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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)

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 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)

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 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

12.40.430.1 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: 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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