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Periodically Modulated Thermal Convection.

TL;DR: In this paper, the influence of temporal modulations on global transport properties has hardly been studied, and the authors performed numerical simulations of Rayleigh-B\'enard convection with time periodic modulation in the temperature boundary condition and reported how this modulation can lead to a significant heat flux (Nusselt number Nu) enhancement.
Abstract: Many natural and industrial turbulent flows are subjected to time-dependent boundary conditions. Despite being ubiquitous, the influence of temporal modulations (with frequency $f$) on global transport properties has hardly been studied. Here, we perform numerical simulations of Rayleigh-B\'enard convection with time periodic modulation in the temperature boundary condition and report how this modulation can lead to a significant heat flux (Nusselt number Nu) enhancement. Using the concept of Stokes thermal boundary layer, we can explain the onset frequency of the Nu enhancement and the optimal frequency at which Nu is maximal, and how they depend on the Rayleigh number Ra and Prandtl number Pr. From this, we construct a phase diagram in the 3D parameter space ($f$, Ra, Pr) and identify the following: (i) a regime where the modulation is too fast to affect Nu; (ii) a moderate modulation regime, where Nu increases with decreasing $f$, and (iii) slow modulation regime, where Nu decreases with further decreasing $f$. Our findings provide a framework to study other types of turbulent flows with time-dependent forcing.
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Periodically Modulated Thermal Convection
Rui Yang ,
1,2,
Kai Leong Chong ,
1,
Qi Wang ,
1,3
Roberto Verzicco ,
1,4,5
Olga Shishkina ,
2
and Detlef Lohse
1,2,*
1
Physics of Fluids Group, Max Planck Center for Complex Fluid Dynamics, MESA+ Institute and J.M.Burgers Center for Fluid
Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands
2
Max Planck Institute for Dynamics and Self-Organisation, Am Fassberg 17, 37077 Göttingen, Germany
3
Department of Modern Mechanics, University of Scie nce and Tech nology of China, Hefei 230027, China
4
Dipartimento di Ingegneria Industriale, University of Rome Tor Vergata, Via del Politecnico 1, Roma 00133, Italy
5
Gran Sasso Science InstituteViale F. Crispi, 7, 67100 LAquila, Italy
(Received 29 April 2020; accepted 9 September 2020; published 9 October 2020)
Many natural and industrial turbulent flows are subjected to time-dependent boundary conditions .
Despite being ubiquitous, the influence of temporal modulations (with frequency f) on global transport
properties has hardly been studied. Here, we perform numerical simulations of Rayleigh-B´enard
convection with time periodic modulation in the temperature boundary condition and report how this
modulation can lead to a significant heat flux (Nusselt number Nu) enhancement. Using the concept of
Stokes thermal boundary layer, we can explain the onset frequency of the Nu enhancement and the optimal
frequency at which Nu is maximal, and how they depend on the Rayleigh number Ra and Prandtl number
Pr. From this, we construct a phase diagram in the 3D parameter space (f, Ra, Pr) and identify the
following: (i) a regime where the modulation is too fast to affect Nu; (ii) a moderate modulation regime,
where Nu increases with decreasing f, and (iii) slow modulation regime, where Nu decreases with further
decreasing f. Our findings provide a framework to study other types of turbulent flows with time-
dependent forcing.
DOI: 10.1103/PhysRevLett.125.154502
Turbulent flows driven by time-dependent forcing are
common in nature and industrial applications [1,2].For
example, Earths atmosphere circulation is driven by
periodical heating from solar radiation, the ocean tidal
current by periodical gravitational attractions from both the
Moon and the Sun, and the blood circulation by the
beating heart.
In periodically driven turbulence in shear flows, a mean-
field theory has been used to analyze the resonance maxima
of the Reynolds number [3,4]. Periodic forcing in other
turbulent systems, for example, in the homogeneous
isotropic turbulence [58], pipe flow [912], channel flow
[13,14], Taylor-Couette flow [15,16], and Rayleigh-B´enard
(RB) convection [1719], is also shown to have highly
nontrivial response properties.
Here we picked turbulent Rayleigh-B´enard convection as
a model system to study how time periodic modulation of
temperature boundary condition influences global heat
transport. The RB system, consisting of a fluid layer heated
from below and cooled from above, has been extensively
studied as the paradigmatic and well-defined system for
convective thermal turbulence [2022]. Also, several
modulation methods have been studied for that system,
such as bottom temperature modulation [17,19,23], rotation
modulation [18,24], and gravity modulation [25,26].
Intuitively, one may expect that the modulation effect on
time-averaged global quantities is limited because the net
force averaged over a cycle vanishes. Indeed, with bottom
temperature modulation in experiments, only a small
enhancement (7%) of the heat flux has been observed
so far [17,19]. However, in those experiments, the effects of
modulation in temperature have not yet been fully explored
because of the experimental challenge in having a broad
range of modulation frequency due to thermal inertia of the
plates. Note that also in numerical simulations thermal
inertial can straightforwardly be treated [27,28], but in this
study, for conceptional clarity, we keep the problems of
thermal transport in the RB cell and in the plates disen-
tangled and assume perfect conductivity of the plates.
In this Letter, we numerically study modulated RB
convection within a wide range (more than 4 orders of
magnitude) of modulation frequency at the bottom plate
temperature and observe a significant (25%) enhance-
ment in heat transport. To explain our findings, we show the
relevance of the Stokes thermal boundary layer (BL), which
is analogous to the classical one for an oscillating plate
[29], in determining the transitional frequency for the heat
transport enhancement and the optimal frequency for the
maximal heat transport. In particular, we calculate the
transition between the different regimes in phase space and
show how they depend on the Rayleigh and Prandtl
numbers, which represent the ratios between buoyancy
and viscosity and between momentum diffusivity and
thermal diffusivity, respectively. Our modulation method
PHYSICAL REVIEW LETTERS 125, 154502 (2020)
0031-9007=20=125(15)=154502(6) 154502-1 © 2020 American Physical Society

is complementary to hitherto used concepts of using
additional body force or modifying the spatial structure
of the system to enhance heat transport, for example,
adding surface roughness [3032], shaking the convection
cell [33], including additional stabilizing forces through
geometrical modification [3436], rotation [37], inclination
[38,39], or a second stabilizing scalar field [40].
Next to the aspect ratio of the horizontal and vertical
extensions of the container, the dimensionless control
parameters are the Rayleigh number Ra ¼ αgH
3
Δ=ðνκÞ
and the Prandtl number Pr ¼ ν=κ, with α, ν, and κ being,
respectively, the thermal expansion coefficient, kinematic
viscosity and thermal diffusivity of the fluid, g the
gravitational acceleration, and Δ the temperature difference
between the bottom and top boundaries. The time, length,
and temperature are made dimensionless by the free-fall
time τ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H=αgΔ
p
, the height H of the container, and the
temperature difference Δ, respectively. In the following, all
quantities are dimensionless, if not otherwise explicitly
stated. In the periodically modulated RB, we give a
sinusoidal modulation signal to the bottom temperature as
θ
bot
¼ 1 þ A cosð2πftÞ: ð1Þ
For modulated RB, two more parameters have to be
introduced, namely the modulation frequency f and its
amplitude A, which is kept fixed in this study, A ¼ 1. The
efficiency of the heat transport and flow strength in the
system are represented in terms of the Nusselt number Nu
(the dimensionless heat flux) and the Reynolds number
Re (the ratio between inertia and viscous forces). Direct
numerical simulation (DNS) for incompressible Oberbeck-
Bousinesq flow are employed [41]; the numerical details
are provided in the Supplemental Material [42]. The DNS
are conducted in a two-dimensional square box with no-slip
and impermeable boundary conditions (BCs) for all walls.
The explored parameter range spans 10
7
Ra 10
9
,
1 Pr 8, and 10
4
f 4. We are aware of the
limitation of the two-dimensionality of the system on
which we focus, but in particular for Pr 1 two- and
three-dimensional RB convections show very close simi-
larities and features [44]. To support that our results are also
relevant for 3D RB, we conduct a set of three-dimensional
DNS in a cubic box at Ra ¼ 10
8
and Pr ¼ 4.3 with various
frequencies.
Figure 1(a) shows how the global convective heat
flux Nu depends on the modulation frequency f at fixed
Pr ¼ 4.3 (corresponding to water). The dependence of Nu
on f exhibits a universal trend for both, two- and three-
dimensional results, which is independent of Ra: When f is
large enough, Nu is not sensitive to the modulation
frequency, and the value is close to the value Nu
0
for
the case without modulation. However, when f is below a
certain onset frequency (denoted as f
onset
), there exists an
intermediate regime with significantly enhanced heat flux
as compared to Nu
0
. With f decreasing further, one
observes an optimal frequency f
opt
at which Nu is maximal
with an enhancement of approximately 25%. Such a
large enhancement of Nu is highly nontrivial because
the time-averaged temperature of the bottom plate is still
fixed at 1, and we only have changed the bottom
temperature from a steady value to a time periodic signal.
In Fig. 1(b), we further examine the NuðfÞ dependence for
different Pr, with Ra fixed at 10
8
. One can see that
both f
onset
and f
opt
are much more sensitive to Pr than
to Ra.
We first examine whether the transition is related to the
strength of the large-scale circulation (LSC). Figure 1(c)
shows the global Reynolds number Re as function of f for
various Ra, from which we can see that Re is maximized at
a Ra-dependent frequency f
opt;Re
(see Reynolds resonance
in Supplemental Material [42] for further analysis of
f
opt;Re
). However, when comparing the Nu and Re behav-
ior, one observes that the position of the strongest LSC does
not correspond to that of the maximum heat transport
(a)
(b)
(c)
FIG. 1. (a) Modulated frequency dependence of the Nusselt
number NuðfÞ, normalized by Nu
0
¼ Nuðf ¼ 0Þ, for different
Rayleigh numbers and fixed Pr ¼ 4.3. (b) NuðfÞ=Nu
0
for differ-
ent Prandtl numbers and fixed Ra ¼ 10
8
. (c) Global ReðfÞ
normalized by the Re
0
¼ Reðf ¼ 0Þ for different Rayleigh
numbers and fixed Pr ¼ 4.3.
PHYSICAL REVIEW LETTERS 125, 154502 (2020)
154502-2

(f
opt
f
opt;Re
). What physics then governs the transitions
between the regimes of heat flux?
To gain insight into this problem, we analyze how the
flow structure is changed under modulation. Figure 2(a)
shows the temperature fields at different phases of modu-
lation at f ¼ 10
3
. During the heating phase (θ
bot
> 1),
frequent plume emissions are observed near the bottom
plate. On the contrary, during the cooling phase (θ
bot
< 1),
there are no plume emissions from the bottom plate because
of the stable stratification near that surface, and the
resulting weakening of the circulation.
We further calculate the conditional average of the
temperature profiles at different phases, and compare
these profiles for different modulation frequencies in
Figs. 2(b)2(d). Without modulation, we recover traditional
RB with a mean bulk temperature of 0.5 [Fig. 2(b)].
When f ¼ 10
1
as shown in Fig. 2(c), the temperature
adjacent to the bottom is significantly affected by modu-
lation, whereas the bulk value is still close to 0.5. However,
the overall influence of the modulation is limited because it
is too fast to be sensed by the system. With decreasing
modulation frequency, the bulk temperature is more and
more influenced by the modulation (see Fig. 1 of the
Supplemental Material [42]). This suggests that there
exists a certain length scale which characterizes how deep
the influence of the modulation can penetrate into the
convective flow.
To better understand this length scale, we recall the
classical Stokes problem. In this flow, a BL is created by
an oscillating solid surface with modulating velocity
U cosð2πftÞ. Likewise, in modulated RB, we can draw
the analogy between an oscillating velocity and the
oscillating temperature θ
0
, where θ
0
¼ θ
¯
θðzÞ, with
¯
θðzÞ being the temporally averaged temperature at height
z. The governing equation and corresponding BCs are
θ
0
=t ¼ðRaPrÞ
1=2
2
θ
0
=z
2
;
θ
0
ð0;tÞ¼A cosð2π ftÞ; θ
0
ð;tÞ¼0: ð2Þ
The analytical solution of this PDE is an exponential
profile:
θ
0
ðz; tÞ¼Ae
z=λ
S
cos ð2πft z=λ
S
Þ; ð3Þ
with the so-called Stokes thermal BL thickness
λ
S
¼ π
1=2
f
1=2
Ra
1=4
Pr
1=4
; ð4Þ
which is the penetration depth of the disturbance created by
the oscillating temperature at the boundary. The distortion
[Eq. (3)] travels as a transverse wave through the fluid.
From Eq. (4) one can see that the thickness λ
S
of the
Stokes thermal BL decreases with increasing modulation
frequency. Depending on the relative thicknesses of λ
S
, that
of the thermal BL λ
θ
, and that of the momentum BL λ
u
,we
can obtain three regimes shown in Fig. 2(e). Here we have
restricted our discussion to 1 Pr 8, where λ
u
λ
θ
.
Regime (i): for λ
S
< λ
θ
< λ
u
, the effect of modulation is
confined inside the thermal BL, which is also shown by the
temperature profiles in Fig. 2(c). In such case, the effect of
modulation is negligible and the heat transport is almost
unaffected.
Regime (ii): for λ
θ
λ
S
< λ
u
, the plume emission, which
occurs at the edge of the thermal BL, can now be influenced
by the modulation [Fig. 2(e)], leading to the enhancement
of heat transport. We note that in thermal convection with a
rough plate, a Nu enhancement can also be observed when
the thermal BL is perturbed by roughness [45,46]. Here, we
understand the enhancement in Nu by the following
mechanism: In the heating phase (θ
bot
> 1), there is a
(a)
(b) (c) (d)
(e) (f)
FIG. 2. (a) Instantaneous temperature fields at different phases in one modulation period for Ra ¼ 10
8
,Pr¼ 4.3, f ¼ 10
3
.
(b)(d) Phase-averaged temperature profiles during one period for Ra ¼ 10
8
,Pr¼ 4.3 and different modulation frequencies, namely
(b) without modulation; (c) f ¼ 10
1
; (d) f ¼ 10
4
. The horizontal axis is the temperature and the vertical axis is the height. The
colorbar shows the bottom temperature (phase angle) from 0ðπ=2Þ to 2ðπ=2Þ. (e) Sketch of the relations between the three BLs [Stokes
thermal BL (λ
S
), thermal BL (λ
θ
), momentum BL (λ
u
)] for the three regimes (Pr ¼ 4.3): (i) λ
u
> λ
θ
> λ
S
; (ii) λ
u
> λ
S
> λ
θ
;
(iii) λ
S
> λ
u
> λ
θ
. Arrows represent the flow in the bulk. (b) Sket ch of two different phases during one period for regime ii: (a) heating
phase when θ
bot
> 1 and (b) cooling phase when θ
bot
< 1. (f) Phase-averaged center temperature for Ra ¼ 10
8
,Pr¼ 4.3. The red (blue)
curve represents the phase when the bottom temperature is maximal (minimal). The dashed lines (from right to left) correspond to f
onset
and f
opt
for Nu.
PHYSICAL REVIEW LETTERS 125, 154502 (2020)
154502-3

stronger convective flow and more energetic plumes, as
compared to the case without modulation. This can be seen
from the value of Nu at the top plate (see Supplementary
Material [42]), where it increases to the values above that
without the modulation during the heating phase. However,
in the cooling phase, Nu starts to decline but still remains
at values comparable to that without modulation, due to
the remaining convective flow. Therefore, there is a net
increase in Nu after one cycle, as compared to the value of
Nu without time-dependent modulation.
Regime (iii): for λ
θ
< λ
u
λ
S
, the effect of temperature
modulation penetrates into the bulk region occupied by the
LSC. The role of the bulk flow is to efficiently bring the
injected hot/cold fluid near the plates during the heating/
cooling phase to the center of the system. Therefore, the
center temperature also varies with the phases as seen in
Fig. 2(f), in contrast to the situation in regimes (i) and (ii).
As a result, at the peak of the heating phase (θ
bot
¼ 2), the
temperature difference between the bottom plate and the
bulk cannot be maintained at Δθ 1.5. The thermal driving
in the heating phase becomes weaker for smaller f, and the
global Nu is expected to decrease for decreasing f. When
the frequency decreases further and goes to 0, the limiting
value of Nu should be higher than without modulation. This
is because the asymptotic value of Nu is the integral of the
NuðRaÞ (ranging from 0 to 2Ra). However, the relation
between Nu and Ra is nonlinear [47].
According to the physical picture of the three regimes, we
compare the relative BL thickness to obtain the boundaries
of the regimes, i.e., f
onset
ðRa; PrÞ and f
opt
ðRa; PrÞ.First,we
make use of the relations λ
θ
Nu
1
and λ
u
Re
1=2
for the
thermal and momentum BL thicknesses. Then we use the
Grossmann-Lohse model for the scaling of NuðRa; PrÞ and
ReðRa; PrÞ in the I
regime (for large Pr) [47,48]:Nu
Pr
0
Ra
1=3
and Re Pr
1
Ra
2=3
. The onset frequency f
onset
corresponds to the transition between regime i and regime ii
(λ
S
λ
θ
), and we obtain
f
onset
Ra
1=6
Pr
1=2
: ð5Þ
The optimal frequency f
opt
corresponds to the transition
between regime ii and regime iii (λ
S
λ
u
), and we have
f
opt
Ra
1=6
Pr
3=2
: ð6Þ
To check these predictions for f
onset
and f
opt
, we replot
NuðfÞ for various Ra but now versus the rescaled fre-
quency fRa
1=6
; see Fig. 3(a) (Pr ¼ 4.3 fixed). Indeed, the
figure shows rather good collapses around the onset. Next,
we vary Pr for a fixed Ra ¼ 10
8
and plot Nu versus the
correspondingly rescaled frequencies, namely f Pr
1=2
for
the onset [Fig. 3(b)] and f Pr
3=2
for the optimum [Fig. 3(c)].
Indeed, one can see the rescaled frequencies (horizontal
axis) collapse well, indicating that equations (5) and (6)
correctly predict the onset frequency and the optimal
frequency for all Pr.
Finally, we present the phase diagram in the f vs Ra
and the f vs Pr parameter spaces in Figs. 3(d) and 3(e) .
(a)
(d) (e)
(b)
(c)
FIG. 3. (a) Normalized Nu as a function of fRa
1=6
, for different Ra and Pr ¼ 4.3, (b) f Pr
1=2
. (c) f Pr
3=2
for different Pr and
Ra ¼ 10
8
. Dashed lines show the onset frequency [where NuðfÞ starts to be affected, NuðfÞ=Nu
0
¼ 1.01] or optimal frequency [where
NuðfÞ reaches the maximum], averaged for different Ra or Pr. Phase diagram (a) in the f vs Ra and (b) in the f vs Pr parameter spaces. In
(a), the lower dashed line shows the optimal frequency f
opt
¼ 0.65Ra
0.22
that corresponds to the maximal Nu. The upper dashed line
shows the onset frequency f
onset
¼ 0.015Ra
0.14
that corresponds to the onset of the heat flux enhancement. In (b), the lower dashed line
shows the optimal frequency f
opt
¼ 0.06 Pr
1.35
, while the upper one shows the onset frequency f
onset
¼ 0.45 Pr
0.65
. The prefactors
originate from fits to the DNS data for f
opt
and f
onset
[set to occur when NuðfÞ=Nu
0
¼ 1.01]; see Supplemental Material [42] for details
on the fitting.
PHYSICAL REVIEW LETTERS 125, 154502 (2020)
154502-4

We classify three regimes: classical RB regime (i), modu-
lation-enhancement regime (ii), and modulation-reduction
regime (iii). The boundary between the regimes is found
by fitting the numerically obtained f
onset
and f
opt
. The
fitting scaling relations for onset and optimum
(f
onset
Ra
0.14
Pr
0.65
, f
opt
Ra
0.22
Pr
1.35
) show a good
agreement with the derived ones (f
onset
Ra
1=6
Pr
1=2
,
f
opt
Ra
1=6
Pr
3=2
) except f
opt
vs Ra, corresponding to
λ
S
λ
u
. We notice that in our model, λ
S
is obtained based
on a diffusion equation. The neglected advection term can
become significant, particularly in regime iii where the
Stokes BL may penetrate into the bulk. It, therefore,
imposes uncertainty in estimating the weak Ra dependence
of λ
opt
. Our explored parameter range only spans 1 < Pr < 8
due to extreme costs to explore a wider range. But our
model is general for various Pr, as long as the boundary
layers exist and follow the given scaling relations. These
obviously no longer hold for extreme Pr values (i.e., very
large Pr when the flow becomes laminar and very small Pr,
when λ
u
< λ
θ
). Moreover, our model indicates the relation
of the magnitude of Nu enhancement with Ra and Pr. From
Figs. 1(a) and 1(b), the maximal Nu enhancement increases
as Pr increases while it is independent of Ra. This is
because f
onset
and f
opt
have the same scaling with Ra but
different scalings with Pr, as shown in Eqs. (5) and (6).As
Pr increases, the gap between f
onset
and f
opt
becomes
larger, and Nu keeps increasing in between. Therefore, the
maximal Nu increases with increasing Pr.
In conclusion, our results have substantial implications
for the investigation of modulated convection systems. For
a wide range of parameters in the three-dimensional
parameter space (modulation frequency f, Rayleigh num-
ber Ra, and Prandtl number Pr), we have demonstrated how
the global heat transport efficiency can be enhanced
through temperature modulation in both two- and three-
dimensional simulations. The high similarity between 2D
and 3D DNS results supports that our results are applicable
in both cases and robust. Based on the heat transfer
enhancement, we can identify three different regimes:
the classical RB regime for fast modulation, an inter-
mediate regime in which the modulation leads to increasing
Nu enhancement, and the slow modulation regime in which
it leads to decreasing Nu enhancement. The transitions
between the regimes are well predicted by the relative
thicknesses of thermal, momentum, and Stokes thermal
BLs. Our concept of explaining global transport properties
in modulated BL flows by the relative thicknesses of the
three relevant BLs can also be extended to the angular
velocity transfer in modulated turbulent Taylor-Couette
flow, or to the kinetic energy transfers in modulated
turbulent pipe flow.
This work was supported by the Priority Programme
SPP 1881 Turbulent Superstructures of the Deutsche
Forschungsgemeinschaft and by NWO via the
Zwaartekrachtprogramma MCEC and an ERC-Advanced
Grant under Project No. 740479. This work was partly
carried out on the national e-infrastructure of SURFsara.
We also gratefully acknowledge support by the Balzan
Foundation.
*
d.lohse@utwente.nl
These authors contributed equally to this work.
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PHYSICAL REVIEW LETTERS 125, 154502 (2020)
154502-5

Citations
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Journal ArticleDOI
TL;DR: In this paper, the effect of the width-to-height aspect ratio on the stability of Rayleigh-Benard convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations.
Abstract: Rayleigh–Benard (RB) convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations. Two configurations are considered, one is two-dimensional (2-D) RB convection and the other one three-dimensional (3-D) RB convection with a rotating axis parallel to the plate, which for strong rotation mimics 2-D RB convection. For the 2-D simulations, we explore the parameter range of Rayleigh numbers from to and Prandtl numbers from to . The effect of the width-to-height aspect ratio is investigated for . We show that zonal flow, which was observed, for example, by Goluskin et al. (J. Fluid. Mech., vol. 759, 2014, pp. 360–385) for , is only stable when is smaller than a critical value, which depends on and . The regime in which only zonal flow can exist is called the first regime in this study. With increasing , we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger , in a third regime, only convection roll states are statistically stable and zonal flow is not sustained. How many convection rolls form (or in other words, what the mean aspect ratio of an individual roll is), depends on the initial conditions and on and . For instance, for and , the aspect ratio of an individual, statistically stable convection roll can vary in a large range between and . A convection roll with a large aspect ratio of , or more generally already with , can be seen as ‘localized’ zonal flow, and indeed carries over various properties of the global zonal flow. For the 3-D simulations, we fix and , and compare the flow for and . We first show that with increasing rotation rate both the flow structures and global quantities like the Nusselt number and the Reynolds number increasingly behave like in the 2-D case. We then demonstrate that with increasing aspect ratio , zonal flow, which was observed for small by von Hardenberg et al. (Phys. Rev. Lett., vol. 15, 2015, 134501), completely disappears for . For such large , only convection roll states are statistically stable. In-between, here for medium aspect ratio , the convection roll state and the zonal flow state are both statistically stable. What state is taken depends on the initial conditions, similarly as we found for the 2-D case.

47 citations

Journal ArticleDOI
28 Sep 2021
TL;DR: In this paper, the critical Rayleigh number for the onset of convection in confined geometries and the optimal shape of the container is estimated. But, with the increasing height of the cell, the critical number grows much slower.
Abstract: In an effort to achieve very large Rayleigh numbers when studying turbulence on a Rayleigh--Baposenard configuration, one can carry out simulations and experiments in as high convection cells as possible which involves using convection cells with the smallest possible aspect ratio. However, with the increasing height of the cell, the Rayleigh number grows much slower than the critical Rayleigh number for the onset of convection in the same container. This article discusses how to estimate accurately the critical Rayleigh number for the onset of convection in confined geometries and the optimal shape of the container.

22 citations

Journal ArticleDOI
TL;DR: In this article, a unified theory for turbulent purely internally heated convection was proposed, based on the splitting of the kinetic and thermal dissipation rates in respective boundary and bulk contributions.
Abstract: We offer a unifying theory for turbulent purely internally heated convection, generalizing the unifying theories of Grossmann and Lohse (2000, 2001) for Rayleigh--Benard turbulence and of Shishkina, Grossmann and Lohse (2016) for turbulent horizontal convection, which are both based on the splitting of the kinetic and thermal dissipation rates in respective boundary and bulk contributions. We obtain the mean temperature of the system and the Reynolds number (which are the response parameters) as function of the control parameters, namely the internal thermal driving strength (called, when nondimensionalized, the Rayleigh--Roberts number) and the Prandtl number. The results of the theory are consistent with our direct numerical simulations.

15 citations

Journal ArticleDOI
TL;DR: In this paper, a unified theory for turbulent, purely internally heated convection was proposed, which generalizes the unifying theories of Grossmann and Lohse (2000, https://doi.org/10.1103/PhysRevLett.86.3316) for Rayleigh-Benard turbulence and of Shishkina et al. (2016, 2019) for turbulent horizontal convection.
Abstract: We offer a unifying theory for turbulent, purely internally heated convection, generalizing the unifying theories of Grossmann and Lohse (2000, https://doi.org/10.1017/S0022112099007545; 2001, https://doi.org/10.1103/PhysRevLett.86.3316) for Rayleigh-Benard turbulence and of Shishkina et al. (2016, https://doi.org/10.1002/2015GL067003) for turbulent horizontal convection, which are both based on the splitting of the kinetic and thermal dissipation rates in respective boundary and bulk contributions. We obtain the mean temperature of the system and the Reynolds number (which are the response parameters) as function of the control parameters, namely the internal thermal driving strength (called, when nondimensionalized, the Rayleigh-Roberts number) and the Prandtl number. The results of the theory are consistent with our direct numerical simulations of the Boussinesq equations.

14 citations

Journal ArticleDOI
TL;DR: In this article, an experimental study of the Prandtl-number effects in quasi-two-dimensional (quasi-2-D) Rayleigh-Benard convection was performed.
Abstract: We report an experimental study of the Prandtl-number effects in quasi-two-dimensional (quasi-2-D) Rayleigh–Benard convection. The experiments were conducted in four rectangular convection cells over the Prandtl-number range of are in agreement with the predictions by Grossmann & Lohse (Phys. Fluids, vol. 16, 2004, pp. 4462–4472). These results enrich our understanding of quasi-2-D thermal convection, and its similarities and differences compared to 2-D and 3-D systems.

13 citations

References
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Journal ArticleDOI
TL;DR: In this article, the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr.
Abstract: The progress in our understanding of several aspects of turbulent Rayleigh-Benard convection is reviewed. The focus is on the question of how the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the dynamics of the large scale convection roll are addressed as well. The review ends with a list of challenges for future research on the turbulent Rayleigh-Benard system.

1,372 citations

Journal ArticleDOI
TL;DR: In this article, a systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh-Benard convection is suggested and shown to be compatible with recent experiments.
Abstract: A systematic theory for the scaling of the Nusselt number Nu and of the A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh–Benard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large-scale convection roll (‘wind of turbulence’) and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number Ra versus Prandtl number Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the kinetic boundary layer is thicker. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra [less, similar] 1011) the leading terms are Nu [similar] Ra1/4Pr1/8, Re [similar] Ra1/2Pr[minus sign]3/4 for Pr [less, similar] 1 and Nu [similar] Ra1/4Pr[minus sign]1/12, Re [similar] Ra1/2Pr[minus sign]5/6 for Pr [greater, similar] 1. In most measurements these laws are modified by additive corrections from the neighbouring regimes so that the impression of a slightly larger (effective) Nu vs. Ra scaling exponent can arise. The most important of the neighbouring regimes towards large Ra are a regime with scaling Nu [similar] Ra1/2Pr1/2, Re [similar] Ra1/2Pr[minus sign]1/2 for medium Pr (‘Kraichnan regime’), a regime with scaling Nu [similar] Ra1/5Pr1/5, Re [similar] Ra2/5Pr[minus sign]3/5 for small Pr, a regime with Nu [similar] Ra1/3, Re [similar] Ra4/9Pr[minus sign]2/3 for larger Pr, and a regime with scaling Nu [similar] Ra3/7Pr[minus sign]1/7, Re [similar] Ra4/7Pr[minus sign]6/7 for even larger Pr. In particular, a linear combination of the ¼ and the 1/3 power laws for Nu with Ra, Nu = 0.27Ra1/4 + 0.038Ra1/3 (the prefactors follow from experiment), mimics a 2/7 power-law exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through the non-normal-nonlinear transition to turbulence and another regime emerges.

933 citations

Journal ArticleDOI
TL;DR: In this article, the properties of the structure functions and other small-scale quantities in turbulent Rayleigh-Benard convection are reviewed from an experimental, theoretical, and numerical point of view.
Abstract: The properties of the structure functions and other small-scale quantities in turbulent Rayleigh-Benard convection are reviewed, from an experimental, theoretical, and numerical point of view. In particular, we address the question of whether, and if so where in the flow, the so-called Bolgiano-Obukhov scaling exists, i.e., Sθ(r) ∼ r2/5 for the second-order temperature structure function and Su(r) ∼ r6/5 for the second-order velocity structure function. Apart from the anisotropy and inhomogeneity of the flow, insufficiently high Rayleigh numbers, and intermittency corrections (which all hinder the identification of such a potential regime), there are also reasons, as a matter of principle, why such a scaling regime may be limited to at most a decade, namely the lack of clear scale separation between the Bolgiano length scale LB and the height of the cell.

750 citations

Journal ArticleDOI
TL;DR: Key emphasis is given to the physics and structure of the thermal and velocity boundary layers which play a key role for the better understanding of the turbulent transport of heat and momentum in convection at high and very high Rayleigh numbers.
Abstract: Recent experimental, numerical and theoretical advances in turbulent Rayleigh-Benard convection are presented. Particular emphasis is given to the physics and structure of the thermal and velocity boundary layers which play a key role for the better understanding of the turbulent transport of heat and momentum in convection at high and very high Rayleigh numbers. We also discuss important extensions of Rayleigh-Benard convection such as non-Oberbeck-Boussinesq effects and convection with phase changes.

630 citations

Journal ArticleDOI
TL;DR: In this article, a finite-difference scheme for direct simulation of the incompressible time-dependent three-dimensional Navier-Stokes equations in cylindrical coordinates is presented.

612 citations

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Q1. What are the contributions mentioned in the paper "Periodically modulated thermal convection" ?

Yang, Kai Leong Chong, Qi Wang, Roberto Verzicco, Olga Shishkina, and Detlef Lohse this paper.