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Abstract

The large-scale circulation (LSC) of fluid is one of the main concepts in turbulent thermal convection as it is known to be important in global heat and mass transport in the system. In turbulent Rayleigh-Bénard convection (RBC) in slender containers, the LSC is formed of several dynamically changing convective rolls that are stacked on top of each other. The present study reveals the following two important facts: (i) the mechanism which causes the twisting and breaking of a single-roll LSC into multiple rolls is the elliptical instability and (ii) the heat and momentum transport in RBC, represented by the Nusselt (Nu) and Reynolds (Re) numbers, is always stronger (weaker) for smaller (larger) number n of the rolls in the LSC structure. Direct numerical simulations support the findings for n=1,…,4 and the diameter-to-height aspect ratio of the cylindrical container Γ=1/5, the Prandtl number Pr=0.1 and Rayleigh number Ra=5×105. Thus, Nu and Re are, respectively, 2.5 and 1.5 times larger for a single-roll LSC (n=1) than for a LSC with n=4 rolls.
Elliptical Instability and Multiple-Roll Flow Modes of the Large-Scale Circulation
in Confined Turbulent Rayleigh-B´enard Convection
Lukas Zwirner *
Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
Andreas Tilgner
Institute for Geophysics, Georg-August University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
Olga Shishkina
Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
(Received 17 February 2020; accepted 13 July 2020; published 30 July 2020)
The large-scale circulation (LSC) of fluid is one of the main concepts in turbulent thermal convection as
it is known to be important in global heat and mass transport in the system. In turbulent Rayleigh-B´enard
convection (RBC) in slender containers, the LSC is formed of several dynamically changing convective
rolls that are stacked on top of each other. The present study reveals the following two important facts:
(i) the mechanism which causes the twisting and breaking of a single-roll LSC into multiple rolls is the
elliptical instability and (ii) the heat and momentum transport in RBC, represented by the Nusselt (Nu) and
Reynolds (Re) numbers, is always stronger (weaker) for smaller (larger) number nof the rolls in the LSC
structure. Direct numerical simulations support the findings for n¼1;;4and the diameter-to-height
aspect ratio of the cylindrical container Γ¼1=5, the Prandtl number Pr ¼0.1and Rayleigh number
Ra¼5×105. Thus, Nu and Re are, respectively, 2.5 and 1.5 times larger for a single-roll LSC (n¼1) than
for a LSC with n¼4rolls.
DOI: 10.1103/PhysRevLett.125.054502
In thermally driven turbulent flows, one of the most
prominent features is the large-scale circulation (LSC) of a
fluid, which contributes significantly to the heat and mass
transport in the system. The capability of the LSC to
transport heat and mass is influenced by its shape and rich
dynamics. Rayleigh-B´enard convection (RBC), where a
fluid is confined between a heated plate (at temperature Tþ)
from below and a cooled plate (at temperature T) from
above, is a paradigmatic system in thermal convection
studies [14]; it is characterized by the Rayleigh number,
Ra αgΔH3=ðκνÞ(thermal driving), Prandtl number,
Pr κ=ν(fluid property), and geometry of the convection
cell [5].
Although the LSC in RBC has been known for a long
time, recent investigations aim to provide a deeper under-
standing of its versatile dynamics, e.g., reversals, preces-
sion, sloshing, and twisting [612]. One particular factor
that influences the LSC is the geometry of the convection
cell. For example, for an annular cell, Xie et al. [13] found
experimentally a bifurcation between quadrupole and
dipole LSC states; the less symmetric dipole state was
shown to be less efficient in heat transport. For a more
sophisticated geometry, where the convection cell was
partitioned by several vertical walls, Bao et al. [14] found
a significant increase of the heat transport due to a
reorganization of the LSC. Several studies focused on
how lateral confinement in one direction influences the heat
transport and flow structures [1519], though only a few
studies focused on lateral confinement in two directions,
e.g., slender cylindrical cells of small diameter-to-height
aspect ratio Γ¼D=H. Not only a single-roll mode (SRM)
of the LSC, but also a double-roll mode (DRM)
composed of two rolls on top of each otherwas found
for cylindrical cells with Γ¼1,1=2,1=3, and 1=5[2022].
Experimental studies with water (Pr 5) found that the
SRM is characterized by a slightly enhanced heat transport
(0.5%) compared to the DRM [20,21]. It was also found
that small-Γsystems spend more time in the DRM than in
the SRM. Direct numerical simulations (DNS) [22] for
Ra ¼106,Pr¼0.1, and Γ¼1=5showed that the heat
transport of the DRM is only 80% compared to the SRM.
In 2D DNS [23,24], up to four vertically stacked rolls
were found for Γ¼0.4. Less heat transport was observed
in the case of more rolls and the comparison at different Pr
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PHYSICAL REVIEW LETTERS 125, 054502 (2020)
0031-9007=20=125(5)=054502(6) 054502-1 Published by the American Physical Society
revealed that the Γdependence is more pronounced at
lower Pr. It remains unclear, however, whether in 3D there
exist multiple-roll flow modes (with three or more rolls on
top of each other), what is their efficiency in heat transport
and which mechanism creates these modes.
In this Letter, we explore multiple-roll modes of the LSC
in RBC and propose the elliptical instability as a plausible
mechanism to trigger their formation [25]. This inertial
instability also plays a role, e.g., in the precession-driven
motion of the Earths core [26] and in the dynamics of
vortex pairs [27,28]. In the study [29] of the elliptical
instability under an imposed radial temperature gradient, it
was found that its growth rate decreases with increasing Ra,
but this effect is less pronounced at low Pr. In this Letter, we
also show a close relation between the LSC roll number and
the mean heat transport in the system, whereby the
strongest heat transport is provided by a single-roll LSC
structure that suffers most from the elliptical instability.
Understanding of the driving mechanisms and properties of
the LSC in turbulent RBC in low Pr fluids is an important
step towards control and optimization of turbulent heat
transport in numerous engineering applications like cool-
ants in nuclear and fusion reactors and space power
plants [30].
Numerical method.We conduct DNS using the
high-order finite-volume code
GOLDFISH
[31], which
solves the momentum and energy equations in Oberbeck-
Bousinessq approximation, for an incompressible flow
(·u¼0):
tuþu·u¼pþν2uþαgθˆ
z;ð1Þ
tθþu·θ¼κ2θ:ð2Þ
The DNS were conducted at Ra ¼5×106,Pr¼0.1,
and Γ¼1=5, using a mesh of 256 ×128 ×22 nodes in
z,φ, and rdirections, which is of sufficient resolution
[22,32]. We consider the volume-averaged instantaneous
heat transport NuðtÞ½huzðtÞθðtÞi κhzθðtÞi=ðκΔ=HÞ
(Nusselt number) and the Reynolds number, ReðtÞ
Hffiffiffiffiffiffiffiffi
hu2i
p=ν, which is based on the kinetic energy. Here
and in the following h·idenotes volume average, ¯· time
average, and h·iShorizontal area average.
Properties of different flow modes.Inside the slender
cylindrical cell of Γ¼1=5, we observe flow modes
consisting of up to n¼4distinct rolls, which are vertically
stacked [Figs. 1(a)1(d)]. These n-roll flow modes endure
for a few free fall time units, tfH= ffiffiffiffiffiffiffiffiffiffiffiffi
αgΔH
p, before they
transition into another mode. From time to time, the SRM is
strongly twisted [Fig. 1(a)], before it breaks up into two
distinct rolls [Fig. 1(b)]. Also, the rolls of the DRM may
break up into more rolls or are only twisted for a certain
time period. Whether the rolls are twisted or break up can
be distinguished by the shape of the profiles along the
cylinder axis of different horizontally averaged quantities,
in particular, of the normalized horizontal and vertical
components of the squared velocity, u2
hðt; zÞ=U2¼hu2
rþ
u2
φiS=hu·uiand u2
vðt; zÞ=U2¼hu2
ziS=hu·ui, respec-
tively, and of the temperature θðt; zÞ. Note that the profiles
are averaged over horizontal slices and depend on time t
and the vertical coordinate z. In Figs. 1(a)1(d), the profiles
are presented next to the corresponding snapshots of the
flow modes. Additionally, the enstrophy profiles, ω2
iðt; zÞ,
are shown, which will be discussed below. Note, that the
profiles u2
vðzÞand u2
hðzÞof the DRM [Fig. 1(b)] have,
respectively, a characteristic local minimum and maximum
at the junction of the rolls (z3=5H) in contrast to the
twisted SRM [Fig. 1(a)], where these extrema are absent.
Moreover, the temperature profile of the DRM shows a
characteristic steplike behavior at junction height; there, the
temperature gradient is locally increased, resembling a
thermal boundary layer. These shapes of the vertical
profiles are characteristic and independent from the number
of rolls. Based on the analysis of these profiles, we
developed an algorithm to extract the distinct rolls at
any time step and performed conditional averaging on
either each n-roll flow mode or each roll individually [33].
Note that the rolls are not necessarily equally distributed
within the cell. Thus, two smaller rolls and one larger roll
can form a three-roll mode. This is similar to the findings
for water [20], where a DRM, consisting of a larger roll and
a smaller one, was observed. Although one might expect a
five-roll mode for the aspect ratio Γ¼1=5as well, such a
mode was not observed during the simulated time interval.
However, it cannot be excluded that a five-roll mode exists,
as it is presumably a rare mode. Note that Xi and Xia [20]
also did not observe a triple-roll mode in their cell
of Γ¼1=3.
Furthermore, we examine the enstrophy ω2, which is the
squared vorticity, ω≡∇×u, and splits similarly to the
squared velocities, into the horizontal, ω2
h¼ω2
rþω2
φand
vertical, ω2
v¼ω2
zcontributions. These are normalized with
Ω2¼ hω·ωi. The horizontal component of the enstrophy
profile, ω2
hðzÞ, is strong within the region of a distinct roll
and shows a local minimum at the juncture of two rolls and
close to the cooled and heated plates [Figs. 1(a)1(d)].
On the other hand, the vertical component of the
enstrophy is approximately one order of magnitude weaker
(Table I).
As discussed above, the vertical profiles allow detection
and systematic analysis of all n-roll flow modes. One of the
primary quantities of interest in a thermally convective
system is the global heat transport (Nu). Table Ilists,
among other quantities, the Nusselt number of each flow
mode, and it shows that Nu decreases as the number of rolls
increases. This is a consistent extension of previous studies
[2022], where only SRM and DRM were observed. In
contrast to high-Pr experiments [20,34], where the differ-
ence in the heat transport between the SRM and DRM was
PHYSICAL REVIEW LETTERS 125, 054502 (2020)
054502-2
FIG. 1. Instantaneous flow fields, for a LSC composed of a different number nof rolls: (a) n¼1, (b) n¼2, (c) n¼3, (d) n¼4.
Trajectories of passive tracer particles in two perpendicular perspectives, obtained with the ParaView Particle Tracerfilter (pink for
upward and blue for downward flows), the normalized horizontally averaged profiles of the squared vertical (pink solid) and horizontal
(dashed blue) components of the velocity uiand vorticity ωiand the horizontally averaged profiles of the temperature θare shown for
each case. (e) Temporal evolution of the normalized volume-averaged heat flux NuðtÞ=Nu. The times of the snapshots (a)(d) are
marked by vertical dashed lines. Parameters are Ra ¼5×106,Pr¼0.1,Γ¼1=5.
PHYSICAL REVIEW LETTERS 125, 054502 (2020)
054502-3
only 0.5%, the decrease of Nu in the DRM is apparently
much larger (30%) at low Pr.
Besides that, the heat transport also varies strongly in
time [Fig. 1(e)], the standard deviation of Nu is 2.6 and the
distribution has a strong positive skewness (31.7), which
means a long tail at high Nu. The Reynolds number, Re,
varies less strongly with time [Fig. 1(e)]. The system is
most likely to be in a DRM (40.6%). Additionally, Table I
gives the lifetimes, τnof each flow mode. The mean
lifetime of any flow mode is approximately 2tf.
Mechanism of the mode transitions.The elliptical
instability refers to the linear instability mechanism that
arises from 2D elliptical streamlines and generates a 3D
flow [25]. In its simplest form, the elliptical instability
appears for an unbounded strained vortex in inviscid flow,
u¼ðξηÞzˆ
xðξþηÞxˆ
z, where ˆ
xand ˆ
zare the unit
vectors in xand zdirections, respectively [Fig. 2(a)]. The
strain is denoted by ηand this vortex has a constant
vorticity ω¼2ξˆ
yand is characterized by the aspect ratio
Γ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðξηÞ=ðξþηÞ
p. Since the SRM in a slender cylin-
drical cell resembles such an elliptical vortex [Fig. 1(a)],
this instability presumably triggers its break up, and thus
the emergence of the multiple-roll flow modes. Assuming
that the interior of the LSC is nearly isothermal, the stability
analysis of the LSC is identical to the stability analysis of
an elliptical vortex [35,36]. The unstable mode contains
vorticity along the zdirection. Thus, an indicator of the
elliptical instability is the growth of vorticity in the
direction orthogonal to the vorticity of the elliptical flow,
Ω. In Fig. 2(b) a snapshot of the trajectories of passive
tracer particles is shown, and a prominent azimuthal flow is
visible, which twists and/or breaks up the single-roll LSC.
A necessary requirement for the elliptical instability to
emerge is that the growth rate, σ, is much larger than the
damping rate due to viscous dissipation, which is of the
order ν=H2. To estimate σ, it is assumed that the aspect ratio
of the elliptical SRM is the same as that of the cell, hence
Γ¼1=5. The vorticity, 2ξ, of the SRM is approximated by
taking the square root of the averaged horizontal enstrophy
ffiffiffiffiffiffiffiffiffi
hω2
hi
q7=tf(Table I). The inviscid growth rate for the
aspect ratio 1=5is then approximated as σ0.3ξ(Fig. 1
in [36])orσ1=tf. However, the viscous damping is
ν=H21.4×104=tfand thus about four orders of
magnitude smaller than the growth rate. Therefore, the
elliptical instability is strong enough to grow. Furthermore,
for the studied combination of Ra and Pr, the damping
by the turbulent eddy viscosity is also not sufficiently
strong to suppress the elliptical instability, since ντ=νis
not larger than about 103, where ντis the turbulent eddy
viscosity [37,38]. This implies that the growth rate of
the elliptical instability, σ, is larger by at least an order
of magnitude than the turbulent eddy viscosity damp-
ing, ντ=H2.
FIG. 2. (a) Sketch of the primary elliptical LSC, showing the
vorticity Ωof the SRM. (b) A snapshot illustrating a strong
azimuthal motion, due to the elliptical instability (colors as in
Fig. 1). (c) Upper panel: time signals of several quantities
(indicated by the legend), χis used as a placeholder. From each
quantity χthe respective time average ¯
χis subtracted and then
normalized by its standard deviation σχ. Each signal is shifted in
time by its correlation time tcwith respect to NuðtÞ. For better
visibility the graphs are vertically shifted by steps of þ1ordered
by increasing tc[e.g., hω2
ztÞhas the largest delay with respect to
NuðtÞ]. Lower panel: the number of the rolls nas a function of
time t(without time shift, tc¼0). DNS for Ra ¼5×106,
Pr ¼0.1,Γ¼1=5.
TABLE I. Lifetimes τn, probabilities Pn, mean heat transport
Nun, mean Reynolds number Ren, horizontal hω2
hiand vertical
enstrophy hω2
vi, of the n-roll flow modes, for Ra ¼5×106,
Pr ¼0.1,Γ¼1=5.
nτn=tfPn=%NunRenhω2
hit2
fhω2
vit2
f
12.40.430.1 7.80.3 990 30 55 27.00.3
21.50.240.6 5.20.3 820 20 36 25.90.3
31.30.224.0 3.80.3 720 20 27 25.70.3
41.30.45.3 3.10.3 640 30 21 24.80.3
Avg 1.60.15.50.3 830 20 39 26.20.3
PHYSICAL REVIEW LETTERS 125, 054502 (2020)
054502-4
In RBC, the following relationships of the energy
dissipation rates and Nu hold: hεuνhω2
ν3H4RaPr2ðNu 1Þand hεθκΔ2H2Nu. Although,
these equations are only fulfilled for the time averaged
quantities, their respective time series are highly correlated
as well. An example from these time series and their shift
can be seen in Fig. 2(c). This temporal correlation also
holds, if one considers the vertical and horizontal enstrophy
components separately. Here, we calculate the correlations
in time with respect to NuðtÞto find the temporal sequence
of the underlying processes. An increase of NuðtÞis
followed by an increase of the kinetic energy or ReðtÞ
approximately 0.55tflater. After that, the thermal dissipa-
tion rate, εθ, increases (0.76tflater). Shortly after that, the
horizontal enstrophy, ω2
h, increases (0.84tflater), which
is due to the strengthening of the LSC. Finally, the vertical
enstrophy, ω2
v, increases (2tflater), which is presumably
caused by the elliptical instability. The delay of 2tfis,
compared to the mean lifetime, τn,ofan-roll flow mode
(Table I), of similar duration. Note, that the kinetic energy
dissipation rate, εu, which is the sum of the horizontal and
vertical enstrophy, has a correlation time of 1tfwhich
lies, as expected, in between the correlation times of each
component. The average time period of the fluctuations of
NuðtÞis TNu 12tf, hence, the elliptical instability arises
delayed by TNu=6. During one period, the LSC can
undergo several mode transitions. This demonstrates the
temporal interplay of the heat transport, circulation strength
and growth of the instability.
Conclusions.We found that in laterally confined tur-
bulent RBC at moderate Ra, a LSC with several rolls
stacked on top of each other can form, whereas the LSC
with more (less) rolls generally transports less (more) heat
and mass, i.e., is characterized by smaller (larger) Nu and
Re. The emergence of the multiple-roll modes as well as the
twisting and breaking of a single-roll LSC into multiple
rolls is caused by the elliptical instability.
By example of turbulent RBC, we have shown that wall-
bounded turbulent flows can consecutively take different
states and have studied the mechanism how one state
becomes unstable so that the system moves on to the next
state. Our approach is, however, much more general than
for RBC only; it can be extended to study the mechanism
that causes formation of turbulent superstructures in any
wall-bounded turbulent flows, including other paradig-
matic flows such as Taylor-Couette flows, geophysical
flows and flows in engineering applications.
We greatly appreciate valuable discussions with Detlef
Lohse and acknowledge the support by the Deutsche
Forschungsgemeinschaft (DFG) under Grant No. Sh405/
7 (SPP 1881 Turbulent Superstructures) and the Leibniz
Supercomputing Centre (LRZ).
*lukas.zwirner@ds.mpg.de
olga.shishkina@ds.mpg.de; http://www.lfpn.ds.mpg.de/
shishkina/index.html
[1] E. Bodenschatz, W. Pesch, and G. Ahlers, Annu. Rev. Fluid
Mech. 32, 709 (2000).
[2] G. Ahlers, S. Grossmann, and D. Lohse, Rev. Mod. Phys.
81, 503 (2009).
[3] D. Lohse and K.-Q. Xia, Annu. Rev. Fluid Mech. 42, 335
(2010).
[4] F. Chill`a and J. Schumacher, Eur. Phys. J. E 35, 58 (2012).
[5] Here, αis the isobaric thermal expansion coefficient, νthe
kinematic viscosity, κthe thermal diffusivity, gthe accel-
eration due to gravity, ΔTþTand Tþand Tare the
temperatures of heated and cooled plates.
[6] D. Funfschilling and G. Ahlers, Phys. Rev. Lett. 92, 194502
(2004).
[7] D. Funfschilling, E. Brown, and G. Ahlers, J. Fluid Mech.
607, 119 (2008).
[8] H.-D. Xi, S.-Q. Zhou, Q. Zhou, T. S. Chan, and K.-Q. Xia,
Phys. Rev. Lett. 102, 044503 (2009).
[9] K. Sugiyama, R. Ni, R. J. A. M. Stevens, T. S. Chan, S.-Q.
Zhou, H.-D. Xi, C. Sun, S. Grossmann, K.-Q. Xia, and D.
Lohse, Phys. Rev. Lett. 105, 034503 (2010).
[10] M. Assaf, L. Angheluta, and N. Goldenfeld, Phys. Rev. Lett.
107, 044502 (2011).
[11] K. L. Chong, Y. Yang, S.-D. Huang, J.-Q. Zhong, R. J. A.
M. Stevens, R. Verzicco, D. Lohse, and K.-Q. Xia, Phys.
Rev. Lett. 119, 064501 (2017).
[12] L. Zwirner, R. Khalilov, I. Kolesnichenko, A. Mamykin,
S. Mandrykin, A. Pavlinov, A. Shestakov, A. Teimurazov,
P. Frick, and O. Shishkina, J. Fluid Mech. 884, A18
(2020).
[13] Y.-C. Xie, G.-Y. Ding, and K.-Q. Xia, Phys. Rev. Lett. 120,
214501 (2018).
[14] Y. Bao, J. Chen, B.-F. Liu, Z.-S. She, J. Zhang, and Q. Zhou,
J. Fluid Mech. 784, R5 (2015).
[15] S. Wagner and O. Shishkina, Phys. Fluids 25, 085110
(2013).
[16] S.-D. Huang, M. Kaczorowski, R. Ni, and K.-Q. Xia, Phys.
Rev. Lett. 111, 104501 (2013).
[17] K. L. Chong and K.-Q. Xia, J. Fluid Mech. 805, R4 (2016).
[18] K. L. Chong, S.-D. Huang, M. Kaczorowski, and K.-Q. Xia,
Phys. Rev. Lett. 115, 264503 (2015).
[19] K. L. Chong, S. Wagner, M. Kaczorowski, O. Shishkina,
and K.-Q. Xia, Phys. Rev. Fluids 3, 013501 (2018).
[20] H.-D. Xi and K.-Q. Xia, Phys. Fluids 20, 055104 (2008).
[21] S. Weiss and G. Ahlers, J. Fluid Mech. 715, 314 (2013).
[22] L. Zwirner and O. Shishkina, J. Fluid Mech. 850, 984
(2018).
[23] E. P. van der Poel, R. J. A. M. Stevens, and D. Lohse, Phys.
Rev. E 84, 045303(R) (2011).
[24] E. P. van der Poel, R. J. A. M. Stevens, K. Sugiyama, and D.
Lohse, Phys. Fluids 24, 085104 (2012).
[25] R. R. Kerswell, Annu. Rev. Fluid Mech. 34, 83 (2002).
[26] S. Lorenzani and A. Tilgner, J. Fluid Mech. 492, 363
(2003).
[27] T. Leweke and C. H. K. Williamson, J. Fluid Mech. 360,85
(1998).
[28] T. Leweke, S. L. Dizes, and C. H. K. Williamson, Annu.
Rev. Fluid Mech. 48, 507 (2016).
PHYSICAL REVIEW LETTERS 125, 054502 (2020)
054502-5
[29] M. L. Bars and S. L. Dizes, J. Fluid Mech. 563, 189
(2006).
[30] P. Frick, R. Khalilov, I. Kolesnichenko, A. Mamykin, V.
Pakholkov, A. Pavlinov, and S. A. Rogozhkin, Europhys.
Lett. 109, 14002 (2015).
[31] G. L. Kooij, M. A. Botchev, E. M. Frederix, B. J.
Geurts, S. Horn, D. Lohse, E. P. van der Poel, O. Shishkina,
R. J. A. M. Stevens, and R. Verzicco, Comput. Fluids 166,1
(2018).
[32] O. Shishkina, R. J. A. M. Stevens, S. Grossmann, and D.
Lohse, New J. Phys. 12, 075022 (2010).
[33] See the Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.125.054502 for the roll
extraction algorithm.
[34] P. Wei and K.-Q. Xia, J. Fluid Mech. 720, 140 (2013).
[35] F. Waleffe, Phys. Fluids 2, 76 (1990).
[36] M. J. Landman and P. G. Saffman, Phys. Fluids 30, 2339
(1987).
[37] O. Shishkina, S. Horn, S. Wagner, and E. S. C. Ching, Phys.
Rev. Lett. 114, 114302 (2015).
[38] M. S. Emran and J. Schumacher, J. Fluid Mech. 776,96
(2015).
PHYSICAL REVIEW LETTERS 125, 054502 (2020)
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... [2,3,[15][16][17][18]. For Γ < 1, the LSC forms multiple rolls arranged on top of each other [4,[19][20][21]. The particular LSC configuration determines the magnitude of transferred heat which is quantified by the Nusselt number N u [17,21,22]. ...
... For Γ < 1, the LSC forms multiple rolls arranged on top of each other [4,[19][20][21]. The particular LSC configuration determines the magnitude of transferred heat which is quantified by the Nusselt number N u [17,21,22]. Furthermore, most theories of turbulent heat transfer [23,24] rely on the existence of a mean wind, another notion for the LSC, that provides the major fraction of kinetic energy dissipation close to the plates and allows to separate this region from the interior. ...
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... [2,3,[15][16][17][18]. For Γ < 1, the LSC forms multiple rolls arranged on top of each other [4,[19][20][21]. The particular LSC configuration determines the magnitude of transferred heat which is quantified by the Nusselt number N u [17,21,22]. ...
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The large-scale flow structure and the turbulent transfer of heat and momentum are directly measured in highly turbulent liquid metal convection experiments for Rayleigh numbers varied between $4 \times 10^5$ and $\leq 5 \times 10^9$ and Prandtl numbers of $0.025~\leq~Pr~\leq ~0.033$. Our measurements are performed in two cylindrical samples of aspect ratios $\Gamma =$ diameter/height $= 0.5$ and 1 filled with the eutectic alloy GaInSn. The reconstruction of the three-dimensional flow pattern by 17 ultrasound Doppler velocimetry sensors detecting the velocity profiles along their beamlines in different planes reveals a clear breakdown of the large-scale circulation for $\Gamma = 0.5$. As a consequence, the scaling laws for heat and momentum transfer inherit a dependence on the aspect ratio. We show that this breakdown of coherence is accompanied with a reduction of the Reynolds number $Re$. The scaling exponent $\beta$ of the power law $Nu\propto Ra^{\beta}$ crosses over from $\beta=0.221$ to 0.124 when the liquid metal flow at $\Gamma=0.5$ reaches $Ra\gtrsim 2\times 10^8$.
... The initial conditions are tailored to trigger the desired amount of Taylor rolls (N R ) necessary to maintain their aspect ratio (Γ R = N R /Γ = 1). This we do to not alter the large-scale dynamics and the transport behaviour of the system, which are known to depend on Γ R [11,[24][25][26]. The highest friction Reynolds number (Re τ = uτ d /ν) measured in all DNS is 408 (Tab. ...
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Turbulent flows exhibit wide ranges of time and length scales. Their small-scale dynamics is well understood, but less is known about the mechanisms governing the dynamics of large-scale coherent motions in the flow field. We perform direct numerical simulations of axisymmetric Taylor--Couette flow and show that beyond a critical domain size, the largest structures (turbulent Taylor rolls) undergo erratic drifts evolving on a viscous time scale. We estimate a diffusion coefficient for the drift and compare the dynamics to analogous motions in Rayleigh--B\'enard convection and Poiseuille flow. We argue that viscous processes govern the lateral displacement of large coherent structures in wall-bounded turbulent flows.
... Data for Pr ¼ 0.74 (gas N 2 ) and Pr ¼ 0.84 (gas SF 6 ) were taken using the same apparatus as in [47] but were not published there. The inset shows an enlargement at the highest Ra in normal representation for both axes (see also Supplemental Material [9], which includes [10][11][12]). ...
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... 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 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. ...
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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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We report direct numerical simulations (DNS) of the Nusselt number $Nu$ , the vertical profiles of mean temperature $\varTheta (z)$ and temperature variance $\varOmega (z)$ across the thermal boundary layer (BL) in closed turbulent Rayleigh–Bénard convection (RBC) with slippery conducting surfaces ( $z$ is the vertical distance from the bottom surface). The DNS study was conducted in three RBC samples: a three-dimensional cuboid with length $L = H$ and width $W = H/4$ ( $H$ is the sample height), and two-dimensional rectangles with aspect ratios $\varGamma \equiv L/H = 1$ and $10$ . The slip length $b$ for top and bottom plates varied from $0$ to $\infty$ . The Rayleigh numbers $Ra$ were in the range $10^{6} \leqslant Ra \leqslant 10^{10}$ and the Prandtl number $Pr$ was fixed at $4.3$ . As $b$ increases, the normalised $Nu/Nu_0$ ( $Nu_0$ is the global heat transport for $b = 0$ ) from the three samples for different $Ra$ and $\varGamma$ can be well described by the same function $Nu/Nu_0 = N_0 \tanh (b/\lambda _0) + 1$ , with $N_0 = 0.8 \pm 0.03$ . Here $\lambda _0 \equiv L/(2Nu_0)$ is the thermal boundary layer thickness for $b = 0$ . Considering the BL fluctuations for $Pr>1$ , one can derive solutions of temperature profiles $\varTheta (z)$ and $\varOmega (z)$ near the thermal BL for $b \geqslant 0$ . When $b=0$ , the solutions are equivalent to those reported by Shishkina et al. ( Phys. Rev. Lett. , vol. 114, 2015, 114302) and Wang et al. ( Phys. Rev. Fluids , vol. 1, 2016, 082301(R)), respectively, for no-slip plates. For $b > 0$ , the derived solutions are in excellent agreement with our DNS data for slippery plates.
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The fully developed turbulent Boussinesq convection is known to form large-scale rolls, often termed the 'large-scale circulation' (LSC). It is an interesting question how such a large-scale flow is created, in particular in systems when the energy input occurs at small scales, when inverse cascade is required in order to transfer energy into the large-scale modes. Here, the small-scale driving is introduced through stochastic, randomly distributed heat source (say radiational). The mean flow equations are derived by means of simplified renormalization group technique, which can be termed 'weakly nonlinear renormalization procedure' based on consideration of only the leading order terms at each step of the recursion procedure, as full renormalization in the studied anisotropic case turns out unattainable. The effective, anisotropic viscosity is obtained and it is shown, that the inverse energy cascade occurs via an effective 'motive force' which takes the form of transient negative, vertical diffusion.
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We investigate by direct numerical simulation Rayleigh-B\'enard convection in a rotating rectangular cell with rotation vector and gravity perpendicular to each other. The flow is two dimensional near the onset of convection with convection rolls aligned parallel to the rotation axis of the boundaries. At a sufficiently large Rayleigh number, the flow becomes unstable to three dimensional disturbances which changes the scaling of heat transport and kinetic energy with Rayleigh number. The mechanism leading to the instability is identified as an elliptical instability. At the transition, the Reynolds and Rossby numbers $\mathrm{Re}$ and $\mathrm{Ro}$ based on the kinetic energy of the flow are related by $\mathrm{Re} \propto \mathrm{Ro}^{-2}$ at small $\mathrm{Ro}$ with a geometry dependent prefactor.
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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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Computational codes for direct numerical simulations of Rayleigh–Bénard (RB) convection are compared in terms of computational cost and quality of the solution. As a benchmark case, RB convection at Ra=10⁸ and Pr=1 in a periodic domain, in cubic and cylindrical containers is considered. A dedicated second-order finite-difference code (AFID/RBFLOW) and a specialized fourth-order finite-volume code (GOLDFISH) are compared with a general purpose finite-volume approach (OPENFOAM) and a general purpose spectral-element code (NEK5000). Reassuringly, all codes provide predictions of the average heat transfer that converge to the same values. The computational costs, however, are found to differ considerably. The specialized codes AFID/RBFLOW and GOLDFISH are found to excel in efficiency, outperforming the general purpose flow solvers NEK5000 and OPENFOAM by an order of magnitude with an error on the Nusselt number Nu below 5%. However, we find that Nu alone is not sufficient to assess the quality of the numerical results: in fact, instantaneous snapshots of the temperature field from a near wall region obtained for deliberately under-resolved simulations using NEK5000 clearly indicate inadequate flow resolution even when Nu is converged. Overall, dedicated special purpose codes for RB convection are found to be more efficient than general purpose codes.
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Many natural and engineering systems are simultaneously subjected to a driving force and a stabilizing force. The interplay between the two forces, especially for highly nonlinear systems such as fluid flow, often results in surprising features. Here we reveal such features in three different types of Rayleigh-B\'enard (RB) convection, i.e. buoyancy-driven flow with the fluid density being affected by a scalar field. In the three cases different {\it stabilizing forces} are considered, namely (i) horizontal confinement, (ii) rotation around a vertical axis, and (iii) a second stabilizing scalar field. Despite the very different nature of the stabilizing forces and the corresponding equations of motion, at moderate strength we counterintuitively but consistently observe an {\it enhancement} in the flux, even though the flow motion is weaker than the original RB flow. The flux enhancement occurs in an intermediate regime in which the stabilizing force is strong enough to alter the flow structures in the bulk to a more organised morphology, yet not too strong to severely suppress the flow motions. Near the optimal transport enhancements all three systems exhibit a transition from a state in which the thermal boundary layer (BL) is nested inside the momentum BL to the one with the thermal BL being thicker than the momentum BL.
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We study the effect of severe geometrical confinement in Rayleigh–Bénard convection with a wide range of width-to-height aspect ratio Γ , 1/128 Γ 1, and Rayleigh number Ra, 3 × 10 4 Ra 1 × 10 11 , at a fixed Prandtl number of Pr = 4.38 by means of direct numerical simulations in Cartesian geometry with no-slip walls. For convection under geometrical confinement (decreasing Γ from 1), three regimes can be recognized (Chong et al., Phys. Rev. Lett., vol. 115, 2015, 264503) based on the global and local properties in terms of heat transport, plume morphology and flow structures. These are Regime I: classical boundary-layer-controlled regime; Regime II: plume-controlled regime; and Regime III: severely confined regime. The study reveals that the transition into Regime III leads to totally different heat and momentum transport scalings and flow topology from the classical regime. The convective heat transfer scaling, in terms of the Nusselt number Nu, exhibits the scaling Nu − 1 ∼ Ra 0.61 over three decades of Ra at Γ = 1/128, which contrasts sharply with the classical scaling Nu − 1 ∼ Ra 0.31 found at Γ = 1. The flow in Regime III is found to be dominated by finger-like, long-lived plume columns, again in sharp contrast with the mushroom-like, fragmented thermal plumes typically observed in the classical regime. Moreover, we identify a Rayleigh number for regime transition, Ra * = (29.37/Γ) 3.23 , such that the scaling transition in Nu and Re can be clearly demonstrated when plotted against Ra/Ra * .
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This article reviews the characteristics and behavior of counter-rotating and corotating vortex pairs, which are seemingly simple flow configurations yet immensely rich in phenomena. Since the reviews in this journal by Widnall (1975) and Spalart (1998), who studied the fundamental structure and dynamics of vortices and airplane trailing vortices, respectively, there have been many analytical, computational, and experimental studies of vortex pair flows. We discuss two-dimensional dynamics, including the merging of same-sign vortices and the interaction with the mutually induced strain, as well as three-dimensional displacement and core instabilities resulting from this interaction. Flows subject to combined instabilities are also considered, in particular the impingement of opposite-sign vortices on a ground plane. We emphasize the physical mechanisms responsible for the flow phenomena and clearly present the key results that are useful to the reader for predicting the dynamics and instabilities of parallel vortices.
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Coherent structures are ubiquitous in turbulent flows and play a key role in transport. The most important coherent structures in thermal turbulence are plumes. Despite being the primary heat carriers, the potential of manipulating thermal plumes to transport more heat has been overlooked so far. Unlike some other forms of energy transport, such as electromagnetic or sound waves, heat flow in fluids is generally difficult to manipulate, as it is associated with the random motion of molecules and atoms. Here we report how a simple geometrical confinement can lead to the condensation of elementary plumes. The result is the formation of highly coherent system-sized plumes and the emergence of a new regime of convective thermal turbulence characterized by universal temperature profiles and significantly enhanced heat transfer. It is also found that the universality of the temperature profiles and heat transport originate from the geometrical properties of the coherent structures, i.e., the thermal plumes. Therefore, in contrast to the classical regime, boundary layers in this plume-controlled regime are being controlled, rather than controlling.
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Enhancement of heat transport across a fluid layer is of fundamental interest as well as great technological importance. For decades, Rayleigh–Bénard convection has been a paradigm for the study of convective heat transport, and how to improve its overall heat-transfer efficiency is still an open question. Here, we report an experimental and numerical study that reveals a novel mechanism that leads to much enhanced heat transport. When vertical partitions are inserted into a convection cell with thin gaps left open between the partition walls and the cooling/heating plates, it is found that the convective flow becomes self-organized and more coherent, leading to an unprecedented heat-transport enhancement. In particular, our experiments show that with six partition walls inserted, the heat flux can be increased by approximately 30 %. Numerical simulations show a remarkable heat-flux enhancement of up to 2.3 times (with 28 partition walls) that without any partitions.
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Any tilt of a Rayleigh–Bénard convection cell against gravity changes the global flow structure inside the cell, which leads to a change of the heat and momentum transport. Especially sensitive to the inclination angle is the heat transport in low-Prandtl-number fluids and confined geometries. The purpose of the present work is to investigate the global flow structure and its influence on the global heat transport in inclined convection in a cylindrical container of diameter-to-height aspect ratio $\unicode[STIX]{x1D6E4}=1/5$ . The study is based on direct numerical simulations where two different Prandtl numbers $Pr=0.1$ and 1.0 are considered, while the Rayleigh number, $Ra$ , ranges from $10^{6}$ to $10^{9}$ . For each combination of $Ra$ and $Pr$ , the inclination angle is varied between 0 and $\unicode[STIX]{x03C0}/2$ . An optimal inclination angle of the convection cell, which provides the maximal global heat transport, is determined. For inclined convection we observe the formation of two system-sized plume columns, a hot and a cold one, that impinge on the opposite boundary layers. These are related to a strong increase in the heat transport.