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Elliptical Instability and Multiple-Roll Flow Modes of the Large-Scale Circulation in Confined Turbulent Rayleigh-Bénard Convection



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)
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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
[31], which
solves the momentum and energy equations in Oberbeck-
Bousinessq approximation, for an incompressible flow
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
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
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
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
vertical, ω2
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)
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)
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,
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
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
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.
12.40.430.1 7.80.3 990 30 55 27.00.3 5.20.3 820 20 36 25.90.3 3.80.3 720 20 27 25.70.3 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)
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).
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PHYSICAL REVIEW LETTERS 125, 054502 (2020)
... Several numerical and experimental studies report that a decrease in aspect ratio Γ below 1 can noticeably degrade the stability of the LSC (Xi & Xia 2007Weiss The 3-D flow structures in turbulent RBC at low Pr = 0.03 & Ahlers 2011;Zwirner, Tilgner & Shishkina 2020;Paolillo et al. 2021;Schindler et al. 2022). Moreover, the flow structure in slender convection cells (Γ < 1) reveals the existence of configurations with several rolls on top of each other. ...
... Later experiments performed in cylinders with aspect ratios of 0.5 and 0.33, respectively, confirmed the robust existence of a DRS regime or even a three-roll structure (TRS) (Xi & Xia 2008;Weiss & Ahlers 2011). Zwirner et al. (2020) suggest the elliptical instability as the mechanism for the breaking up of the SRS followed by the transition to a DRS. They also found that the heat and momentum transport of the flow deteriorates as the number of rolls increases. ...
... However, the use of t to only makes sense if the stability of the SRS is guaranteed at least during this time scale, about which the rapid changes of Re in figure 12(b) raise some doubts. The reconstructions of the 3-D flow structures that we present in § 6.4 will confirm the rapid changes in the flow structure and also show that the shape and size of the SRS is also subject to significant changes even during 974 A48-20 Thus, our measurements at small Pr confirm previous observations that the change from a Γ = 1 cell to smaller aspect ratios destabilizes the SRS-LSC, deteriorates the coherence of the large-scale structures and leads to a much more dynamic behaviour of the flow that apparently results from frequent transitions between different roll states (Xi & Xia 2007Weiss & Ahlers 2011;Xi et al. 2016;Zwirner et al. 2020;Schindler et al. 2022). Numerical simulations by Zwirner et al. (2020) in a Γ = 0.2 cylinder for Ra = 5 × 10 6 and Pr = 0.1 show significant fluctuations of Nu and Re occurring within a few free-fall times. ...
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In this paper we report on an experimental study focusing on the manifestation and dynamics of the large-scale circulation (LSC) in turbulent liquid metal convection. The experiments are performed inside a cylinder of aspect ratio $\varGamma = 0.5$ filled with the ternary alloy GaInSn, which has a Prandtl number of $Pr = 0.03$ . The large-scale flow structures are classified and characterized at Rayleigh numbers of ${Ra} = 9.33 \times 10^6, 5.31 \times 10^7$ and $6.02 \times 10^8$ by means of the contactless inductive flow tomography which enables the full reconstruction of the three-dimensional (3-D) flow structures in the entire convection cell. This is complemented with the multi-thermal-probe method for capturing the azimuthal temperature variation induced by the LSC at the sidewall. We use proper orthogonal decomposition (POD) to identify the dominating modes of the turbulent convection. The analysis reveals that a single-roll structure of the LSC alternates in short succession with double-roll structures or a three-roll structure. This is accompanied by dramatic fluctuations of the Reynolds number, whose instantaneous values can deviate by more than 50 % from the time-average value. No coherent oscillations are observed, whereas a correlation analysis indicates a residual contribution of the torsion and sloshing modes. Results of the POD analysis suggest a stabilization of the single-roll LSC with increasing $Ra$ at the expense of flow structures with multiple rolls. Moreover, the relative lifetime of all identified flow states, measured in units of free-fall times, increases with rising $Ra$ .
... From numerical simulations, based on the 2D and 3D time-resolved data of RBC it has been found that the temporal evolution of the vortical flow structures is correlated to the evolution of heat and momentum transport (Zwirner, Tilgner, & Shishkina, 2020), (Kashanj & Nobes, 2021b). In another work, the knowledge of the temperature distribution of RBC has led to design of a new DNA 20th LISBON Laser Symposium 2022 polymerase chain reaction (PCR) (Khodakov, Li, Zhang, & Zhang, 2021). ...
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Experimental study of Rayleigh-Benard convection (RBC) requires 3D measurement of the velocity and temperature due to the complexity of the physics of this system. Using a scanning system, the application of the 3D particle image velocimetry (PTV) and 3D time-resolved two-color laser-induced fluorescence (LIF) is investigated on a slender RBC cell. Calculation of the out-of-plane velocity component by using the planar velocity components and applying the continuity law is discussed. From these observations that calculation of the dimensionless heat transfer coefficient, the Nusselt number from the 3D reconstructed temperature field close to the boundary of the fluid domain is investigated.
... All the way up to the macroscopic deterministic level, turbulent flows in confined geometries or extended layers can exhibit differently ordered large-scale spatial patterns that are visited for longer transients in a long-term evolution [3][4][5][6]. The (rapid) crossover from one configuration to another is triggered by fluctuations of secondary flow structures, smaller eddies, shear layers, or plumes that can affect the turbulent transport of heat or momentum [7][8][9][10][11][12][13]. The state or phase space of macroscopic flows is infinite-dimensional or at least extremely high-dimensional and requires drastic dimensionality reductions to model the observed large-scale dynamics effectively [14]. ...
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The dynamics in a confined turbulent convection flow is dominated by multiple long-lived macroscopic circulation states that are visited subsequently by the system in a Markov-type hopping process. In the present work, we analyze the short transition paths between these subsequent macroscopic system states by a data-driven learning algorithm that extracts the low-dimensional transition manifold and the related new coordinates, which we term collective variables, in the state space of the complex turbulent flow. We therefore transfer and extend concepts for conformation transitions in stochastic microscopic systems, such as in the dynamics of macromolecules, to a deterministic macroscopic flow. Our analysis is based on long-term direct numerical simulation trajectories of turbulent convection in a closed cubic cell at a Prandtl number Pr=0.7 and Rayleigh numbers Ra=106 and 107 for a time lag of 105 convective free-fall time units. The simulations resolve vortices and plumes of all physically relevant scales, resulting in a state space spanned by more than 3.5 million degrees of freedom. The transition dynamics between the large-scale circulation states can be captured by the transition manifold analysis with only two collective variables, which implies a reduction of the data dimension by a factor of more than a million. Our method demonstrates that cessations and subsequent reversals of the large-scale flow are unlikely in the present setup, and thus it paves the way for the development of efficient reduced-order models of the macroscopic complex nonlinear dynamical system.
... Similarly, the total kinetic energy dissipation ε u is proportional to Ra 1.55 in cylinders [71], to Ra 1.33 in 2D square enclosures [62], and to Ra 1.3 in a cubical boxes [72]. Recently, Zwirner et al. [73] have shown the dissipation is high for the monopole compared to the multiroll state in cylindrical enclosures. In Fig. 16, we have shown time and area-averaged thermal and kinetic energy dissipations as functions of Ra. ...
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Partitions are an essential part of industrial reactors and thermal management devices whose primary purpose is to increase transport rates by obstructing flow in one direction and promoting in the other via secondary and small-scale motions. Inspired by such applications, we have investigated thermal convection in a two-dimensional square enclosure heated at the bottom and cooled at the top, with four additional thin vertical partitions arranged parallel to facilitate organized plume motions in the range of Rayleigh numbers 106 to 109. The large-scale classical circulation observed in thermal convection breaks down into many roll configurations based on the constriction gap (S) between the partitions and the conduction walls. Due to their arrangement, we observed increased plume ejection, impact, and shear near the conduction walls when the partitions disturb the thermal boundary layers. The plume ejection and impact on either end of the constriction gap sets a pressure-driven forced convection on the conduction wall, thus increasing overall heat transport by at least an order of magnitude. We found the maximum heat transport when 0.2δRB<S<0.4δRB, where δRB is the time-averaged thermal boundary layer thickness in classical thermal convection. Using both the numerical simulations and a simple control volume-based analysis, we have estimated that the heat transport increases as S3 for small constriction gaps and as an inverse power of S for the large gap limit. With the help of energy dissipation, we have concluded that increasing plume intensity near the conduction walls leads to the observed high heat transport.
A typical feature of thermal convection is the formation of large-scale flow (LSF) structures of the order of system size. How this structure affects global heat transport is an important issue in the study of thermal convection. We present an experimental study of the coupling between the flow structure and heat transport in liquid metal convection with different degrees of spatial confinement, characterized by the aspect ratio $\varGamma$ of the convection cell. Combining measurements in two convection cells with $\varGamma =1.0$ and 0.5, the study shows that a large-scale circulation (LSC) transports ${\sim }35\,\%$ more heat than a twisted LSC. It is further found that when the LSF is in the form of the LSC state, the system is in a fully developed turbulence state with a $Nu\sim Ra^{0.29}$ scaling for the heat transport. However, the twisted LSC state with a heat transport scaling of $Nu\sim Ra^{0.37}$ appears when the system is not in the fully developed turbulence state. Bistability is observed when the system evolves from the twisted-LSC-dominated to the LSC-dominated state.
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Our understanding of the material organization of complex fluid flows has benefited recently from mathematical developments in the theory of objective coherent structures. These methods have provided a wealth of approaches that identify transport barriers in three-dimensional (3-D) turbulent flows. Specifically, theoretical advances have been incorporated into numerical algorithms that extract the most influential advective, diffusive and active barriers to transport from data sets in a frame-indifferent fashion. To date, however, there has been very limited investigation into these objectively defined transport barriers in 3-D unsteady flows with complicated spatiotemporal dynamics. Similarly, no systematic comparison of advective, diffusive and active barriers has been carried out in a 3-D flow with both thermally driven and mechanically modified structures. In our study, we utilize simulations of turbulent rotating Rayleigh-Bénard convection to uncover the interplay between advective transport barriers (Lagrangian coherent structures), material barriers to diffusive heat transport, and objective Eulerian barriers to momentum transport. For a range of (inverse) Rossby numbers, we identify each type of barrier and find intriguing relationships between momentum and heat transport that can be related to changes in the relative influence of mechanical and thermal forces. Further connections between bulk behaviours and structure-specific behaviours are also developed.
We conduct an experimental study on the flow structures and dynamics of turbulent Rayleigh-Bénard convection in an annular cell with radius ratio η≃0.5 and aspect ratio Γ≃4. The working fluid is water with a Prandtl number of Pr≃5.4, and the Rayleigh number (Ra) ranges from 5.05×107 to 5.05×108. The multithermal-probe method and the particle image velocimetry technique are employed to measure the temperature profiles and the velocity fields, respectively. Two distinct states with multiroll standing waves are observed, which are the quadrupole state (QS) characterized by a four-roll structure and the sextupole state (SS) by a six-roll structure. The scaling exponents of Reynolds number Re with Ra are different for the two states, which are 0.56 for QS and 0.41 for SS. In addition, the standing waves become unstable upon tilting the cell by 1∘ in relation to the horizontal plane, and they evolve into traveling waves. At relatively high Ra, for instance, Ra⩾2.55×108, it is observed that the traveling wave state SS undergoes a transition to the traveling wave state QS. However, the opposite transition from QS to SS is not observed in our experiments. Our findings provide insights into the flow structures and dynamics in the convection flow with rotation symmetry.
We seek to develop a low-dimensional model for the interactions between horizontally adjacent turbulent convection rolls. This was tested in Rayleigh-Bénard convection experiments with two adjacent cubic cells with a partial wall in between. Observed stable states include both counterrotating and corotating states for Rayleigh number 7.6×107< Ra <3.5×109 and Prandtl number 6.41. The stability of each of these states and their dynamics can be modeled low-dimensionally by stochastic ordinary differential equations of motion in terms of the orientation, amplitude, and mean temperature of each convection roll. The form of the interaction terms is predicted based on an effective turbulent diffusion of temperature between the adjacent rolls, which is projected onto the neighboring rolls with sinusoidal temperature profiles. With measurements of a constant coefficient for effective thermal turbulent diffusion, quantitative predictions are made for the nine forcing terms which affect stable fixed points of the corotating and counterrotating states for 5.5×108< Ra <3.5×109. Predictions are found to be accurate within a factor of 3 of experiments. This suggests that the same turbulent thermal diffusivity that describes macroscopically averaged heat transport also controls the interactions between neighboring convection rolls. The surprising stability of corotating states is due to the temperature difference between the neighboring rolls becoming large enough that the heat flux between the rolls stabilizes the temperature profile of aligned corotating states. This temperature difference can be driven with an asymmetry, for example, by heating the plates of the two cells to different mean temperatures. When such an asymmetry is introduced, it also shifts the orientations of the rolls of counterrotating states in opposite directions away from their preferred orientation, which is otherwise due to the geometry of the cell. As the temperature difference between the plates of the different cells is increased, the shift in orientation increases until the counterrotating states become unstable and only corotating states are stable. At very large temperature differences between cells, both the counterrotating and predicted corotating states become unstable; instead we observe a corotating state with much larger temperature difference between the rolls that cannot be explained by turbulent thermal diffusion. Spontaneous switching between corotating and counterrotating states is also observed, including in nominally symmetric systems. Switching to counterrotating states occurs mainly due to cessation (a significant weakening of a convection roll), which reduces damping on changes in orientation, allowing the orientation to change rapidly due to diffusive fluctuations. Switching to corotating states is mainly driven by smaller diffusive fluctuations in the orientation, amplitude, and mean temperature of rolls, which have a positive feedback that destabilizes the counterrotating state.
Magneto-gravity-elliptic instability is addressed here considering an unbounded strained vortex (with constant vorticity $2\varOmega$ and with ellipticity parameter $\varepsilon$ ) of a perfectly conducting fluid subjected to a uniform axial magnetic field (with Alfvén velocity scaled from the basic magnetic field $B)$ and an axial stratification (with a constant Brunt–Väisälä frequency $N$ ). Such a simple model allows us to formulate the stability problem as a system of equations for disturbances in terms of Lagrangian Fourier (or Kelvin) modes which is universal for wavelengths of the perturbation sufficiently small with respect to the scale of variation of the basic velocity gradients. It can model localised patches of elliptic streamlines which often appear in some astrophysical flows (stars, planets and accretion discs) that are tidally deformed through gravitational interaction with other bodies. In the limit case where the flow streamlines are exactly circular ( $\varepsilon =0),$ there are fast and slow magneto-inertia-gravity waves with frequencies $\omega _{1,2}$ and $\omega _{3,4},$ respectively. Under the effect of finite ellipticity, the resonant cases of these waves, $\omega _i-\omega _j=n\varOmega$ $(i\ne j)$ ( $n$ being an integer), can become destabilising. The maximal growth rate of the subharmonic instability (related to the resonance of order $n=2)$ is determined by extending the asymptotic method by Lebovitz & Zweibel ( Astrophys. J. , vol. 609, 2004, pp. 301–312). The domains of the $(k_0B/\varOmega, N/\varOmega )$ plane for which this instability operates are identified ( $1/k_0$ being a characteristic length scale). We demonstrate that the $N\rightarrow 0$ limit is, in fact, singular (discontinuous). The axial stable stratification enhances the subharmonic instability related to the resonance between two slow modes because, at large magnetic field strengths, its maximal growth rate is twice that found in the case without stratification.
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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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We report an experimental observation of a flow topology transition via global bifurcation in a turbulent Rayleigh-Bénard convection. This transition corresponds to a spontaneous symmetry breaking with the flow becomes more turbulent. Simultaneous measurements of the large-scale flow (LSF) structure and the heat transport show that the LSF bifurcates from a high heat transport efficiency quadrupole state to a less symmetric dipole state with a lower heat transport efficiency. In the transition zone, the system switches spontaneously and stochastically between the two long-lived metastable states.
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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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We study, using direct numerical simulations, the effect of geometrical confinement on heat transport and flow structure in Rayleigh-B\'enard convection in fluids with different Prandtl numbers. Our simulations span over two decades of Prandtl number $Pr$, $0.1 \leq Pr \leq 40$, with the Rayleigh number $Ra$ fixed at $10^8$. The width-to-height aspect ratio $\Gamma$ spans between $0.025$ and $0.25$ while the length-to-height aspect ratio is fixed at one. We first find that for $Pr \geq 0.5$, geometrical confinement can lead to a significant enhancement in heat transport as characterized by the Nusselt number $Nu$. For those cases, $Nu$ is maximal at a certain $\Gamma = \Gamma_{opt}$. It is found that $\Gamma_{opt}$ exhibits a power-law relation with $Pr$ as $\Gamma_{opt}=0.11Pr^{-0.06}$, and the maximal relative enhancement generally increases with $Pr$ over the explored parameter range. As opposed to the situation of $Pr \geq 0.5$, confinement-induced enhancement in $Nu$ is not realized for smaller values of $Pr$, such as $0.1$ and $0.2$. The $Pr$ dependence of the heat transport enhancement can be understood in its relation to the coverage area of the thermal plumes over the thermal boundary layer (BL) where larger coverage is observed for larger $Pr$ due to a smaller thermal diffusivity. We further show that $\Gamma_{opt}$ is closely related to the crossing of thermal and momentum BLs, and find that $Nu$ declines sharply when the thickness ratio of the thermal and momentum BLs exceeds a certain value of about one. In addition, through examining the temporally averaged flow fields and 2D mode decomposition, it is found that for smaller $Pr$ the large-scale circulation is robust against the geometrical confinement of the convection cell.
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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.
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.