted at Pr =0.03.
The computational mesh, used in the simulations, is clustered near the boundaries and
the largest simulation (RaH=108and Γ=1) has a resolution of 6743points, while the
averaging time is usually tavg150 tf. We start to collect the statistics after the flow
reaches a stationary state as indicated by the fluctuations of Nu(t)around a constant
long-time average. The number of points within the viscous BL is always Nδu4and
due to the small Prandtl number, the thermal BL is much thicker than the viscous one
and therefore both BLs are well resolved. Additionally, we conducted a mesh convergence
study and also ensure that our typical mesh distance hresolves the Kolmogorov scale η=
3u)1/4(where εuis the kinetic energy dissipation rate), i.e. hη, which represents
the smallest scales of the flow. A detailed list of the DNS resolution can be found in the
Appendix,table 1.
Important characteristic output quantities in VC are the Nusselt number, the Reynolds
number and the thermal/viscous BL thicknesses. The Nusselt number represents the heat
transport through the system and is defined as
NuHuxTκ∂x¯
TAyz
κΔ/H,(2.5)
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Vertical convection of liquid metals
where ·Ayz means averaging over a slice parallel to the hot plate and ¯· averaging over time.
The thickness of the thermal BL is defined as usual: δθH/(2Nu). For the definition of
the Reynolds number, we follow Shishkina (2016) and introduce the wind velocity in a
similar manner as
Uwind max
xuzAyz ,(2.6)
and with this the wind-based Reynolds number
Rewind =LUwind
ν.(2.7)
Furthermore, we define the thickness of the viscous BL using the slope method (Zhou &
Xia 2010)
δuUwind
xuzAyz |x=0.(2.8)
Note that, for laminar flow,δuRe1/2, and using (1.1) we conclude that δuRa1/4.
For an example of the horizontal profile uzwe refer the reader to figure 8(c,d).
2.2. Laboratory set-up
Figure 1 shows a schematic drawing of the experimental set-up. The experiments were
performed in two rectangular vessels with a square vertical cross-section of L2=
200 mm ×200 mm and a distance between the vertical heated/cooled boundaries of H=
40 and H=66 mm, which gives aspect ratios of Γ=L/H=5andΓ3, respectively.
On the two square vertical surfaces of the vessel, the heat is introduced and removed via
two copper plates, which are tempered by a circulating water bath and the temperature is
controlled via thermocouples inside the copper plates.
The maximal thermal power input is P=1500 W in this study. The other sidewalls are
madeof30mmthickPolyvinylchloride. To minimize heat losses, the whole convection
cell is wrapped in a 30 mm thick closed-cell foam, which has a thermal conductivity
of approximately 0.036 W mK1and is 660 times lower that of the liquid metal
(24 W mK1).
The vessel is filled with the eutectic alloy GaInSn that has a melting temperature of Ts=
10.5C. At room temperature, it has the density ρ=6350 kg m3, the thermal expansion
coefficient α=1.24 ×1041K
1, the thermal conductivity λ=24.05 W mK1,
the kinematic viscosity ν=3.38 ×107m2s1and the thermal diffusivity κ=
1.05 ×105m2s1(Plevachuk et al. 2014). The corresponding Prandtl number is
Pr 0.03.
For the majority of the measurements, we keep the mean fluid temperature constant
at approximately 21 C. Only for the highest Ra do we have to increase the mean fluid
temperature up to 35 C. However, the temperature dependence of the material parameters
is comparatively low, thus the Prandtl number changes only by approximately 10%
(2.97 Pr/1023.25). Therefore, we consider possible non-Oberbeck–Boussinesq
effects to have a rather weak influence on the flow.
2.3. Measuring technique
In this study, the flow velocities are measured using ultrasonic Doppler velocimetry (UDV)
which is an established measurement technique for opaque fluids like liquid metals (Aubert
et al. 2001;Britoet al. 2001; Eckert & Gerbeth 2002; Gillet et al. 2007; Nataf et al. 2008;
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L. Zwirner and others
Vogt, Räbiger & Eckert 2014; Vogt et al. 2018a,b; Zürner et al. 2019,2020; Yang, Vogt &
Eckert 2021; Vogt, Horn & Aurnou 2021a; Vogt et al. 2021b). The used UDV system is a
DOP3010 (from Signal Processing SA, Lausanne) equipped with 8 MHz transducers. The
UDV transducers send a pulsed ultrasonic signal into the liquid metal which is reflected
by microscopic particles such as oxides. The position and velocity of the particles can
be determined from the transit time of the ultrasound and the phase shift of the echo
from subsequent echo pulses. This allows us to determine the beam-parallel velocity
distribution along the ultrasonic beam (Takeda 2012). The two ultrasonic sensors used in
this study measure velocities along the vertical direction (cf. figure 1). The distances from
the sensors to the heated plate are xp/H=0.25 and 0.75 for Γ=5, and xp/H=0.15
and 0.85 for the Γ=3. The measuring system records velocity profiles with a time
resolution of approximately 0.3s. The spatial resolution is approximately 1 mm in the
beam direction and 5 mm in the lateral direction due to the diameter of the ultrasound
emitting piezoelectric transducer.
The experiment is equipped with 22 thermocouples with nine being embedded in each
copper plate to measure the temperature drop across the fluid layer. The remaining four
thermocouples are attached to the water channels that temperate the copper plates, in order
to determine the temperature change of the circulating water (Tin Tout). Together with
the measurements of the water flow rate ˙
V, this enables the calculation of the convective
heat transport, which is expressed non-dimensionally by the Nusselt number
Nu =˙
Qtot
˙
Qcond =ρcp˙
V(Tin Tout)
λL2Δ/H,(2.9)
where cpis the isobaric heat capacity of water. There are small differences between the
measurements at the hot and cold plates due to thermal losses. The Nusselt numbers
presented here are the averages of the hot and cold measurement (Γ=5), or shown
separately (Γ=3).
2.4. Extraction of the Reynolds number
One objective of this work is to compare the simulations with the experiments, and
therefore we need to define a Reynolds number based on a certain velocity that is easily
measurable in both simulations and experiments. In the experiments, only limited velocity
data are available from the UDV probes, while the simulations provide access to the
complete velocity fields. Therefore, we can directly compare the two-dimensional velocity
fields uz(t,z)only at the Doppler probe positions (xp/H=1/4and3/4). Examples of
these data are visualized in figure 2(a,c,e). There are multiple possibilities to define a
characteristic velocity based on these data. Here, we investigate two candidates in detail,
a straightforward one (the maximal velocity) and a more sophisticated one (based on the
autocorrelation) which we define in the following.
The maximal velocity is defined as
Umax max
zuz(t,z),(2.10)
and the corresponding Reynolds number
Remax
LLUmax
ν.(2.11)
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Vertical convection of liquid metals
(f)
(a)(b)
(c)(d)
(e)
z/Lζ/L
0
0.2
0.4
0
0.2
–0.2
–0.4
0.4
0
0.2
–0.2
–0.4
0.4
ζ/L
ζ/L
0
0.2
–0.2
–0.4 –4 –2
–1
02
01
4
–1.0 –0.5 0 0.5 1.0
0.4
0.6
0.8
1.0
z/L
0
0.2
0.4
0.6
0.8
1.0
z/L
0255075
t/tfτ/tf
Cuz (τ,ζ)uz/Uf
100 125 150
0.2
0.4
0.6
0.8
1.0
Figure 2. Examples of time–space plots of the vertical velocity uz(t,z)for: (a) DNS data at probe A,Γ=5,
RaL=1.25 ×107,(c) experimental data at probe B,RaL=1.05 ×107and (e) noisy experimental data at
probe B,RaL=6.94 ×106.(b,d,e) The autocorrelation function Cuz(τ, ζ ) for (a,c,e) respectively, cf. (2.12).
The dashed lines indicate the characteristic velocity Uobtained from the autocorrelation as described in § 2.4.
A more sophisticated approach is based on the two-dimensional autocorrelation of
uz(t,z), which we calculate in a discrete manner as
Cuz(τ, ζ ) =1
NMσ2
N,M
i,j
u
z(ti,zj)·u
z(tiτ,zjζ), (2.12)
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L. Zwirner and others
where u
z(t,z)uz(t,z)−uzzand σis the standard deviation of uz.Notethat,in(2.12),
the fraction in front of the sum normalizes Cuzto values between 1 and 1. Examples of
Cuz(τ, ζ ) are shown in figure 2(b,d,f). The next step is to fit a straight line,
ζ(τ) =Uτ(2.13)
through the points
i
i):Cuzi
i)=max
τ
Cuz(τ, ζi).(2.14)
The slope of this line gives the characteristic velocity U. Additionally, we restrict the
domain of the fit to the relevant points around the origin. How this extraction algorithm
works with experimental and numerical data is shown by several examples in figure 2.At
this point, we can define the Reynolds number based on Uas
Re
L=LU
ν.(2.15)
The advantage of evaluating the velocity in this way is that this method provides very
robust results even in the case of low-quality signals. Figure 2(e) shows an example of
a measurement with a very noisy signal. This is due to weak echo amplitudes caused
by an imperfect acoustic contact between the transducer and the liquid metal or an
insufficient number of reflective tracers in the measurement volume. In this case, the local
velocity values are subject to a high measurement uncertainty, in particular with increasing
distance from the sensor. However, the method proposed here uses almost all the available
information from the measurements, and by taking into account the spatio-temporal nature
of the velocity field, the deficit of poor signal quality can be essentially compensated.
3. Results and discussion
In the following, we will present the results and discuss the evidence, that the relevant
length scale in VC is based on the plate size rather than on the distance between the hot and
cold plates. Furthermore, we are going to explore the flow structures and BLs, and finally
we discuss the new method based on the autocorrelation to determine Reynolds numbers
from UDV data. A summary of the measured quantities can be found in the Appendix,
tables 2 and 3, however, this is only for reference, as all the quantities are presented in
various figures.
3.1. Global flow organization
To provide a deeper insight into the general flow organization, we show in figure 3
instantaneous flow fields at different RaLand aspect ratios Γ. We calculate the full heat
transport vector
Ω(uTκT)/(κΔ/H), (3.1)
and then find regions where its magnitude is greater or equal to the Nusselt number
(|Ω|Nu). This prominently shows the large-scale circulation and captures the major
regions, where heat is transported. Here, we choose the magnitude of the full heat flux
vector as the basis and not just the xcomponent. In this way we can capture the regions
of vertical heat transport close to the tempered walls and also the heat transport between
them. Additionally, this surface is coloured by temperature, to distinguish the hot and cold
streams of the flow. This way of visualization helps to overcome two difficulties: on the one
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Vertical convection of liquid metals
Temperature
T+
T
(a)(b)
(c)
(e)(d)
Figure 3. Instantaneous heat transport and temperature. The shown superstructures enclose regions, where
the instantaneous heat transport |Ω(t)|Nu, and they are coloured with the local temperature. Panels (ac)
are at RaH=5×104and different aspect ratios Γ=1,2,5, respectively. Panels (d,e,c) are at similar RaL=
5×106,4×106,6.25 ×106, respectively.
hand, vector fields are difficult to render in three dimensions and even streamlines appear
rather cluttered, and on the other hand, temperature isosurfaces appear rather smooth due
to the low Prandtl number and therefore hide the vigorous nature of turbulent convection
in liquid metals.
3.2. The characteristic length scale
There are two prominent length scales in a rectangular box with square hot/cold plates:
the distance Hbetween the plates and the height of the side hot/cold square plates L. One
of the questions we are going to answer in this section is: On which length scale (Hor L)
should the scaling dimensionless quantities like Ra and Re be based?
We denote quantities that are based on Lor Hwith the subscript Lor H, respectively,
and they convert as
RaL=RaHΓ3,NuL=NuHΓand ReL=ReHΓ. (3.2ac)
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L. Zwirner and others
1
2
5
10
20
RaH
N
uH
NuL
NuH RaH
−0.25
Γ = 1 (DNS)
Γ = 2 (DNS)
Γ = 3 (EXP)
Γ = 5 (DNS)
Γ = 5 (EXP)
0.10
0.15
0.20
1
2
5
10
20
104105106107108
RaL
104105106107108
104105106107108
NuL RaL
−0.25
0.12
0.14
0.16
104105106107108
(a)
(b)
Figure 4. Nusselt number, Nu, vs Rayleigh number, Ra, based on length scales: (a)distanceofhotandcold
plate Hand (b) plate length L.ForΓ=3 the Nusselt number is measured at the hot plate (red pluses) and at the
cold plate (blue pluses). The insets show the compensated Nusselt number based on a Ra1/4-scaling (indicated
by dashed lines).
A comparison of the length scales for the Nusselt number is shown in figure 4(a,b)andthe
data collapse for the length scale L, while they do not do so for length scale H. In general,
one can see this collapse for the Reynolds numbers in figure 5(a,b) and the BL thickness
in figure 6. This means, that the relevant length scale in VC is the plate size, L,rather
than the plate distance, H, for global quantities like Nu and Re. Further below, will analyse
these quantities and figures in detail.
The Nusselt number for the Γ=3 experiments is only 70 % of the value observed in
the numerical simulations. Interestingly, this appears to be a systematic difference in liquid
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Vertical convection of liquid metals
RaL
104
102
103
104
102
103
104
105106107108
RaL
104105106107108
104105106107108
(a)
(b)
Γ = 1, Rewind (DNS)
Γ = 2, Rewind (DNS)
Γ = 5, Rewind (DNS)
Γ = 5, Probe A (DNS)
Γ = 5, Probe B (DNS)
Noisy experiments
Γ = 5, Probe A (EXP)
Γ = 5, Probe B (EXP)
Γ = 3, Probe A (EXP)
Γ = 3, Probe B (EXP)
0.8
1.0
1.2
1.4
ReL
max RaL
−0.5
104105106107108
0.5
1.0
1.5
R
eL
max
ReL
ReL
RaL
−0.5
Figure 5. (a) Reynolds numbers based on the characteristic velocity Re,and(b) Reynolds numbers based on
the maximal vertical velocity vs the Rayleigh number. The data of red/blue triangles (DNS) and crosses/pluses
(experiments) are obtained by probes at position A/B, respectively. The orange background of data points
indicates noisy measurements, cf. figure 2(e). The dashed lines show the theoretical scaling Re Ra1/2
(Shishkina 2016) and the grey band represents the uncertainty margin of ±20 %. The wind-based Reynolds
numbers Rewind from DNS data are shown by grey solid symbols for different aspect ratios Γ=1(circle),2
(square) and 5 (triangle). The insets show compensated plots.
metal simulations and experiments. Zwirner et al. (2020a) compare DNS and experiments
of VC with liquid sodium as fluid inside a cylinder (Γ=1). There, the absolute
measurement of Nu also shows similar differences. Since the DNS results are in agreement
with the literature (Scheel & Schumacher 2016), we think this systematic deviation is due
to some physical effects not considered within the Navier–Stokes equations, and should be
investigated in the future.
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L. Zwirner and others
RaL
104
103
10–3
10–2
10–1
100
105106107108
108
106
104
108
106
104
δ/L
2.0
2.5
3.0
3.5
0.20
0.30
0.40
δuRaL
0.25/L
δθRa–0.25/L
δθ
δu
Figure 6. Thermal BL thickness δθ=H/(2Nu)(open symbols, black crosses) and viscous BL thickness δu
based on slope criterion (filled symbols, blue crosses, red pluses) vs the Rayleigh number. The dashed and
dash-dotted lines show the theoretical scaling law δRa0.25 for both BLs. The insets show the compensated
data. Experimental data for Γ=3 (pluses) and Γ=5 the (crosses). DNS data (circles, squares, triangles) as
in figure 4.
As next step, we take a closer look at the local flow organization, which reveals a more
complex picture. Here, we focus on the time-averaged heat transport across the central
vertical cross-section ¯
Ωx(figure 7a) and the respective profile along the vertical direction
(figure 7b). The profiles are obtained from the same simulations as the instantaneous
snapshots in figure 3.Note that the three red curves correspond to figure 3(d,e,c)and
have a similar RaL. Although, the slopes of the profiles close to the boundary are similar,
the behaviour in the bulk is qualitatively different for different aspect ratios. The smaller
aspect ratios, Γ=1 and 2, have a steep rise and a steep fall of the heat transport around its
maximum close to the wall, where the heat transport is approximately 5 times larger than
the average Nu, while for Γ=5, the rise and fall are gentle and the maximum is slightly
above the average heat transport. The shape of the heat transport profile for the Γ=5 case
shares some similarities with the profiles at smaller aspect ratios Γ=1 and 2 for similar
RaH(blue curves). Therefore, one may conclude that the local flow organization is strongly
influenced by the aspect ratio and the profiles show a rather complex behaviour. This is
further supported by the shapes of the time-averaged horizontal profiles of the temperature
and vertical velocity component (figure 8).
In figure 9 we show the vertical profiles of the thermal BL thickness at the heated
plate, for the same cases discussed in the previous paragraph. Here, one notices that,
for the same RaH, the profiles are quite different for different aspect ratios (figure 9a),
while, for similar RaL5×106,the thermal BLs are of similar structure (figure 9b). At
the top (z/L=0) and bottom (z/L=1) one can prominently notice the influence of the
adiabatic boundaries. The thermal BL grows from the bottom to top, first
slowly until z/L0.7 and then faster, being influenced by the adiabatic top
boundary.
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Vertical convection of liquid metals
z
y
0
2
4
6
z/L
Ωx (x = 0.5 H)y/Nu
Γ = 1 RaH = 5 × 104
Γ = 2 RaH = 5 × 104
Γ = 5 RaH = 5 × 104
Γ = 1 RaH = 5 × 106
Γ = 2 RaH = 5 × 105
00.1 0.2 0.3 0.4 0.5
(a)
(b)
Figure 7. (a) Cross-section of ¯
Ωx(x=0.5H)/Nu for Γ=1andRaH=5×106and (b) profiles of the local
heat transport ¯
Ωx(x=0.5H)averaged in the y-direction as functions of vertical coordinate z. For the red lines
RaL5×106and these data are collected from the same simulations as presented in figure 3(ce). All DNS
data.
0 0.1 0.2 0.3 0.4 0.50 0.1 0.2 0.3 0.4 0.5
x/H
0.1 0.2 0.3 0.4 0.5
x/H
0.1 0.2 0.3 0.4 0.5
(a)(b)
(c)(d)
uzAyz /Uf
T0
T+
T
Ayz /Δ
T0
T+
0
0.2
0.4
0.6
0
0.2
0.4
0.6
Γ = 1 RaH = 5 × 106
Γ = 2 RaH = 5 × 105
Γ = 5 RaH = 5 × 104
Γ = 1 RaH = 5 × 104
Γ = 2 RaH = 5 × 104
Γ = 5 RaH = 5 × 104
Figure 8. Time-averaged horizontal profiles of (a,b) the temperature and (c,d) the vertical velocity
component for (a,c)RaH=5×104and (b,d)RaL5×106. All DNS data.
3.3. The Reynolds number in experiments and DNS
In § 2.4, we introduced two different methods to extract the Reynolds number and
in the following we discuss the robustness of these methods and compare the results
from experiments and DNS. We need to take into account that the beam of the
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L. Zwirner and others
0.1 0.2 0.3 0.4
(a)(b)
δθ(z)/L
0.2 0.4 0.6
δθ(z)/L
0
0.2
0.4
0.6
0.8
1.0
z
/L
0
0.2
0.4
0.6
0.8
1.0
Figure 9. Profiles of the thermal BL δθ/L=−(2xT)1at the hot plate x=0averagedinthe y-direction as a
function of the vertical coordinate z. Note that this definition of δθis analogous to δθ/L=1/(2Nu),andatthe
plate heat is transported exclusively by conduction;(a)Γ=1,2,5andRaH=5×104and (b)Γ=1,2,5at
similar RaL5×106. The legend is identical to figure 7(b). All DNS data.
(a)(b)
0.1 0.2 0.3 0.4 0.5
xp/H
02468
0.98
1.00
1.02
1.04
100 Ap/H2
ReL/ReL (Ap = 0)
ReL/ReL (xp = 0.25)
0
0.5
1.0
1.5
2.0
Figure 10. Dependence of the normalized Reynolds numbers Remax
L(squares) and Re
L(circles) on (a)the
horizontal probe position and (b) the beam thickness (represented by the cross-sectional area Ap) for the DNS
at RaH=104(filled symbols), RaH=5×105(open symbols) and Γ=5.
Doppler-velocimetry probe has a certain horizontal extent of diameter less than H/4.
Analysing the DNS data, we can mimic this horizontal extent by locally averaging over
a certain area. Due to the Cartesian geometry used in the simulations, we average over a
beam with a square cross-section of area ApandasidelengthofuptoH/4 to account for
the finite extent of the Doppler-velocimetry beam.
To verify the robustness of our measurements of the vertical velocity component we
separately analyse the influence of two major aspects: (i) the horizontal position of the
probe and (ii) the thickness of the ultrasonic beam. The Reynolds number Re
Ldepends
weakly on the horizontal position xpof the probe, while Remax
Ldepends strongly on the
horizontal position of the probe (figure 10a). By changing the averaging area of the square
beam from our numerical probes in the DNS, we find that the extend of the probing beam
has a rather small effect of 2 % on the Reynolds numbers (figure 10b).
The Re
Ldata of regular and noisy experimental data agree very well (figure 5a),
but the Remax
Ldata show systematically lower values for noisy experiments (figure 5b).
This demonstrates the second advantage of the proposed method for extracting the
Reynolds number, besides the weak dependence on the probe position. Note that this weak
dependence on the probe position is quite important, as the positions in the experimental
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Vertical convection of liquid metals
set-ups for Γ=3 and 5 differ. The experimental Re
L-data collapse nicely for both aspect
ratios (figure 5a).
From the discussion above, we conclude that Re
Land thus Uis an appropriate measure
of the characteristic velocity in the system. To finalize this section, we give a physical
interpretation of U. From the definition of U(2.13)andfigure 2(a,c,e) it becomes clear
that Uquantifies the vertical advection component of the velocity fluctuations along the
line of measurement, or in other words: it is the advection velocity of the fluctuations.
3.4. The scaling relations of global Nu and Re
In Shishkina (2016) the scaling relations for heat and momentum transport in laminar
VC were derived based on similarity solutions for the BL equations. Furthermore, the
validity of this theoretical scaling was shown by DNS data for VC inside a cylindrical
domain of aspect ratio one. In figures 4,5and 6the theoretical scalings for Nu Ra1/4
and Re Ra1/2are indicated by dashed lines. These results agree well with our DNS and
experimental data.
A general remark on the uncertainty of the scaling relations: the Prandtl number in the
experiments might be slightly larger than 0.03 and a good estimate is up to 10 %, i.e. 0.033.
To ensure that this does not affect our conclusions, we estimate how it would influence the
data in the double logarithmic plots. The theoretical scaling from Shishkina (2016) agrees
well with our data, hence we use it as the basis for our analysis. The prefactor for the Nu
and Re scalings in our plots thus depends on the Prandtl number, which, for the slightly
larger Prandtl number of Pr =0.033 results in a factor of 1.024 and 1.049, respectively
(cf. (1.1)). From this one may conclude that the experimental data of Nu and Re might be
shifted by 2.4%and4.9 % towards higher Nu and Re, respectively, which would provide a
even better collapse with the DNS data. Although one should keep in mind that this is by
far not the only source of uncertainty in our data, nevertheless this estimation seems to be
a reasonable explanation of the systematic deviation between experiments and DNS in the
Nu and Re data.
One observation from the experiments is that the scaling exponents of Ra are slightly
larger for the Nusselt number in the case of Γ=5, but slightly smaller for the Reynolds
number compared with the expected scaling exponents from theory and also DNS.
The major challenge in the experiments is the measurement of very small temperature
differences, especially at small Ra. Therefore, the determination of Nu in this range
is highly error prone. From this, we noticed that the uncertainty of Nusselt number is
especially large for low Rayleigh numbers and, together with the fact that the experiments
cover just one order of magnitude in Rayleigh number, this could lead to a large uncertainty
of the scaling exponent.
The deviations of the Reynolds number in the experiments compared with the theoretical
scalings one can interpret as follows. With increasing thermal forcing, not only does the
amplitude of the velocity increase, but also the flow structure changes in the convection
cell. Thus, the location, where the maximum vertical velocity is achieved, moves towards
the tempered plates and away from the beam of the UDV measurements. Although the
measurement of Re
Lshows only a weak dependence on the probe position (especially
for Γ=5), the evaluation of the DNS data for all aspect ratios suggests that the scaling
exponent of Re
Lbecomes slightly smaller as the velocity is measured further away from
the tempered plates. The measuring position of the sensor, however, remains unchanged.
It seems to be also plausible that this effect is more strongly manifested for Γ=3
than for Γ=5. Unfortunately, in the experiments, we cannot change the sensor position
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L. Zwirner and others
ReL
DNS Γ = 1, ReL
wind
DNS Γ = 2, ReL
wind
DNS Γ = 5, ReL
wind
DNS Γ = 5, ReL
104
103
102
0.2
0.3
0.4
0.5
0.6
0.7
a = δuReL/L
Figure 11. The parameter afrom the relation δu/L=aRe (cf. Prandtl 1905) vs the Reynolds number. All
data from DNS at different aspect ratios Γfor the Reynolds number based on the wind velocity (2.7)andRe
L.
easily, but scenarios are conceivable in which the measurements at a fixed position
underestimate the increase in Re
Lwith Ra and therefore underestimates the scaling
exponent.
The thickness of the thermal and viscous BLs can be obtained from δθ/L=1/(2NuL)
and δu/L=a/ReL(Prandtl 1905), respectively. In figure 6 we show the respective BLs,
and, while the thermal BL thickness is obtained straightforwardly, the viscous BL needs
an estimation of the parameter awhich is widely accepted in case of RBC to be 0.482 for
a cylindrical cell of unit aspect ratio (Grossmann & Lohse 2002), but in general dependent
on Γ. From the DNS data, using Re
L, we can estimate this parameter for Γ=5tobe
a0.38 and then apply this value of ato the experimental data for both aspect ratios
Γ=3 and 5. Thus, we calculate the average ratio of Re
L1/2and δu, which is obtained
by (2.8). The estimate of aworks well for both aspect ratios Γ=3and5usedbythe
experiments (figure 6). This gives an estimate that the viscous BL at RaL107has a
thickness of approximately 1 mm, while the distance between the plates is H=40 mm
(Γ=5).
This procedure can also be applied to the different aspect ratios, and by using Rewind
instead of Reas the Reynolds number. A full comparison is shown in figure 11.Atfirst
glance these data show no conclusive trend, however,one needs to take into account the
above discussed uncertainty margin of ±20 % for the Reynolds numbers. With this in
mind, figure 11 is similar to the compensated plots shown as insets of figure 5(a,b)and
these data show similar scatter.
4. Conclusions
We studied VC of a liquid metal with low Prandtl number (Pr 0.03), more
precisely, the eutectic alloy GaInSn, using the complementary results from experimental
measurements and DNS. The agreement between DNS and experiments is reasonable and
deviations are within the acceptable range. Furthermore, we extensively discussed these
deviations.
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Vertical convection of liquid metals
Our study shows quantitatively, with respect to the global heat and momentum transport,
and qualitatively by the instantaneous heat transport superstructures (figure 3), that the
plate size L, rather than the distance between the plates H, is the relevant length scale in
VC.
Furthermore, we found that the bulk of the flow contributes only little to the global
heat transport. The heat is mainly transported by horizontal layers close to the top/bottom
boundary of the convection cell, while the bulk heat transport is far below the mean heat
transport or even slightly negative. These heat transporting layers have a thickness in the
range of approximately 0.1Lto 0.25 Leach.
Most importantly, we reported a novel method to extract the wind velocity
from experimental data, in particular from time–space plots of Doppler-velocimetry
measurements, using the two-dimensional autocorrelation defined in (2.12). With the
support from the DNS, we show that the Reynolds numbers obtained by our novel
method are very similar to the wind-based Reynolds numbers, while they are only weakly
dependent on the horizontal position of the probe. Additionally, we showed that even for
noisy experimental data our method delivers reasonable Reynolds numbers. Note that
this cannot be achieved in direct measurements of the velocity, since they are strongly
influenced by the probe position and lead to systematically smaller Reynolds numbers.
Further, we report a value of a0.38 to calculate the viscous BL thickness via δu=
aL/ReLfor VC.
Although the results are obtained only at a single value of Pr 0.03, one would expect
similar behaviour for fluids with Pr 1 in general. Altogether, we gained important
knowledge on the turbulent wind extraction, in particular in liquid metals. This is
important for accurate calculations of the Reynolds number from experimental data and
also because the scaling theories (Grossmann & Lohse 2000; Shishkina 2016) are built
upon the notion of wind. One challenge for further theoretical studies is that, in VC, unlike
RBC, there is a non-closed term in the exact relation between the global kinetic energy
dissipation rate and the vertical convective heat transport (Ng et al. 2015; Zwirner &
Shishkina 2018). Also, for the future, a deeper investigation of the local flow organization
is necessary, because it, unlike the global quantities, strongly depends on the aspect
ratio.
Acknowledgements. The authors acknowledge the Leibniz Supercomputing Centre (LRZ) for providing
computing time.
Funding. This work is funded by the Deutsche Forschungsgemeinschaft (DFG) under the grants Sh405/7
(Priority Programme SPP 1881 ‘Turbulent Superstructures’), VO 2331/1-1 and VO 2331/4-1.
Declaration of interests. The authors report no conflict of interest.
Author ORCIDs.
Lukas Zwirner https://orcid.org/0000-0002-8805-7292;
Sanjay Singh https://orcid.org/0000-0002-5305-7524;
Sven Eckert https://orcid.org/0000-0003-1639-5417;
Tobias Vogt https://orcid.org/0000-0002-0022-5758;
Olga Shishkina https://orcid.org/0000-0002-6773-6464.
Appendix
The following tables contain relevant experimental and numerical data.
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L. Zwirner and others
DNS resolution
L/HRa
HNxNyNzTavg/tfNδuNδθ
15×103114 114 114 25 2 7 41
11×104114 114 114 251 6 34
15×104114 114 114 25 2 4 22
11×105170 170 170 292 4 27
15×105226 226 226 335 4 25
15×106338 338 338 161 5 31
11×107450 450 450 186 5 35
11×108674 674 674 66 7 47
25×103114 142 142 326 6 41
21×104114 142 142 328 5 35
25×104114 170 17 0 5 31 5 26
21×105170 226 226 332 5 26
25×105226 338 338 205 5 26
55×103114 226 226 286 6 48
51×104114 226 226 286 6 43
55×104170 338 338 299 7 49
51×105226 450 450 320 6 49
52×105226 450 450 162 6 40
55×105226 450 450 219 5 30
Table 1. Simulation parameters: the aspect ratio (L/H), Rayleigh number (RaH), grid size (Nx×Ny×Nz),
averaging time (Tavg) and number of points in the viscous (Nδu)andthermalBLs(Nδθ). The Prandtl number is
fixed to Pr =0.03.
DNS
L/HRa
HRaLNuHRemax
LRe
LRewind δu/Hδθ/H
15×1035×1031.42 144 0.0467 0.352
11×1041×1041.75 218 0.0401 0.286
15×1045×1042.76 481 0.0250 0.181
11×1051×1053.30 659 0.0198 0.151
15×1055×1055.05 1407 0.0122 0.099
15×1065×1069.05 4391 0.0062 0.055
11×1071×10710.82 6294 0.0052 0.046
11×1081×10819.82 19 874 0.0027 0.025
25×1034×1041.42 380 0.0395 0.352
21×1048×1041.69 553 0.0333 0.296
25×1044×1052.70 1437 0.0238 0.185
21×1058×1053.35 2237 0.0212 0.149
25×1054×1064.85 4662 0.0141 0.103
55×1036.25 ×1051.20 1257 1908 1281 0.0441 0.417
51×1041.25 ×1061.35 1851 2525 1858 0.0377 0.371
55×1046.25 ×1061.99 4270 5638 4339 0.0255 0.252
51×1051.25 ×1072.44 6305 7849 6373 0.0217 0.205
52×1052.50 ×1073.03 8557 10 858 9475 0.0186 0.165
55×1056.25 ×1074.02 12 686 14 974 16 522 0.0152 0.124
Table 2. Summary of the measured quantities in the DNS for Pr =0.03: the aspect ratio (L/H), the Rayleigh
number (Ra) based on Land H, the Nusselt number (Nu), the various Reynolds numbers (Re)denedby(2.11),
(2.15)and(2.7) and the viscous and thermal BL thicknesses (δu,δθ). Note that Re
Land Remax
Lare the averages
of probes Aand B.
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Vertical convection of liquid metals
Experiments
L/HRa
HRaLNuHRemax
LRe
L
31.30 ×1043.60 ×105— 2911 2836
34.51 ×1041.25 ×106—37453542
39.29 ×1042.59 ×106— 4661 4355
32.31 ×1056.42 ×106— 7405 5488
34.72 ×1051.31 ×107—6817 6758
37.09 ×1051.97 ×107— 7489 7904
39.49 ×1052.64 ×107— 8007 8590
31.68 ×1064.68 ×107—8312 9148
31.19 ×1063.32 ×107—8755 9948
31.44 ×1064.01 ×107 9101 10 850
31.93 ×1065.36 ×107 9401 11 025
53.30 ×1044.13 ×106— 2740 3982
55.53 ×1046.91 ×1061.63 2898 4889
55.55 ×1046.94 ×1061.54 1924 5471
55.80 ×1047.26 ×1061.65 3625 5197
56.55 ×1048.19 ×1061.80 3939 5375
56.59 ×1048.24 ×1061.72 2199 5610
57.11 ×1048.89 ×10 61.69 —
58.40 ×1041.05 ×107— 4468 6236
59.97 ×1041.25 ×1072.02 4011 6419
59.99 ×1041.25 ×1071.93 2425 6015
51.46 ×1051.83 ×1072 .41 3189 7557
51.60 ×1052.00 ×1072.59 4443 7545
52.04 ×1052.55 ×1072.88 6640 8468
52.06 ×1052.57 ×1072.82 4370 8358
52.42 ×1053.03 ×1073.12 7075 9129
52.58 ×1053.23 ×1073.21 6647 9092
53.29 ×1054.12 ×1073.51 7754 9848
53.31 ×1054.14 ×10 73.41 6705 19 784
Table 3. Summary of the measured quantities in the experiments: the Rayleigh number (Ra) based on Land
H, the Nusselt number (Nu), and the two Reynolds numbers (Re)denedby(2.11)and(2.15). Note that Re
L
and Remax
Lare the averages of probes Aand B.Rowsmarkedbyinvolve noisy UDV data. The experiments are
conducted at Pr 0.03 and the aspect ratios L/H=3 and 5.
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L. Zwirner and others
Experiments Γ=3
RaHRaLNuH(hot) NuH(cold) RaHRaLNuH(hot) NuH(cold)
5.31 ×1041.43 ×1061.74 — 2.36 ×1056.38 ×106—2.31
5.41 ×1041.46 ×106—1.412.41 ×1056.51 ×1062.28 —
5.76 ×1041.55 ×106—1.322.44 ×1056.60 ×106—2.29
6.02 ×1041.62 ×1061.65 — 2.50 ×1056.75 ×106—2.31
6.11 ×1041.65 ×1061.82 — 2.58 ×1056.98 ×1062.32 —
6.15 ×1041.66 ×106—1.472.60 ×1057.02 ×1062.33 —
6.65 ×1041.79 ×1061.76 — 2.66 ×1057.18 ×1062.39 —
6.96 ×1041.88 ×106—1.492.74 ×1057.39 ×106—2.36
7.43 ×1042.01 ×106—1.592.81 ×1057.58 ×1062.37 —
7.66 ×1042.07 ×106—1.523.00 ×1058.11 ×106—2.39
7.94 ×1042.14 ×1061.81 — 3.09 ×1058.35 ×1062.43 —
8.32 ×1042.25 ×106—1.583.21 ×1058.67 ×1062.47 —
8.52 ×1042.30 ×106—1.683.53 ×1059.54 ×106—2.48
8.63 ×1042.33 ×106—1.723.62 ×1059.79 ×1062.50 —
9.22 ×1042.49 ×106—1.773.82 ×1051.03 ×1072.54 —
9.26 ×1042.50 ×106—1.754.05 ×1051.09 ×107—2.56
9.27 ×1042.50 ×1061.86 — 4.13 ×1051.12 ×1072.63 —
9.51 ×1042.57 ×106—1.704.56 ×1051.23 ×107—2.62
9.64 ×1042.60 ×106—1.814.74 ×1051.28 ×1072.69 —
9.88 ×1042.67 ×1061.57 — 4.98 ×1051.34 ×107—2.68
1.04 ×1052.82 ×106—1.835.09 ×1051.37 ×107—2.68
1.06 ×1052.86 ×1061.91 — 5.22 ×1051.41 ×1072.76 —
1.07 ×1052.89 ×1061.88 — 7.42 ×1052.00 ×107—2.90
1.07 ×1052.90 ×1061.96 — 7.71 ×1052.08 ×107—2.92
1.08 ×1052.91 ×1061.79 — 7.77 ×1052.10 ×1073.00 —
1.08 ×1052.92 ×106—1.781.00 ×1062.71 ×107—3.08
1.19 ×1053.23 ×1061.91 — 1.03 ×1062.78 ×1073.19 —
1.20 ×1053.23 ×106—1.871.04 ×1062.82 ×107—3.08
1.32 ×1053.57 ×1061.98 — 1.26 ×1063.41 ×107—3.23
1.43 ×1053.86 ×1062.03 — 1.29 ×1063.50 ×1073.34 —
1.45 ×1053.90 ×106—1.981.32 ×1063.55 ×107—3.23
1.56 ×1054.22 ×1062.06 — 1.52 ×1064.12 ×107—3.34
1.58 ×1054.25 ×1062.13 — 1.56 ×1064.21 ×1073.47 —
1.64 ×1054.42 ×106—2.171.58 ×1064.27 ×107—3.34
1.68 ×1054.53 ×1062.10 — 1.79 ×1064.82 ×107—3.44
1.68 ×1054.53 ×1062.11 — 1.85 ×1065.00 ×107—3.46
1.79 ×1054.85 ×1062.14 — 1.85 ×1065.01 ×107—3.46
1.97 ×1055.33 ×106—2.191.86 ×1065.02 ×1073.59 —
2.08 ×1055.61 ×1062.27 — 1.89 ×1065.12 ×1073.59 —
2.11 ×1055.69 ×1062.23 — 1.97 ×1065.31 ×107—3.49
2.22 ×1056.00 ×106—2.231.97 ×1065.32 ×107—3.50
2.30 ×1056.20 ×1062.27 — 2.02 ×1065.47 ×107—3.51
Table 4. Experiments with Γ=3: the Rayleigh number (Ra) based on Land H, the Nusselt number (Nu)
based on Hmeasured at the hot or cold plate.
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Vertical convection of liquid metals
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... The work devoted to the study of natural convection in liquid gallium for the crystal growth applications can be distinguished as one of the early ones [20]. In some studies on inclined convection, the position of a cylindrical container at the extreme horizontal point will also correspond to the case of vertical convection [8,9] The most complete study from the fluid mechanics point of view on vertical convection of liquid metal in a box-shaped container was published in [21]. ...
... Lx/H = 8; Ly/H = 4 -∼ 10 2 − 10 4 [19] 10 −2 < P r < 30 10 5 < Ra < 10 10 L/D= 1 ∼ 10 0 − 10 2 < 1.8 [26] 10 −3 < P r < 10 10 3 < Gr < 5 · 10 7 L/H = 10/6 -- [9] P r ∼ 0.009 Ra > 10 7 L/D = 1 ∼ 6 − 7 ∼ 10 4 [5] P r = 10 10 7 < Ra < 10 14 H/L = 1 17 − 1908 ∼ 10 1 − 10 4 [21] P r = 0.03 5 · 10 3 < Ra < 10 8 1, 2, 3 and 5. ∼ 1 − 20 ∼ 10 2 − 2 · 10 4 ...
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