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

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.

Full text

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 [1–4]; 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 [6–12]. 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 [15–19], 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 other—was found
for cylindrical cells with Γ¼1,1=2,1=3, and 1=5[20–22].
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 Earth’s 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·∇u¼−∇pþν∇2uþαgθˆ
z;ð1Þ
∂tθþu·∇θ¼κ∇2θ: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 NuðtÞ≡½huzðtÞθðtÞi −κh∂zθðtÞi=ðκΔ=HÞ
(Nusselt number) and the Reynolds number, ReðtÞ≡
Hffiffiffiffiffiffiffiffiffi
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, tf≡H= ffiffiffiffiffiffiffiffiffiffiffiffiffi
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 (z≈3=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
[20–22], 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)
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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 Tracer”filter (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,
u¼ðξ−ηÞzˆ
x−ðξþηÞxˆ
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
ðξ−ηÞ=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
q≈7=tf(Table I). The inviscid growth rate for the
aspect ratio 1=5is then approximated as σ≈0.3ξ(Fig. 1
in [36])orσ≈1=tf. However, the viscous damping is
ν=H2≈1.4×10−4=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
ziðtÞhas 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
12.40.430.1 7.80.3 990 30 55 27.00.3
21.50.240.6 5.20.3 820 20 36 25.90.3
31.30.224.0 3.80.3 720 20 27 25.70.3
41.30.45.3 3.10.3 640 30 21 24.80.3
Avg 1.60.1 5.50.3 830 20 39 26.20.3
PHYSICAL REVIEW LETTERS 125, 054502 (2020)
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In RBC, the following relationships of the energy
dissipation rates and Nu hold: hεui¼νhω2i¼
ν3H−4RaPr−2ðNu −1Þand hεθi¼κΔ2H−2Nu. 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
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